Vector examples

WEBVTT
Kind: captions
Language: en

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In the last video I was a little
formal in defining what

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Rn is, and what a vector is,
and what vector addition or

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scalar multiplication is.

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In this video I want to kind of
go back to basics and just

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give you a lot of examples.

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And give you a more tangible
sense for what vectors are and

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how we operate with them.

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So let me define a couple
of vectors here.

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And I'm going to do, most of my
vectors I'm going to do in

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this video are going
to be in R2.

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And that's because they're
easy to draw.

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Remember R2 is the set
of all 2-tuples.

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Ordered 2-tuples where each of
the numbers, so you know you

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could have x1, my 1 looks like a
comma, x1 and x2, where each

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of these are real numbers.

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So you each of them, x1 is a
member of the reals, and x2 is

00:00:58.290 --> 00:00:59.580
a member of the reals.

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And just to give you a sense
of what that means, if this

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right here is my coordinate
axes, and I wanted a plot all

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my x1's, x2's.

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You know you could view this
as the first coordinate.

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We always imagine that
as our x-axis.

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And then our second coordinate
we plotted on

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the vertical axis.

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That traditionally is our
y-axis, but we'll just call

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that the second number
axis, whatever.

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You could visually represent all
of R2 by literally every

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single point on this plane if
we were to continue off to

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infinity in every direction.

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That's what R2 is.

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R1 would just be points
just along one of

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these number lines.

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That would be R1.

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So you could immediately
see that R2 is

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kind of a bigger space.

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But anyway, I said that I
wouldn't be too abstract, that

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I would show you examples.

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So let's get some vectors
going in R2.

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So let me define my vector a.

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I'll make it nice and bold.

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My vector a is equal to,
I'll make some numbers

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up, negative 1, 2.

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And my vector b, make it nice
and bold, let me make that, I

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don't know, 3, 1.

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Those are my two vectors.

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Let's just add them up
and see what we get.

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Just based on my definition
of vector addition.

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I'll just stay in one color
for now so I don't have to

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keep switching back and forth.

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So a, nice deep a, plus bolded
b is equal to, I just add up

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each of those terms.
Negative 1 plus 3.

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And then 2 plus 1.

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That was my definition
of vector addition.

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So that is going to be
equal to 2 and 3.

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Fair enough that just
came out of my

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definition of vector addition.

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But how can we represent
this vector?

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So we already know that if we
have coordinates, you know, if

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I have the coordinate, and this
is just a convention.

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It's just the way
that we do it.

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The way we visualize things.

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If I wanted to plot the
point 1, 1, I go to

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my coordinate axes.

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The first point I go along
the horizontal, what we

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traditionally call our x-axis.

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And I go 1 in that direction.

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And then convention is, the
second point I go 1 in the

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vertical direction.

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So the point 1, 1.

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Oh, sorry, let me
be very clear.

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This is 2 and 2, so one
is right here, and

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one is right there.

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So the point 1, 1 would
be right there.

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That's just the standard
convention.

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Now our convention for
representing vectors are, you

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might be tempted to say, oh,
maybe I just represent this

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vector at the point
minus 1, 2.

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And on some level
you can do that.

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I'll show you in a second.

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But the convention for vectors
is that you can

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start at any point.

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Let's say we're dealing with
two dimensional vectors.

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You can start at any
point in R2.

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So let's say that you're
starting at the

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point x1, and x2.

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This could be any point in R2.

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To represent the vector, what
we do is we draw a line from

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that point to the point x1.

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And let me call this, let's say
that we wanted to draw a.

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So x1 minus 1.

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So this is, I'm representing
a.

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So this is, I want to represent
the vector a.

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x1 minus 1, and then
x1 plus 2.

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Now if that seems confusing to
you, when I draw it, it'll be

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very obvious.

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So let's say I just want to
start at the point, let's just

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say for quirky reasons, I just
pick a random point here.

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I just pick a point.

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That one right there.

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That's my starting point.

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So minus 4, 4.

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Now if I want to represent my
vector a, what I just said is

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that I add the first term
in vector a to my first

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coordinate.

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So x1 plus minus 1
or x1 minus 1.

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So my new one is going to be,
so this is my x1 minus 4.

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So now it's going to be, let's
see, I'm starting at the point

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minus 4 comma 4.

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If I want to represent a, what
I do is, I draw an arrow to

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minus 4 plus this first
term, minus 1.

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And then 4 plus the
second term.

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4 plus 2.

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And so this is what?

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This is minus 5 comma 6.

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So I go to minus 5 comma 6.

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So I go to that point right
there and I just draw a line.

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So my vector will
look like this.

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I draw a line from
there to there.

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And I draw an arrow
at the end point.

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So that's one representation
of the vector minus 1, 2.

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Actually let me do it
a little bit better.

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Because minus 5 is actually
more, a little

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closer to right here.

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Minus 5 comma 6 Is right
there, so I draw

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my vector like that.

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But remember this point minus
4 comma 4 was an arbitrary

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place to draw my vector.

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I could have started
at this point here.

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I could have started at the
point 4 comma 6 and done the

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same thing.

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I could have gone minus 1 in
the horizontal direction,

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that's my movement in the
horizontal direction.

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And then plus 2 in the
vertical direction.

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So I could have drawn, so minus
1 in the horizontal and

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plus 2 in the vertical
gets me right there.

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So I could have just as easily
drawn my vector like that.

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These are both interpretations
of the same vector a.

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I should draw them in the
color of vector a.

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So vector a was this light
blue color right there.

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So this is vector a.

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This is vector a.

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Sometimes there'll
be a little arrow

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notation over the vector.

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But either of those vectors.

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I could draw an infinite
number of vector a's.

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I could draw vector a here.

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I could draw it like that.

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Vector a, it goes
back 1 and up 2.

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So vector a could
be right there.

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Similarly vector b.

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What does vector b do?

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I could pick some arbitrary
point for vector b.

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It goes to the right 3, so it
goes to the right 1, 2, 3 and

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then it goes up 1.

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So vector b, one representation
of vector b,

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looks like this.

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Another represention.

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I can start it right here.

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I could go to the right 3,
1, 2, 3, and then up 1.

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This would be another
representation of my vector b.

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There's an infinite number of
representations of them.

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But the convention is to often
put them in what's called the

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standard position.

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And that's to start
them off at 0, 0.

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So your initial point, let
me write this down.

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Standard position is just to
start the vectors at 0, 0 and

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then draw them.

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So vector a in standard
position, I'd start at 0, 0

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like that and I would go
back 1 and then up 2.

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So this is vector a in standard
position right there.

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And then vector b in
standard position.

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Let me write that.

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That's a.

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And then vector b in standard
position is 3, go to the 3

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right and then up 1.

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These are the vectors in
standard position, but any of

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these other things we drew
are just as valid.

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Now let's see if we can get
an interpretation of what

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happened when we
added a plus b.

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Well if I draw that vector in
standard position, I just

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calculated, it's 2, 3.

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So I go to the right
2 and I go up 3.

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So if I just draw it in
standard position

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it looks like this.

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This vector right there.

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And at first when you look at
it, this vector right here is

00:09:01.270 --> 00:09:05.730
the vector a plus b in
standard position.

00:09:05.730 --> 00:09:08.300
When you draw it like that,
it's not clear what the

00:09:08.300 --> 00:09:11.220
relationship is when
we added a and b.

00:09:11.220 --> 00:09:14.020
But to see the relationship what
you do is, you put a and

00:09:14.020 --> 00:09:15.190
b head to tails.

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What that means is, you put
the tail end of b to the

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front end of a.

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Because remember, all
of these are valid

00:09:21.780 --> 00:09:23.800
representations of b.

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All of the representations
of the vector b.

00:09:26.380 --> 00:09:28.620
They all have, they're all
parallel to each other, but

00:09:28.620 --> 00:09:29.720
they can start from anywhere.

00:09:29.720 --> 00:09:32.990
So another equally valid
representation of vector b is

00:09:32.990 --> 00:09:37.310
to start at this point right
here, kind of the end point of

00:09:37.310 --> 00:09:40.010
vector a in standard position,
and then draw vector b

00:09:40.010 --> 00:09:40.620
starting from there.

00:09:40.620 --> 00:09:41.860
So you go 3 to the right.

00:09:41.860 --> 00:09:46.470
So you go 1, 2, 3.

00:09:46.470 --> 00:09:48.970
And then you go up 1.

00:09:48.970 --> 00:09:52.210
So vector b could also be
drawn just like that.

00:09:54.830 --> 00:09:56.210
And then you should
see something

00:09:56.210 --> 00:09:57.760
interesting had happened.

00:09:57.760 --> 00:10:01.070
And remember, this vector b
representation is not in

00:10:01.070 --> 00:10:03.650
standard position, but it's just
an equally valid way to

00:10:03.650 --> 00:10:05.350
represent my vector.

00:10:05.350 --> 00:10:06.630
Now what do you see?

00:10:06.630 --> 00:10:13.320
When I add a, which is right
here, to b what do I get if I

00:10:13.320 --> 00:10:17.220
connect the starting point of
a with the end point of b?

00:10:17.220 --> 00:10:19.830
I get the addition.

00:10:19.830 --> 00:10:23.340
I have added the two vectors.

00:10:23.340 --> 00:10:24.300
And I could have done
that anywhere.

00:10:24.300 --> 00:10:25.870
I could have started
with a here.

00:10:25.870 --> 00:10:29.000
And then I could have
done the end point.

00:10:29.000 --> 00:10:33.010
I could have started b here and
gone 3 to the right, 1, 2,

00:10:33.010 --> 00:10:34.940
3 and then up 1.

00:10:34.940 --> 00:10:39.560
And I could have drawn b
right there like that.

00:10:39.560 --> 00:10:42.600
And then if I were to add a plus
b, I go to the starting

00:10:42.600 --> 00:10:46.380
point of a, and then
the end point of b.

00:10:46.380 --> 00:10:48.220
And that should also
be the visual

00:10:48.220 --> 00:10:52.420
representation of a plus b.

00:10:52.420 --> 00:10:54.580
Just to make sure it confirms
with this number, what I did

00:10:54.580 --> 00:10:57.520
here was I went 2 to
the right, 1, 2 and

00:10:57.520 --> 00:10:58.730
then I went 3 up.

00:10:58.730 --> 00:11:03.300
1, 2, 3 and I got a plus b.

00:11:03.300 --> 00:11:04.870
Now let's think about
what happens when

00:11:04.870 --> 00:11:06.740
we scale our vectors.

00:11:06.740 --> 00:11:12.180
When we multiply it times
some scalar factor.

00:11:12.180 --> 00:11:13.860
So let me pick new vectors.

00:11:13.860 --> 00:11:15.670
Those have gotten monotonous.

00:11:15.670 --> 00:11:18.860
Let me define vector v.

00:11:18.860 --> 00:11:20.550
v for vector.

00:11:20.550 --> 00:11:25.620
Let's say that it is
equal to 1, 2.

00:11:25.620 --> 00:11:28.980
So if I just wanted to draw
vector v in standard position,

00:11:28.980 --> 00:11:32.130
I would just go 1 to
the horizontal and

00:11:32.130 --> 00:11:34.180
then 2 to the vertical.

00:11:34.180 --> 00:11:34.870
That's it.

00:11:34.870 --> 00:11:37.210
That's the vector in
standard position.

00:11:37.210 --> 00:11:39.620
If I wanted to do it in a non
standard position, I could do

00:11:39.620 --> 00:11:40.160
it right here.

00:11:40.160 --> 00:11:42.980
1 to the right up 2,
just like that.

00:11:42.980 --> 00:11:45.810
Equally valid way of
drawing vector v.

00:11:45.810 --> 00:11:48.950
Equally valid way of doing it.

00:11:48.950 --> 00:11:51.740
Now what happens if I
multiply vector v.

00:11:51.740 --> 00:11:58.570
What if I have, I don't know,
what if I have 2 times v?

00:11:58.570 --> 00:12:03.060
2 times my vector v is now going
to be equal to 2 times

00:12:03.060 --> 00:12:06.740
each of these terms. So it's
going to be 2 times 1 which is

00:12:06.740 --> 00:12:11.800
2, and then 2 times
2 which is 4.

00:12:11.800 --> 00:12:15.120
Now what does 2 times
vector v look like?

00:12:15.120 --> 00:12:17.650
Well let me just start from
an arbitrary position.

00:12:17.650 --> 00:12:20.390
Let me just start
right over here.

00:12:20.390 --> 00:12:22.940
So I'm going to go 2
to the right, 1, 2.

00:12:22.940 --> 00:12:23.880
And I go up 4.

00:12:23.880 --> 00:12:26.220
1, 2, 3, 4.

00:12:26.220 --> 00:12:30.530
So this is what 2 times
vector v looks like.

00:12:30.530 --> 00:12:33.970
This is 2 times my vector v.

00:12:33.970 --> 00:12:36.160
And if you look at it, it's
pointing in the exact same

00:12:36.160 --> 00:12:38.690
direction but now it's
twice as long.

00:12:38.690 --> 00:12:42.180
And that makes sense because we
scaled it by a factor of 2.

00:12:42.180 --> 00:12:44.910
When you multiply it by a
scalar, or you're not changing

00:12:44.910 --> 00:12:45.550
its direction.

00:12:45.550 --> 00:12:48.180
Its direction is the exact same
thing as it was before.

00:12:48.180 --> 00:12:50.250
You're just scaling
it by that amount.

00:12:50.250 --> 00:12:51.530
And I could draw
this anywhere.

00:12:51.530 --> 00:12:52.270
I could have drawn
it right here.

00:12:52.270 --> 00:12:54.580
I could have drawn 2v
right on top of v.

00:12:54.580 --> 00:12:56.780
Then you would have seen it,
I don't want to cover it.

00:12:56.780 --> 00:13:00.330
You would have seen that it
goes, it's exactly, in this

00:13:00.330 --> 00:13:01.570
case when I draw it in standard

00:13:01.570 --> 00:13:03.100
position, it's colinear.

00:13:03.100 --> 00:13:05.630
It's along the same line,
it's just twice as far.

00:13:05.630 --> 00:13:06.750
it's just twice as long
but they have

00:13:06.750 --> 00:13:08.700
the exact same direction.

00:13:08.700 --> 00:13:15.450
Now what happens if I were
to multiply minus 4

00:13:15.450 --> 00:13:16.880
times our vector v?

00:13:19.530 --> 00:13:23.060
Well then that will be equal
to minus 4 times 1,

00:13:23.060 --> 00:13:24.440
which is minus 4.

00:13:24.440 --> 00:13:28.720
And then minus 4 times
2, which is minus 8.

00:13:28.720 --> 00:13:30.000
So this is on my new vector.

00:13:30.000 --> 00:13:31.270
Minus 4, minus 8.

00:13:31.270 --> 00:13:34.020
This is minus 4 times
our vector v.

00:13:34.020 --> 00:13:36.500
So let's just start at
some arbitrary point.

00:13:36.500 --> 00:13:38.630
Let's just do it in
standard position.

00:13:38.630 --> 00:13:40.850
So you go to the right 4.

00:13:40.850 --> 00:13:41.910
Or you go to the left 4.

00:13:41.910 --> 00:13:45.460
So so you go to the left
4, 1, 2, 3, 4.

00:13:45.460 --> 00:13:46.890
And then down 8.

00:13:46.890 --> 00:13:48.520
Looks like that.

00:13:48.520 --> 00:13:52.140
So this new vector is going
to look like this.

00:13:52.140 --> 00:13:54.670
Let me try and draw a relatively
straight line.

00:13:54.670 --> 00:13:56.000
There you go.

00:13:56.000 --> 00:13:59.290
So this is minus 4 times
our vector v.

00:13:59.290 --> 00:14:00.730
I'll draw a little arrow
on it to make sure

00:14:00.730 --> 00:14:01.540
you know it's a vector.

00:14:01.540 --> 00:14:03.110
Now what happened?

00:14:03.110 --> 00:14:05.640
Well we're kind of in
the same direction.

00:14:05.640 --> 00:14:07.480
Actually we're in the exact
opposite direction.

00:14:07.480 --> 00:14:09.970
But we're still along the
same line, right?

00:14:09.970 --> 00:14:11.940
But we're just in the exact
opposite direction.

00:14:11.940 --> 00:14:16.370
And it's this negative right
there that flipped us around.

00:14:16.370 --> 00:14:18.850
If we just multiplied negative
1 times this, we would have

00:14:18.850 --> 00:14:21.680
just flipped around to
right there, right?

00:14:21.680 --> 00:14:24.150
But we multiplied it
by negative 4.

00:14:24.150 --> 00:14:27.150
So we scaled it by 4, so you
make it 4 times as long, and

00:14:27.150 --> 00:14:29.830
then it's negative, so
then it flips around.

00:14:29.830 --> 00:14:32.070
It flips backwards.

00:14:32.070 --> 00:14:35.500
So now that we have that notion,
we can kind of start

00:14:35.500 --> 00:14:40.510
understanding the idea of
subtracting vectors.

00:14:40.510 --> 00:14:44.320
Let me make up 2 new
vectors right now.

00:14:44.320 --> 00:14:51.800
Let's say my vector x, nice and
bold x, is equal to, and

00:14:51.800 --> 00:14:54.140
I'm doing everything in R2, but
in the last part of this

00:14:54.140 --> 00:14:57.210
video I'll make a few examples
in R3 or R4.

00:14:57.210 --> 00:15:04.170
Let's say my vector x
is equal to 2, 4.

00:15:04.170 --> 00:15:06.720
And let's say I have
a vector y.

00:15:06.720 --> 00:15:08.750
y, make it nice and bold.

00:15:08.750 --> 00:15:16.410
And then that is equal to
negative 1, minus 2.

00:15:16.410 --> 00:15:20.460
And I want to think about
the notion of what x

00:15:20.460 --> 00:15:22.110
minus y is equal to.

00:15:26.790 --> 00:15:30.970
Well we can say that this is the
same thing as x plus minus

00:15:30.970 --> 00:15:34.250
1 times our vector y.

00:15:34.250 --> 00:15:34.980
Right?

00:15:34.980 --> 00:15:37.230
So x plus minus 1 times
our vector y.

00:15:37.230 --> 00:15:38.700
Now we can use our
definitions.

00:15:38.700 --> 00:15:40.260
We know how to multiply
by a scalar.

00:15:40.260 --> 00:15:42.540
So we'll say that this
is equal to,

00:15:42.540 --> 00:15:43.200
let me switch colors.

00:15:43.200 --> 00:15:44.760
I don't like this color.

00:15:44.760 --> 00:15:48.440
This is equal to our
x vector is 2, 4.

00:15:48.440 --> 00:15:51.480
And then what's minus
1 times y?

00:15:51.480 --> 00:15:56.070
So minus 1 times y is minus
1 times minus 1 is 1.

00:15:56.070 --> 00:16:00.160
And then minus 1 times
minus 2 is 2.

00:16:00.160 --> 00:16:03.030
So x minus y is going to be
these two vectors added to

00:16:03.030 --> 00:16:03.620
each other, right?

00:16:03.620 --> 00:16:05.340
I'm just adding the
minus of y.

00:16:05.340 --> 00:16:09.500
This is minus vector y.

00:16:09.500 --> 00:16:16.430
So this x minus y is going to
be equal to 3 and 3 and 6.

00:16:16.430 --> 00:16:18.500
So let's see what that looks
like when we visually

00:16:18.500 --> 00:16:19.690
represent them.

00:16:19.690 --> 00:16:22.160
Our vector x was 2, 4.

00:16:22.160 --> 00:16:25.980
So 2, 4 in standard position
it looks like this.

00:16:29.870 --> 00:16:31.720
That's my vector x.

00:16:31.720 --> 00:16:34.420
And then vector y in standard
position, let me do it in a

00:16:34.420 --> 00:16:37.610
different color, I'll
do y in green.

00:16:37.610 --> 00:16:39.865
Vector y is minus 1, minus 2.

00:16:48.960 --> 00:16:50.980
It looks just like this.

00:16:56.105 --> 00:16:58.980
And actually I ended up
inadvertently doing collinear

00:16:58.980 --> 00:17:01.570
vectors, but, hey, this
is interesting too.

00:17:01.570 --> 00:17:03.540
So this is vector y.

00:17:03.540 --> 00:17:05.020
So then what's their
difference?

00:17:05.020 --> 00:17:07.020
This is 3, 6.

00:17:07.020 --> 00:17:08.720
So it's the vector 3, 6.

00:17:08.720 --> 00:17:10.030
So it's this vector.

00:17:10.030 --> 00:17:11.720
Let me draw it someplace else.

00:17:11.720 --> 00:17:15.130
If I start here I go 1, 2, 3.

00:17:15.130 --> 00:17:16.800
And then I go up 6.

00:17:16.800 --> 00:17:18.920
So then up 6.

00:17:18.920 --> 00:17:21.440
It's a vector that
looks like this.

00:17:21.440 --> 00:17:24.050
That's the difference between
the two vectors.

00:17:24.050 --> 00:17:26.900
So at first you say,
this is x minus y.

00:17:26.900 --> 00:17:29.550
Hey, how is this the difference
of these two?

00:17:29.550 --> 00:17:31.170
Well if you overlay this.

00:17:31.170 --> 00:17:34.040
If you just shift this over
this, you could actually just

00:17:34.040 --> 00:17:36.190
start here and go straight up.

00:17:36.190 --> 00:17:38.710
And you'll see that it's really
the difference between

00:17:38.710 --> 00:17:39.310
the end points.

00:17:39.310 --> 00:17:41.250
You're kind of connecting
the end points.

00:17:41.250 --> 00:17:44.250
I actually didn't want to
draw collinear vectors.

00:17:44.250 --> 00:17:45.860
Let me do another example.

00:17:45.860 --> 00:17:47.120
Although that one's kind
of interesting.

00:17:47.120 --> 00:17:48.700
You often don't see that
one in a book.

00:17:48.700 --> 00:17:57.450
Let me to define vector x
in this case to be 2, 3.

00:17:57.450 --> 00:18:09.870
And let me define vector y
to be minus 4, minus 2.

00:18:09.870 --> 00:18:11.980
So what would be x in
standard position?

00:18:11.980 --> 00:18:13.950
It would be 2, 3.

00:18:13.950 --> 00:18:15.250
It'd look like that.

00:18:15.250 --> 00:18:18.710
That is our vector x if we
start at the origin.

00:18:18.710 --> 00:18:19.990
So this is x.

00:18:19.990 --> 00:18:21.640
And then what does vector
y look like?

00:18:21.640 --> 00:18:23.580
I'll do y in orange.

00:18:23.580 --> 00:18:26.740
Minus 4, minus 2.

00:18:26.740 --> 00:18:28.500
So vector y looks like this.

00:18:34.540 --> 00:18:36.325
Now what is x minus y?

00:18:41.750 --> 00:18:44.020
Well you know, we could
view this, 2 plus

00:18:44.020 --> 00:18:45.410
minus 1 times this.

00:18:45.410 --> 00:18:47.090
We could just say
2 minus minus 4.

00:18:47.090 --> 00:18:48.700
I think you get the idea now.

00:18:48.700 --> 00:18:51.320
But we just did it the first
way the last time because I

00:18:51.320 --> 00:18:53.540
wanted to go from my basic
definitions of scalar

00:18:53.540 --> 00:18:55.030
multiplication.

00:18:55.030 --> 00:19:01.420
So x minus y is just going to
be equal to 2 plus minus 1

00:19:01.420 --> 00:19:04.390
times minus 4, or
2 minus minus 4.

00:19:04.390 --> 00:19:07.610
That's the same thing as
2 plus 4, so it's 6.

00:19:07.610 --> 00:19:13.420
And then it's 3 minus
minus 2, so it's 5.

00:19:13.420 --> 00:19:13.740
Right?

00:19:13.740 --> 00:19:17.360
So the difference between the
two is the vector 6, 5.

00:19:17.360 --> 00:19:19.870
So you could draw it
out here again.

00:19:19.870 --> 00:19:23.670
So you could go, add 6 to 4, go
up there, then to 5, you'd

00:19:23.670 --> 00:19:24.360
go like that.

00:19:24.360 --> 00:19:26.850
So the vector would look
something like this.

00:19:26.850 --> 00:19:31.260
It shouldn't curve like that,
so that's x minus y.

00:19:31.260 --> 00:19:33.850
But if we drew them between,
like in the last example, I

00:19:33.850 --> 00:19:36.540
showed that you could draw it
between their two heads.

00:19:36.540 --> 00:19:38.770
So if you do it here, what
does it look like?

00:19:38.770 --> 00:19:45.510
Well if you start at this point
right there and you go 6

00:19:45.510 --> 00:19:50.780
to the right and then up 5,
you end up right there.

00:19:50.780 --> 00:19:55.240
So the difference between the
two vectors, let me make sure

00:19:55.240 --> 00:19:57.300
I get it, the difference
between the two

00:19:57.300 --> 00:19:59.610
vectors looks like that.

00:19:59.610 --> 00:20:01.440
It looks just like that.

00:20:01.440 --> 00:20:04.173
Which kind of should make
sense intuitively.

00:20:04.173 --> 00:20:05.900
x minus y.

00:20:05.900 --> 00:20:08.710
That's the difference between
the two vectors.

00:20:08.710 --> 00:20:10.780
You can view the difference as,
how do you get from one

00:20:10.780 --> 00:20:12.300
vector to another
vector, right?

00:20:12.300 --> 00:20:15.860
Like if, you know, let's go
back to our kind of second

00:20:15.860 --> 00:20:17.310
grade world of just scalars.

00:20:17.310 --> 00:20:21.410
If I say what 7 minus 5 is, and
you say it's equal to 2,

00:20:21.410 --> 00:20:25.210
well that just tells you that
5 plus 2 is equal to 7.

00:20:25.210 --> 00:20:27.950
Or the difference between
5 and 7 is 2.

00:20:27.950 --> 00:20:31.410
And here you're saying, look the
difference between x and y

00:20:31.410 --> 00:20:33.710
is this vector right there.

00:20:33.710 --> 00:20:35.680
It's equal to that vector
right there.

00:20:35.680 --> 00:20:39.980
Or you could say look, if I
take 5 and add 2 I get 7.

00:20:39.980 --> 00:20:43.850
Or you could say, look, if I
take vector y, and I add

00:20:43.850 --> 00:20:47.780
vector x minus y, then
I get vector x.

00:20:47.780 --> 00:20:49.750
Now let's do something else
that's interesting.

00:20:49.750 --> 00:20:52.640
Let's do what y minus
x is equal to.

00:20:52.640 --> 00:20:54.630
y minus x.

00:20:54.630 --> 00:20:56.780
What is that equal to?

00:20:56.780 --> 00:20:59.380
Do it in another color
right here.

00:20:59.380 --> 00:21:03.760
Well we'll take minus 4, minus
2 which is minus 6.

00:21:03.760 --> 00:21:05.850
And then you have minus
2, minus 3.

00:21:05.850 --> 00:21:08.190
It's minus 5.

00:21:08.190 --> 00:21:11.810
So y minus x is going to be,
let's see, if we start here

00:21:11.810 --> 00:21:13.430
we're going to go down 6.

00:21:13.430 --> 00:21:16.210
1, 2, 3, 4, 5, 6.

00:21:16.210 --> 00:21:17.470
And then back 5.

00:21:17.470 --> 00:21:20.350
So back 2, 4, 5.

00:21:20.350 --> 00:21:23.530
So y minus x looks like this.

00:21:27.080 --> 00:21:28.860
It's really the exact
same vector.

00:21:28.860 --> 00:21:30.480
Remember, it doesn't matter
where we start.

00:21:30.480 --> 00:21:32.590
It's just pointing in the
opposite direction.

00:21:32.590 --> 00:21:34.360
So if we shifted it here.

00:21:34.360 --> 00:21:36.460
I could draw it right
on top of this.

00:21:36.460 --> 00:21:38.990
It would be the exact as x
minus y, but just in the

00:21:38.990 --> 00:21:40.290
opposite direction.

00:21:40.290 --> 00:21:42.500
Which is just a general
good thing to know.

00:21:42.500 --> 00:21:46.930
So you can kind of do them as
the negatives of each other.

00:21:46.930 --> 00:21:48.760
And actually let me make
that point very clear.

00:21:48.760 --> 00:21:51.790
You know we drew y.

00:21:51.790 --> 00:21:57.120
Actually let me draw x, x
we could draw as 2, 3.

00:21:57.120 --> 00:21:59.280
So you go to the right
2 and then up 3.

00:21:59.280 --> 00:22:00.250
I've done this before.

00:22:00.250 --> 00:22:02.980
This is x in non standard
position.

00:22:02.980 --> 00:22:04.450
That's x as well.

00:22:04.450 --> 00:22:06.160
What is negative x?

00:22:06.160 --> 00:22:08.760
Negative x is minus 2 minus 3.

00:22:08.760 --> 00:22:12.280
So if I were to start here,
I'd go to minus 2, then

00:22:12.280 --> 00:22:14.220
I'd go minus 3.

00:22:14.220 --> 00:22:17.560
So minus x would look
just like this.

00:22:17.560 --> 00:22:18.100
Minus x.

00:22:18.100 --> 00:22:19.260
It looks just like x.

00:22:19.260 --> 00:22:20.100
It's parallel.

00:22:20.100 --> 00:22:21.590
It has the same magnitude.

00:22:21.590 --> 00:22:24.140
It's just pointing in the exact
opposite direction.

00:22:24.140 --> 00:22:26.790
And this is just a good thing
to kind of really get seared

00:22:26.790 --> 00:22:29.360
into your brain is to have an
intuition for these things.

00:22:29.360 --> 00:22:35.510
Now just to kind of finish up
this kind of idea of adding

00:22:35.510 --> 00:22:37.410
and subtracting vectors.

00:22:37.410 --> 00:22:39.400
Everything I did so
far was in R2.

00:22:39.400 --> 00:22:41.270
But I want to show you that
we can generalize them.

00:22:41.270 --> 00:22:44.920
And we can even generalize them
to vector spaces that

00:22:44.920 --> 00:22:50.010
aren't normally intuitive for
us to actually visualize.

00:22:50.010 --> 00:22:52.470
So let me define a couple
of vectors.

00:22:52.470 --> 00:23:03.210
Let me define vector a to be
equal to 0, minus 1, 2, and 3.

00:23:03.210 --> 00:23:13.790
Let me define vector b to be
equal to 4, minus 2, 0, 5.

00:23:13.790 --> 00:23:16.350
We can do the same addition
and subtraction

00:23:16.350 --> 00:23:17.210
operations with them.

00:23:17.210 --> 00:23:18.600
It's just it'll be hard
to visualize.

00:23:18.600 --> 00:23:22.220
We can keep them in
just vector form.

00:23:22.220 --> 00:23:25.320
So that it's still useful to
think in four dimensions.

00:23:25.320 --> 00:23:27.470
So if I were to say 4 times a.

00:23:30.410 --> 00:23:35.670
This is the vector a
minus 2 times b.

00:23:35.670 --> 00:23:36.930
What is this going
to be equal to?

00:23:36.930 --> 00:23:38.290
This is a vector.

00:23:38.290 --> 00:23:40.190
What is this going
to be equal to?

00:23:40.190 --> 00:23:44.310
Well we could rewrite this as
4 times this whole column

00:23:44.310 --> 00:23:47.980
vector, 0, minus 1, 2, and 3.

00:23:47.980 --> 00:23:50.390
Minus 2 times b.

00:23:50.390 --> 00:23:55.106
Minus 2 times 4,
minus 2, 0, 5.

00:23:55.106 --> 00:23:57.650
And what is this going
to be equal to?

00:23:57.650 --> 00:24:04.130
This term right here, 4 times
this, you're going to get, the

00:24:04.130 --> 00:24:07.040
pen tablet seems to not work
well there, so I'm going to do

00:24:07.040 --> 00:24:07.900
it right here.

00:24:07.900 --> 00:24:14.040
4 times this, you're going to
get 4 times 0, 0, minus 4, 8.

00:24:14.040 --> 00:24:14.830
4 times 2 is 8.

00:24:14.830 --> 00:24:17.140
4 times 3 is 12.

00:24:17.140 --> 00:24:24.600
And then minus, I'll do it in
yellow, minus 2 times 4 is 8.

00:24:24.600 --> 00:24:28.560
2 times minus 2 is minus 4.

00:24:28.560 --> 00:24:30.030
2 times 0 is 0.

00:24:30.030 --> 00:24:33.240
2 times 5 is 10.

00:24:33.240 --> 00:24:37.730
This isn't a good part of my
board, so let me just. It

00:24:37.730 --> 00:24:39.430
doesn't write well
right over there.

00:24:39.430 --> 00:24:41.910
I haven't figured out the
problem, but if I were just

00:24:41.910 --> 00:24:45.410
right it over here,
what do we get?

00:24:45.410 --> 00:24:47.270
With 0 minus 8?

00:24:47.270 --> 00:24:48.460
Minus 8.

00:24:48.460 --> 00:24:51.240
Minus 4, minus 4.

00:24:51.240 --> 00:24:52.760
Minus negative 4.

00:24:52.760 --> 00:24:56.000
So that's minus 4 plus
4, so that's 0.

00:24:56.000 --> 00:24:58.170
8 minus 0 is 8.

00:24:58.170 --> 00:25:01.150
12 minus, what was this?

00:25:01.150 --> 00:25:03.140
I can't even read it,
what it says.

00:25:03.140 --> 00:25:04.400
Oh, this is a 10.

00:25:04.400 --> 00:25:05.510
Now you can see it again.

00:25:05.510 --> 00:25:07.270
Something is very bizarre.

00:25:07.270 --> 00:25:09.160
2 times 5 is 10.

00:25:09.160 --> 00:25:13.110
So it's 12 minus
10, so it's 2.

00:25:13.110 --> 00:25:16.940
So when we take this vector
and multiply it by 4, and

00:25:16.940 --> 00:25:20.020
subtract 2 times this vector,
we just get this vector.

00:25:20.020 --> 00:25:23.050
And even though you can't
represent this in kind of an

00:25:23.050 --> 00:25:25.740
easy kind of graph-able format,

00:25:25.740 --> 00:25:26.920
this is a useful concept.

00:25:26.920 --> 00:25:29.860
And we're going to see this
later when we apply some of

00:25:29.860 --> 00:25:33.000
these vectors to
multi-dimensional spaces.


Linear Algebra: Introduction to Vectors

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.170
In this first, what I'll call
this is my first real linear

00:00:04.170 --> 00:00:11.250
algebra video, I want to
introduce you to some concepts

00:00:11.250 --> 00:00:13.660
that you're going to see over
and over in linear algebra,

00:00:13.660 --> 00:00:16.280
and if they're not introduced
to you properly in the

00:00:16.280 --> 00:00:17.980
beginning, they can
be very confusing.

00:00:17.980 --> 00:00:19.930
But hopefully over the course
of this video, you'll

00:00:19.930 --> 00:00:22.400
understand that they're pretty
straightforward ideas.

00:00:22.400 --> 00:00:25.490
If anything, in linear algebra,
the field, or the

00:00:25.490 --> 00:00:28.830
gods of linear algebra, are
kind of experts in to some

00:00:28.830 --> 00:00:31.950
degree stating simple and
obvious things in convoluted

00:00:31.950 --> 00:00:33.050
and Byzantine ways.

00:00:33.050 --> 00:00:34.700
But it does that for a
reason; it gives it

00:00:34.700 --> 00:00:36.620
mathematical rigor.

00:00:36.620 --> 00:00:40.330
So the first thing I want to
introduce you to is the set

00:00:40.330 --> 00:00:44.400
notation for real numbers and
then multiple dimensions of

00:00:44.400 --> 00:00:45.070
real numbers.

00:00:45.070 --> 00:00:48.820
So if you ever see this, an R
with an extra backbone like

00:00:48.820 --> 00:00:53.220
that, or sometimes it's written
as just a really

00:00:53.220 --> 00:00:57.040
bold-faced R if it's maybe
typed in a book, a really

00:00:57.040 --> 00:01:08.810
bold-faced R, this just means,
the set of all real numbers.

00:01:08.810 --> 00:01:11.510
And I probably should make a
few videos where I classify

00:01:11.510 --> 00:01:16.380
numbers by reals and complex
and irrational and rational

00:01:16.380 --> 00:01:20.410
and whatever else, and there's
actually very formal ways of

00:01:20.410 --> 00:01:24.250
defining the set of all real
numbers, but the simple thing

00:01:24.250 --> 00:01:26.460
for me is all numbers
except complex ones.

00:01:26.460 --> 00:01:28.300
So this is the same thing as

00:01:28.300 --> 00:01:35.840
everything but complex numbers.

00:01:35.840 --> 00:01:40.730
So pi is a real number, e is a
real number, square root of 2

00:01:40.730 --> 00:01:44.250
is a real number, 3 is a real
number, minus 3 is a real

00:01:44.250 --> 00:01:50.630
number, minus pi is a real
number, But 1 plus 2i is not a

00:01:50.630 --> 00:01:51.560
real number.

00:01:51.560 --> 00:01:54.120
It's not in the set of reals;
that's a complex number.

00:01:54.120 --> 00:01:55.460
So everything but complex.

00:01:55.460 --> 00:01:58.230
So pretty much all regular
numbers that you ever dealt

00:01:58.230 --> 00:02:00.360
with before you learned
what i was.

00:02:00.360 --> 00:02:02.540
That's the set of
real numbers.

00:02:02.540 --> 00:02:06.660
Now, the next thing you'll see
when you start discussing

00:02:06.660 --> 00:02:10.090
linear algebra is someone will
write-- they won't just write

00:02:10.090 --> 00:02:17.350
the set of all real numbers,
they'll write Rn.

00:02:17.350 --> 00:02:18.570
And you're like, gee,
what is that?

00:02:18.570 --> 00:02:22.000
Are they somehow taking
the exponent of

00:02:22.000 --> 00:02:23.460
all of the real numbers?

00:02:23.460 --> 00:02:26.100
And actually on some level, that
is what they're doing,

00:02:26.100 --> 00:02:30.710
but what they're saying here is
they're taking the set of

00:02:30.710 --> 00:02:32.900
all ordered sets of
real numbers.

00:02:32.900 --> 00:02:34.550
So what do I mean by that?

00:02:34.550 --> 00:02:37.190
So let me do an example where
I don't just have Rn,

00:02:37.190 --> 00:02:45.310
So R2 is the set of all lists
of real numbers, and let me

00:02:45.310 --> 00:02:46.440
make one of the lists.

00:02:46.440 --> 00:02:50.020
So let's say this has
one number and then

00:02:50.020 --> 00:02:51.810
it has another number.

00:02:51.810 --> 00:02:54.330
So it's the set of -- let me
write it this way, so I can

00:02:54.330 --> 00:02:55.460
write in the set of notations.

00:02:55.460 --> 00:03:00.110
It's the set of all-- we could
call these tuples.

00:03:00.110 --> 00:03:07.310
Sometimes it's called a 2-tuple,
which just means a

00:03:07.310 --> 00:03:08.370
list of two numbers.

00:03:08.370 --> 00:03:09.180
That's all it means.

00:03:09.180 --> 00:03:11.920
It just means a list
of two numbers.

00:03:11.920 --> 00:03:16.760
So R2 is a list of all ordered
2-tuples, so all ordered lists

00:03:16.760 --> 00:03:17.340
of two numbers.

00:03:17.340 --> 00:03:19.350
Let me actually write down the
word ordered because that's

00:03:19.350 --> 00:03:25.010
important, so ordered
list of 2-tuples.

00:03:25.010 --> 00:03:27.760
So, for example, this would
be-- well, let me

00:03:27.760 --> 00:03:29.460
write it this way.

00:03:29.460 --> 00:03:32.870
This would be a different-- when
I say ordered, I'm saying

00:03:32.870 --> 00:03:35.690
that this is fundamentally
different than that, that

00:03:35.690 --> 00:03:38.240
these aren't the same 2-tuple.

00:03:38.240 --> 00:03:42.010
Each of these is a 2-tuple,
but R2 is a set of all

00:03:42.010 --> 00:03:48.730
2-tuples such that-- that line
there just means such that or

00:03:48.730 --> 00:03:58.825
maybe for which-- each of
the numbers-- and I'll

00:03:58.825 --> 00:04:00.280
write it this way.

00:04:00.280 --> 00:04:02.250
I could just write-- well, I
could write it a couple of

00:04:02.250 --> 00:04:05.480
ways, but I'll write it in the
harder to understand way, just

00:04:05.480 --> 00:04:06.570
so you get used to it.

00:04:06.570 --> 00:04:12.130
Such that xi is a member
of the reals.

00:04:12.130 --> 00:04:20.140
So member of, that's what this
little character means.

00:04:20.140 --> 00:04:22.010
So that each xi is a member
of the reals.

00:04:22.010 --> 00:04:22.770
So what do you you mean xi?

00:04:22.770 --> 00:04:30.530
Well, xi just means any of these
x's, and I'll write for

00:04:30.530 --> 00:04:35.610
i is less than or equal to 2 and
greater than or equal to

00:04:35.610 --> 00:04:39.550
1, and i is an integer.

00:04:39.550 --> 00:04:43.090
Now, I could have written this
a bunch of different ways.

00:04:43.090 --> 00:04:46.220
This is almost silly how much
effort I'm taking to write it

00:04:46.220 --> 00:04:48.720
this way, but I could have
written it like this, too.

00:04:48.720 --> 00:04:54.290
I could have written that R2
is equal to the set of all

00:04:54.290 --> 00:05:01.180
ordered tuples, x1 x2 such
that-- I could have just

00:05:01.180 --> 00:05:06.780
written such that x1 and x2
are a member of the reals.

00:05:06.780 --> 00:05:07.670
I could have written
it that way.

00:05:07.670 --> 00:05:10.400
That would have been an easier
way to write it.

00:05:10.400 --> 00:05:12.020
But either way, I think
you get the idea.

00:05:12.020 --> 00:05:14.110
It's all of the combinations
of two things.

00:05:14.110 --> 00:05:19.460
So I said in the beginning,
this kind of is a squared

00:05:19.460 --> 00:05:21.720
operator. and why do
they write it as

00:05:21.720 --> 00:05:22.470
a superscript there?

00:05:22.470 --> 00:05:30.750
And my gut sense is if you think
about it, you have an

00:05:30.750 --> 00:05:34.290
infinite number of
real numbers.

00:05:34.290 --> 00:05:37.430
I won't go into things like
accountability and things like

00:05:37.430 --> 00:05:39.820
that, but you have a very large
number of real numbers.

00:05:39.820 --> 00:05:43.830
But let's say that you only
had n real numbers, right?

00:05:43.830 --> 00:05:46.440
Let's say that we weren't
dealing with reals.

00:05:46.440 --> 00:05:48.880
Just for sake of argument, let's
say that you had like

00:05:48.880 --> 00:05:50.910
100 real numbers, right?

00:05:50.910 --> 00:05:52.430
So that means that there
would've been 100

00:05:52.430 --> 00:05:54.880
possibilities there, and then
there's another 100

00:05:54.880 --> 00:05:56.260
possibilities there.

00:05:56.260 --> 00:06:00.200
So now your set of all the
possibilities of 2-tuples,

00:06:00.200 --> 00:06:03.390
well, you'd have 100 times 100
possibilities, so you'd almost

00:06:03.390 --> 00:06:06.030
have the squared number
of possibilities.

00:06:06.030 --> 00:06:10.220
So however many possibilities
you have for R, you have that

00:06:10.220 --> 00:06:13.670
many squared possibilities for
R2, which doesn't make a lot

00:06:13.670 --> 00:06:15.237
of sense, because you have
an infinite number of

00:06:15.237 --> 00:06:17.170
possibilities, but you have that
many more possibilities.

00:06:17.170 --> 00:06:19.470
You have an infinite number of
possibilities here and an

00:06:19.470 --> 00:06:20.840
infinite number of possibilities
here.

00:06:20.840 --> 00:06:24.120
So it's almost a greater
infinity of potential

00:06:24.120 --> 00:06:25.290
possibilities.

00:06:25.290 --> 00:06:27.430
Now, what's R3?

00:06:27.430 --> 00:06:29.070
R3 is just the same thing.

00:06:29.070 --> 00:06:38.940
R3 means you're going to have
three points or three numbers.

00:06:38.940 --> 00:06:40.990
I want to be careful not
to just say points.

00:06:40.990 --> 00:06:46.550
We can graph these things, and,
for example, R2, point in

00:06:46.550 --> 00:06:51.730
R2 can normally be graphed on
graph paper just in Cartesian

00:06:51.730 --> 00:06:52.510
coordinates.

00:06:52.510 --> 00:06:54.930
Points in R3 can be graphed
in three dimensions.

00:06:54.930 --> 00:06:59.300
Points in just R, or we could
just say R1, you could just

00:06:59.300 --> 00:07:01.390
graph those on a number line.

00:07:01.390 --> 00:07:04.430
I can have a number line like
that, and if I had to graph

00:07:04.430 --> 00:07:05.900
pi, you know, that's 0.

00:07:05.900 --> 00:07:07.830
I just say, hey, right
there, that's pi.

00:07:07.830 --> 00:07:09.240
And if i had to graph e,
I'd say, hey, right

00:07:09.240 --> 00:07:10.000
there, that's e.

00:07:10.000 --> 00:07:12.320
And if I had to graph 1,
I'd say, hey, that's 1.

00:07:12.320 --> 00:07:14.330
So you could graph them, but
I want to be careful.

00:07:14.330 --> 00:07:16.370
You don't have to graph them.

00:07:16.370 --> 00:07:18.730
They don't have to correspond
to actual

00:07:18.730 --> 00:07:20.970
physical points on a graph.

00:07:20.970 --> 00:07:23.000
But anyway, when you talk about
R3, you're literally

00:07:23.000 --> 00:07:26.640
just talking about all the
possibilities where you have

00:07:26.640 --> 00:07:28.970
an ordered set of three numbers,
where all three

00:07:28.970 --> 00:07:32.060
numbers are a member of
the real numbers.

00:07:32.060 --> 00:07:33.680
That's all you're saying.

00:07:33.680 --> 00:07:35.370
Now, I want to introduce
you to another

00:07:35.370 --> 00:07:37.275
definition of a vector.

00:07:41.660 --> 00:07:43.780
Now, a vector, you've
seen before.

00:07:43.780 --> 00:07:45.740
You've seen it in physics,
where you said, oh, it's

00:07:45.740 --> 00:07:47.930
something that has a magnitude
and a direction.

00:07:47.930 --> 00:07:51.800
You saw it in calculus, and we
plotted them and whatever not.

00:07:51.800 --> 00:07:54.410
But I'm going to be very formal
and very abstract and

00:07:54.410 --> 00:07:58.700
very broad with the vector right
now because the beauty

00:07:58.700 --> 00:08:02.230
of linear algebra is that it
doesn't just apply to things

00:08:02.230 --> 00:08:06.120
like physics, and it doesn't
just apply-- well, it does

00:08:06.120 --> 00:08:09.100
apply a lot to physics, but it
doesn't just apply to that.

00:08:09.100 --> 00:08:11.060
In linear algebra, you can
apply it to anything.

00:08:11.060 --> 00:08:13.100
It doesn't have to apply to
things that just graphs in

00:08:13.100 --> 00:08:13.780
three dimensions.

00:08:13.780 --> 00:08:15.830
And we'll talk more about that
in the future, but that's why

00:08:15.830 --> 00:08:19.830
I'm trying to stay
pretty abstract.

00:08:19.830 --> 00:08:24.690
So I'm going to define
a vector in Rn, a

00:08:24.690 --> 00:08:26.580
vector to be in Rn.

00:08:26.580 --> 00:08:27.800
Actually, let me be careful.

00:08:27.800 --> 00:08:29.500
I haven't even defined
what Rn is.

00:08:29.500 --> 00:08:34.740
I just defined R2 and R3,
so let me define Rn.

00:08:34.740 --> 00:08:43.409
Rn is equal to the set of all
n-tuples, ordered n-tuples.

00:08:43.409 --> 00:08:48.160
So you have x1, x2, all
the way over to xn.

00:08:48.160 --> 00:08:50.380
So you're going to have n
things; so if this was 100,

00:08:50.380 --> 00:08:52.180
you'd have 100 things here.

00:08:52.180 --> 00:08:58.980
So this is an ordered n-tuple.

00:08:58.980 --> 00:09:06.442
This is an ordered n-tuple, such
that for each xi, where

00:09:06.442 --> 00:09:10.820
xi is one of these, it is a
member of the real, so each of

00:09:10.820 --> 00:09:13.210
these has to be a real number.

00:09:13.210 --> 00:09:19.470
And then I'll just say for i
is less than or equal to n,

00:09:19.470 --> 00:09:21.900
greater than or equal to 1.

00:09:21.900 --> 00:09:25.990
So all that's saying is that x1
is a member of the reals,

00:09:25.990 --> 00:09:28.890
x2 is a member of the reals,
all the way up to xn is a

00:09:28.890 --> 00:09:29.910
member of the reals.

00:09:29.910 --> 00:09:35.440
Rn is the set of all of these
possible ordered n-tuples or

00:09:35.440 --> 00:09:38.360
ordered sets of n numbers.

00:09:38.360 --> 00:09:40.800
So what is a vector?

00:09:40.800 --> 00:09:43.810
What is a vector in Rn?

00:09:43.810 --> 00:09:47.360
Well, a vector in Rn really is
just a particular instance of

00:09:47.360 --> 00:09:48.590
one of these n-tuples.

00:09:48.590 --> 00:09:55.065
I'll just call it an ordered
list of n real numbers.

00:09:59.930 --> 00:10:03.030
And you can represent a vector
in a bunch of different ways.

00:10:03.030 --> 00:10:07.290
You could represent
it like this.

00:10:07.290 --> 00:10:10.850
You could represent it like--
let me do it a bunch of

00:10:10.850 --> 00:10:13.090
different ways.

00:10:13.090 --> 00:10:16.190
A two-dimensional vector, you
could represent like x1, x2,

00:10:16.190 --> 00:10:17.180
kind of like a coordinate.

00:10:17.180 --> 00:10:20.200
You could maybe represent it
with brackets like that.

00:10:20.200 --> 00:10:21.300
This is just syntax.

00:10:21.300 --> 00:10:24.150
This is just different ways
of representing the same

00:10:24.150 --> 00:10:26.510
information, the same idea.

00:10:26.510 --> 00:10:28.440
And for most of what we talk
about in linear algebra, we're

00:10:28.440 --> 00:10:31.690
going to represent them as
essentially these columns, and

00:10:31.690 --> 00:10:33.370
in the very near future,
we'll represent

00:10:33.370 --> 00:10:35.000
them as rows as well.

00:10:35.000 --> 00:10:38.280
But I'm going to define
my vector.

00:10:38.280 --> 00:10:41.670
A vector as an Rn is an
ordered list of n real

00:10:41.670 --> 00:10:44.390
numbers, and I define-- and
this is an example.

00:10:44.390 --> 00:10:54.750
So it would be v1, v2, all the
way to vn, where each of these

00:10:54.750 --> 00:10:56.155
is a member of the
real numbers.

00:11:01.130 --> 00:11:04.830
And just to make sure you
understand the notation or the

00:11:04.830 --> 00:11:06.950
terminology, each of
these is called a

00:11:06.950 --> 00:11:16.360
component of our vector.

00:11:16.360 --> 00:11:18.630
Actually, I want to be very
careful here, because when we

00:11:18.630 --> 00:11:21.850
write vectors, you want
to make sure you

00:11:21.850 --> 00:11:23.130
have a nice bold v.

00:11:23.130 --> 00:11:26.480
I won't make all my v's in the
future that bold, but that's

00:11:26.480 --> 00:11:29.540
how, when you're reading a math
book, they differentiate

00:11:29.540 --> 00:11:32.200
between just a vector and just
a regular quantity , right?

00:11:32.200 --> 00:11:36.090
These regular quantities are
just written non-bolded, while

00:11:36.090 --> 00:11:38.530
the vectors are bolded.

00:11:38.530 --> 00:11:47.850
Now, I'm going to define two
operations for vectors in Rn.

00:11:47.850 --> 00:11:52.460
I'm going to define addition,
and these are definitions.

00:11:52.460 --> 00:11:55.040
I could've defined them in any
arbitrary way, but these I

00:11:55.040 --> 00:11:58.400
think you'll find to be somewhat
natural, so these are

00:11:58.400 --> 00:11:59.650
definitions.

00:12:02.120 --> 00:12:06.000
These are just human abstract
constructs that end up

00:12:06.000 --> 00:12:07.210
becoming very useful.

00:12:07.210 --> 00:12:08.800
So they said, hey, let's
make some definitions.

00:12:08.800 --> 00:12:10.340
Let's define addition.

00:12:13.100 --> 00:12:18.510
So, if I have some vector-- let
me write it as a lowercase

00:12:18.510 --> 00:12:22.460
a, but it's a really bolded
lowercase a, and then that is

00:12:22.460 --> 00:12:25.230
equal to a1, a2.

00:12:25.230 --> 00:12:28.900
That's all of it's components
all the way to an, and I want

00:12:28.900 --> 00:12:31.120
to add that.

00:12:31.120 --> 00:12:34.400
And then I have vector b-- and
I'm going to make that a

00:12:34.400 --> 00:12:41.440
really bolded b right there--
is equal to b1, b2, all the

00:12:41.440 --> 00:12:46.690
way to bn, I'm going to define
addition of these two vectors

00:12:46.690 --> 00:12:51.010
to be just the sum of each
of their components.

00:12:51.010 --> 00:13:04.260
Bolded a, plus bolded b is equal
to a new vector, where

00:13:04.260 --> 00:13:05.740
you just add each of
their components.

00:13:05.740 --> 00:13:08.620
It's going to be a1 plus
b1 is going to

00:13:08.620 --> 00:13:10.420
create this new component.

00:13:10.420 --> 00:13:16.200
a2 plus b2, all the way
down to an plus bn.

00:13:16.200 --> 00:13:20.910
And if you found this somewhat
confusing, just think, I'm

00:13:20.910 --> 00:13:23.360
almost going through pains to
write something very simple in

00:13:23.360 --> 00:13:28.130
very confusing terms. If I have
two vectors in R2, let's

00:13:28.130 --> 00:13:33.810
say I have the vector 2, minus
1, and I want to add that to

00:13:33.810 --> 00:13:41.060
the vector 3, 2, I literally
just add each of the

00:13:41.060 --> 00:13:42.280
corresponding components.

00:13:42.280 --> 00:13:44.460
So I add the first components
to each other.

00:13:44.460 --> 00:13:47.390
So 2 plus 3 is equal to 5.

00:13:47.390 --> 00:13:50.030
Minus 1 plus 2 is equal to 1.

00:13:50.030 --> 00:13:53.740
That's all my definition of
vector addition is for my

00:13:53.740 --> 00:13:55.490
vectors in real numbers.

00:13:55.490 --> 00:13:59.410
And now the next thing I'm going
to define is the idea of

00:13:59.410 --> 00:14:00.660
a scalar multiple.

00:14:07.260 --> 00:14:09.490
You might remember from physics
class, a scalar is

00:14:09.490 --> 00:14:11.930
just a regular number, or if
your multiplying something by

00:14:11.930 --> 00:14:13.920
a scalar, you're just
scaling it up.

00:14:13.920 --> 00:14:16.710
Maybe I shouldn't use the same
word in the definition, but

00:14:16.710 --> 00:14:19.990
I'm going to define it as--
let me say c is a scalar.

00:14:19.990 --> 00:14:24.110
So notice, I didn't
bold the c.

00:14:24.110 --> 00:14:30.070
So c times my vector a-- and
here I'm going to take pains

00:14:30.070 --> 00:14:36.120
to bold it up-- is going to be
equal to all of my components

00:14:36.120 --> 00:14:40.500
of vector a multiplied by
each, multiplied by c.

00:14:40.500 --> 00:14:45.840
So a1, a2, all the way to an,
and they're each going to be

00:14:45.840 --> 00:14:48.790
multiplied by c.

00:14:48.790 --> 00:14:51.800
So that's all I define-- every
term is going to be

00:14:51.800 --> 00:14:52.870
multiplied by c.

00:14:52.870 --> 00:14:57.070
And just to give you an example
like that, if I wrote

00:14:57.070 --> 00:15:04.630
minus 3 times some vector--
I'll just

00:15:04.630 --> 00:15:06.390
draw the vector here.

00:15:06.390 --> 00:15:08.010
Let me draw a vector in R3.

00:15:08.010 --> 00:15:14.370
So let's say the first term is
pi, the second term is minus

00:15:14.370 --> 00:15:18.200
2, and then the third term is 0,
then this is just going to

00:15:18.200 --> 00:15:26.360
be equal to a new vector in R3
or you can kind of view it as

00:15:26.360 --> 00:15:31.100
a three-dimensional vector,
minus 3pi, minus 3 times minus

00:15:31.100 --> 00:15:37.360
2 is positive 6, and then minus
3 times 0 is just 0.

00:15:37.360 --> 00:15:39.100
So it's that straightforward.

00:15:39.100 --> 00:15:41.750
They made this fancy definition,
but it's fairly

00:15:41.750 --> 00:15:43.550
easy to deal with, I think.

00:15:43.550 --> 00:15:48.380
And I'll just make one last
definition, and I'll call that

00:15:48.380 --> 00:15:49.630
the zero vector.

00:15:54.350 --> 00:15:56.760
And depending on what dimension
we're in, it just

00:15:56.760 --> 00:15:58.240
means a vector with
all zeroes.

00:15:58.240 --> 00:16:04.150
So the zero vector in Rn, if
it's arbitrary, is just a

00:16:04.150 --> 00:16:07.010
vector where everything
is zero.

00:16:07.010 --> 00:16:09.145
And if this is Rn, you just
have n components.

00:16:13.080 --> 00:16:14.950
So those are all
my definitions.

00:16:14.950 --> 00:16:19.060
I think I've given you enough
to think about right now in

00:16:19.060 --> 00:16:19.510
this video.

00:16:19.510 --> 00:16:21.140
In the next video, I'm going
to show a bunch more of

00:16:21.140 --> 00:16:25.050
examples, and I'll actually draw
representations of these

00:16:25.050 --> 00:16:27.190
vectors, although you don't
have to, and we'll kind of

00:16:27.190 --> 00:16:28.950
have a little more visual
understanding

00:16:28.950 --> 00:16:30.680
of what they mean.


Using the convolution theorem to solve an initial value prob

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:03.170
Now that we know a little bit
about the convolution integral

00:00:03.170 --> 00:00:05.200
and how it applies to the
Laplace transform, let's

00:00:05.200 --> 00:00:07.950
actually try to solve an actual
differential equation

00:00:07.950 --> 00:00:09.270
using what we know.

00:00:09.270 --> 00:00:12.440
So I have this equation here,
this initial value problem,

00:00:12.440 --> 00:00:17.410
where it says that the second
derivative of y plus 2 times

00:00:17.410 --> 00:00:23.380
the first derivative of y, plus
2 times y, is equal to

00:00:23.380 --> 00:00:26.970
sine of alpha t.

00:00:26.970 --> 00:00:29.340
And they give us some
initial conditions.

00:00:29.340 --> 00:00:34.740
They tell us that y of 0 is
equal to 0, and that y prime

00:00:34.740 --> 00:00:38.090
of 0 is equal to 0.

00:00:38.090 --> 00:00:40.350
And that's nice and convenient
that those initial conditions

00:00:40.350 --> 00:00:42.280
tend to make the problem
pretty clean.

00:00:42.280 --> 00:00:44.170
But let's get to the problem.

00:00:44.170 --> 00:00:46.940
So the first thing we do is we
take the Laplace transform of

00:00:46.940 --> 00:00:48.960
both sides of this equation.

00:00:48.960 --> 00:00:51.470
The Laplace transform of
the second derivative

00:00:51.470 --> 00:00:54.380
of y is just s squared.

00:00:54.380 --> 00:00:56.550
This should be a bit of second
nature to you by now.

00:00:56.550 --> 00:01:01.530
It's s squared times the Laplace
transform of Y, which

00:01:01.530 --> 00:01:06.580
I'll just write as capital Y
of s, minus s-- so we start

00:01:06.580 --> 00:01:08.500
with the same degree as the
number of derivatives we're

00:01:08.500 --> 00:01:12.040
taking, and then we decrement
that every time-- minus s

00:01:12.040 --> 00:01:16.130
times y of 0-- you kind of think
of this as the integral,

00:01:16.130 --> 00:01:18.770
and you take the derivative 1,
so this isn't exactly the

00:01:18.770 --> 00:01:22.286
derivative of that-- minus, you
decrement that 1, you just

00:01:22.286 --> 00:01:25.680
have a 1 there, y prime of o.

00:01:25.680 --> 00:01:28.210
And that's the Laplace transform
of the second

00:01:28.210 --> 00:01:28.730
derivative.

00:01:28.730 --> 00:01:33.370
Now, we have to do the Laplace
transform of 2 times the first

00:01:33.370 --> 00:01:33.860
derivative.

00:01:33.860 --> 00:01:40.470
That's just going to be equal
to plus 2, times sY of s-- s

00:01:40.470 --> 00:01:42.930
times the Laplace transform
of Y; that's that

00:01:42.930 --> 00:01:48.920
there-- minus y of 0.

00:01:48.920 --> 00:01:51.010
And we just have one left.

00:01:51.010 --> 00:01:54.110
The Laplace transform of 2Y.

00:01:54.110 --> 00:01:57.530
That's just equal to plus
2 times the Laplace

00:01:57.530 --> 00:01:59.320
transform of Y.

00:01:59.320 --> 00:02:02.060
And then that's going to be
equal to the Laplace transform

00:02:02.060 --> 00:02:05.910
of sine of alpha t.

00:02:05.910 --> 00:02:08.300
We've done that multiple
times so far.

00:02:08.300 --> 00:02:15.840
That's just alpha over s squared
plus alpha squared.

00:02:15.840 --> 00:02:17.820
Now, the next thing we want to
do is we want to separate out

00:02:17.820 --> 00:02:21.500
the Laplace transform of Y
terms, or the Y of s terms.

00:02:21.500 --> 00:02:23.790
Actually, even better, let's
get rid of these initial

00:02:23.790 --> 00:02:24.640
conditions.

00:02:24.640 --> 00:02:29.460
y of 0, and y prime of 0 is
0, so this term is 0.

00:02:29.460 --> 00:02:33.230
That term is 0, and
that term is 0.

00:02:33.230 --> 00:02:36.570
So our whole expression-- I
can get rid of the colors

00:02:36.570 --> 00:02:40.380
now-- it just becomes-- let me
pick a nice color here--

00:02:40.380 --> 00:02:51.230
becomes s squared times Y of
s, plus 2s, Y of s-- that's

00:02:51.230 --> 00:02:57.580
that term right there-- plus
2Y of s, is equal to the

00:02:57.580 --> 00:03:03.750
right-hand side, is equal to
alpha over s squared plus

00:03:03.750 --> 00:03:04.960
alpha squared.

00:03:04.960 --> 00:03:07.390
Now let's factor out the
Y of s, or the Laplace

00:03:07.390 --> 00:03:09.210
transform of Y.

00:03:09.210 --> 00:03:17.790
And so we get s squared plus 2s,
plus 2, all of that times

00:03:17.790 --> 00:03:22.350
Y of s, is equal to this
right-hand side, is equal to

00:03:22.350 --> 00:03:26.480
alpha over s squared,
plus alpha squared.

00:03:26.480 --> 00:03:29.350
Now we can divide both sides of
this equation by this thing

00:03:29.350 --> 00:03:33.250
right here, by that
right there.

00:03:33.250 --> 00:03:38.870
And we get Y of s, the Laplace
transform of Y is equal to

00:03:38.870 --> 00:03:44.570
this thing, alpha over s
squared, plus alpha squared,

00:03:44.570 --> 00:03:49.770
times-- or, you know, I could
just say times-- 1 over s

00:03:49.770 --> 00:03:52.380
squared, plus 2s, plus 2.

00:03:52.380 --> 00:03:55.040
I could just say divided by
this, but it works out the

00:03:55.040 --> 00:03:56.700
same either way.

00:03:56.700 --> 00:03:57.590
Now, what can we do here?

00:03:57.590 --> 00:04:01.290
Remember, I was doing this in
the context of convolution, so

00:04:01.290 --> 00:04:04.110
I want to look for a Laplace
transform that looks like the

00:04:04.110 --> 00:04:06.820
product of two Laplace
transforms. I know what the

00:04:06.820 --> 00:04:08.330
inverse Laplace transform
of this is.

00:04:08.330 --> 00:04:09.380
In fact, I just took it.

00:04:09.380 --> 00:04:10.890
It's sine of alpha t.

00:04:10.890 --> 00:04:13.150
So if I can figure out the
inverse Laplace transform of

00:04:13.150 --> 00:04:16.950
this, I could at least express
our function y of t at least

00:04:16.950 --> 00:04:19.170
as a convolution integral, even
if I don't necessarily

00:04:19.170 --> 00:04:20.070
solve the integral.

00:04:20.070 --> 00:04:21.870
From there, it's just calculus,
or if it's an

00:04:21.870 --> 00:04:24.180
unsolvable integral, we could
just use a computer or

00:04:24.180 --> 00:04:26.060
something, although you could
actually use a computer to

00:04:26.060 --> 00:04:28.500
solve this so, you might
skip some steps even

00:04:28.500 --> 00:04:29.080
going through this.

00:04:29.080 --> 00:04:31.360
But anyway, let's just try
to get this in terms of a

00:04:31.360 --> 00:04:33.430
convolution integral.

00:04:33.430 --> 00:04:34.930
So what can I do with this?

00:04:34.930 --> 00:04:37.970
This is, let's see, this
isn't a perfect square.

00:04:37.970 --> 00:04:40.250
So if this isn't a perfect
square, the next best thing is

00:04:40.250 --> 00:04:41.870
to try to complete
the square here.

00:04:41.870 --> 00:04:45.990
So let's try to write this as
a, so let's see, if I write

00:04:45.990 --> 00:04:53.640
this as s squared plus 2s,
plus something, plus 2.

00:04:53.640 --> 00:04:55.110
I just rewrote it like this.

00:04:55.110 --> 00:04:58.250
And if I wrote this as s squared
plus 2s, plus 1, that

00:04:58.250 --> 00:05:00.130
becomes s plus 1 squared.

00:05:00.130 --> 00:05:02.210
But if I add a 1, I have
to also subtract a 1.

00:05:02.210 --> 00:05:04.430
I can't just add 1's arbitrarily
to things.

00:05:04.430 --> 00:05:06.990
So if I add 1 I have to subtract
a 1 to cancel out

00:05:06.990 --> 00:05:07.430
with that 1.

00:05:07.430 --> 00:05:09.690
So I really haven't changed
this at all, I just

00:05:09.690 --> 00:05:11.720
rewrote it like this.

00:05:11.720 --> 00:05:14.890
But this now, I can rewrite
this term right

00:05:14.890 --> 00:05:18.900
here as s plus 1 squared.

00:05:18.900 --> 00:05:22.460
And then this becomes plus 1.

00:05:22.460 --> 00:05:23.810
That's this term right here.

00:05:23.810 --> 00:05:25.250
This is the plus 1.

00:05:25.250 --> 00:05:31.390
So I could rewrite my whole Y
of s is now equal to alpha

00:05:31.390 --> 00:05:39.690
over s squared, plus alpha
squared, times 1 over this

00:05:39.690 --> 00:05:45.630
thing, s plus 1 squared,
plus 1.

00:05:45.630 --> 00:05:49.430
Now, I already said, I know
what the inverse Laplace

00:05:49.430 --> 00:05:50.920
transform of this thing is.

00:05:54.760 --> 00:05:56.600
Now I just have to figure out
what the inverse Laplace

00:05:56.600 --> 00:05:58.260
transform of this thing is.

00:05:58.260 --> 00:06:02.160
Of this-- let me pick a nice
color-- of this blue thing in

00:06:02.160 --> 00:06:04.240
the blue box, and then
I can express it as

00:06:04.240 --> 00:06:05.350
a convolution integral.

00:06:05.350 --> 00:06:06.200
And how do I do that?

00:06:06.200 --> 00:06:07.270
I could just do it right now.

00:06:07.270 --> 00:06:11.250
I could just immediately say
that y of t-- let me write

00:06:11.250 --> 00:06:17.830
this down-- y of t, so the
inverse is equal to the

00:06:17.830 --> 00:06:22.740
inverse Laplace transform
of, obviously of Y of s.

00:06:22.740 --> 00:06:26.430
Let me write that
down, Y of s.

00:06:26.430 --> 00:06:29.630
Which is equal to the inverse
Laplace transform

00:06:29.630 --> 00:06:31.420
of these two things.

00:06:31.420 --> 00:06:36.370
The inverse Laplace transform of
alpha over s squared, plus

00:06:36.370 --> 00:06:46.060
alpha squared, times 1 over
s plus 1 squared, plus 1.

00:06:46.060 --> 00:06:50.300
And now the convolution theorem
tells us that this is

00:06:50.300 --> 00:06:55.440
going to be equal to the inverse
Laplace transform of

00:06:55.440 --> 00:06:58.182
this first term in
the product.

00:06:58.182 --> 00:07:02.740
So the inverse Laplace transform
of that first term,

00:07:02.740 --> 00:07:08.960
alpha over s squared, plus
alpha squared, convoluted

00:07:08.960 --> 00:07:14.710
with-- I'll do a little
convolution sign there.

00:07:14.710 --> 00:07:16.510
I was about to say convulsion.

00:07:16.510 --> 00:07:18.610
They're not too different.

00:07:18.610 --> 00:07:20.930
Convoluted with the inverse
Laplace transform of this

00:07:20.930 --> 00:07:27.990
term, the inverse Laplace
transform of 1 over s plus 1

00:07:27.990 --> 00:07:30.950
squared, plus 1.

00:07:30.950 --> 00:07:34.730
If I have the product of two
Laplace transforms, and I can

00:07:34.730 --> 00:07:37.100
take each of them independently
and I can invert

00:07:37.100 --> 00:07:41.410
them, the inverse Laplace
transform of their product is

00:07:41.410 --> 00:07:44.230
going to be the convolution of
the inverse Laplace transforms

00:07:44.230 --> 00:07:48.460
of each of them, each
of the terms.

00:07:48.460 --> 00:07:50.920
And what I just said confused
me a bit, so I don't want to

00:07:50.920 --> 00:07:51.470
confuse you.

00:07:51.470 --> 00:07:52.550
But I think you get the idea.

00:07:52.550 --> 00:07:53.620
I have these two things.

00:07:53.620 --> 00:07:55.590
I recognize these
independently.

00:07:55.590 --> 00:07:58.210
I can independently take the
inverse of each of these

00:07:58.210 --> 00:08:02.840
things, so the inverse Laplace
transform of their products is

00:08:02.840 --> 00:08:07.930
going to be the convolution
of each of their inverse

00:08:07.930 --> 00:08:10.610
transforms. Now what's
this over here?

00:08:10.610 --> 00:08:12.340
Well I had this in the beginning
of the problem?

00:08:12.340 --> 00:08:16.210
The inverse Laplace transform
of this, right here, is sine

00:08:16.210 --> 00:08:17.460
of alpha t.

00:08:21.070 --> 00:08:24.140
And then we're going to
convolute that with the

00:08:24.140 --> 00:08:27.070
inverse Laplace transform
of this right here.

00:08:27.070 --> 00:08:29.980
Let's do a little bit of work on
the side, just to make sure

00:08:29.980 --> 00:08:31.380
we get this right.

00:08:31.380 --> 00:08:41.490
So the Laplace transform of sine
of t is equal to 1 over s

00:08:41.490 --> 00:08:42.960
squared, plus 1.

00:08:42.960 --> 00:08:47.480
That looks like this, but I
was shifted by minus 1.

00:08:47.480 --> 00:08:52.780
You might remember that the
Laplace transform of e to the

00:08:52.780 --> 00:08:58.090
at sine of t, when you multiply
e to the at times

00:08:58.090 --> 00:09:00.350
anything, you're shifting
its Laplace transform.

00:09:00.350 --> 00:09:04.820
So that will be equal
to 1 over s minus a

00:09:04.820 --> 00:09:08.900
squared, plus 1.

00:09:08.900 --> 00:09:10.390
We essentially shifted
it by a.

00:09:10.390 --> 00:09:12.830
So now we have something that
looks very similar to this.

00:09:12.830 --> 00:09:17.110
If we just set our a to be equal
to negative 1, here our

00:09:17.110 --> 00:09:20.400
a is equal to negative 1, then
it fits this pattern.

00:09:20.400 --> 00:09:25.230
This is s minus negative 1.

00:09:25.230 --> 00:09:27.640
So the inverse Laplace transform
of this thing right

00:09:27.640 --> 00:09:34.530
here is just e to the a, which
is minus 1, so minus 1t,

00:09:34.530 --> 00:09:39.800
times sine of t.

00:09:39.800 --> 00:09:42.980
So this is the solution to our
differential equation, even

00:09:42.980 --> 00:09:46.780
though it's not in a pleasant
form to look at.

00:09:46.780 --> 00:09:48.890
And we can, if we want, express
it as an integral.

00:09:48.890 --> 00:09:50.410
I'm not going to actually solve
the integral in this

00:09:50.410 --> 00:09:52.570
problem, because it gets hairy,
and it's not even clear

00:09:52.570 --> 00:09:54.190
that-- well, I won't even
attempt to do it.

00:09:54.190 --> 00:09:56.100
But I just want to get into a
form, and from there it's just

00:09:56.100 --> 00:09:58.020
integral calculus.

00:09:58.020 --> 00:10:01.500
or maybe a computer.

00:10:01.500 --> 00:10:03.110
What's the convolution
of these two things?

00:10:03.110 --> 00:10:10.400
It's the integral from 0 to t,
of sine of the first function

00:10:10.400 --> 00:10:11.810
of t minus tau.

00:10:14.600 --> 00:10:16.800
Well, I could actually switch,
and I haven't shown you this,

00:10:16.800 --> 00:10:20.170
but we can switch the order
either way, but actually let

00:10:20.170 --> 00:10:21.420
me just do it this way.

00:10:24.720 --> 00:10:27.390
I could write this as
sine of [? out ?]

00:10:27.390 --> 00:10:32.950
t minus tau, times alpha-- I'm
taking the sine of all of

00:10:32.950 --> 00:10:42.870
those things-- times e to the
minus tau, sine of tau, dtau.

00:10:42.870 --> 00:10:45.330
That's one way, that if I wanted
to express the solution

00:10:45.330 --> 00:10:47.310
of this differential equation's
integral, I could

00:10:47.310 --> 00:10:48.310
write it like that.

00:10:48.310 --> 00:10:50.660
And it actually should be kind
of obvious to you that this

00:10:50.660 --> 00:10:51.890
could go either way.

00:10:51.890 --> 00:10:57.190
Because when it was a product up
here, obviously order does

00:10:57.190 --> 00:10:57.730
not matter.

00:10:57.730 --> 00:11:00.470
I could write this term first,
or I could write that term

00:11:00.470 --> 00:11:05.450
first. So regardless of which
term is written first, the

00:11:05.450 --> 00:11:07.690
same principle would apply.

00:11:07.690 --> 00:11:09.910
And I'll formally prove
it in a future video.

00:11:09.910 --> 00:11:12.290
So we could have also done
it the other way.

00:11:12.290 --> 00:11:16.260
We could have written this
expression as e to the minus

00:11:16.260 --> 00:11:22.700
t, sine of t, convoluted
with sine of alpha t.

00:11:22.700 --> 00:11:27.510
And that would be equal to the
integral from 0 to t, of e to

00:11:27.510 --> 00:11:38.560
the minus t minus tau, sine of
t, minus tau, times sine of

00:11:38.560 --> 00:11:40.800
alpha tau, dtau.

00:11:40.800 --> 00:11:42.770
So these are equivalent.

00:11:42.770 --> 00:11:46.440
Either of these would be
an acceptable answer.

00:11:46.440 --> 00:11:48.480
And normally on a test like
this, the teacher won't expect

00:11:48.480 --> 00:11:50.240
you to actually evaluate
these integrals.

00:11:50.240 --> 00:11:53.370
The teacher will just say, get
it into an integral just to

00:11:53.370 --> 00:11:56.150
kind of see whether you know
how to convolute things and

00:11:56.150 --> 00:11:59.260
get your solution to the
differential equation at least

00:11:59.260 --> 00:12:02.310
into this form, because from
here it really is just, I

00:12:02.310 --> 00:12:03.860
won't say basic calculus,
but it's

00:12:03.860 --> 00:12:05.135
non-differential equations.

00:12:05.135 --> 00:12:08.560
So hopefully, this second
example with the convolution

00:12:08.560 --> 00:12:10.790
to solve an inverse transform
clarified

00:12:10.790 --> 00:12:12.940
things up a little bit.


The convolution and the laplace transform

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.090
Now that you've had a little
bit of exposure to what a

00:00:03.090 --> 00:00:05.650
convolution is, I can
introduce you to the

00:00:05.650 --> 00:00:14.900
convolution theorem, or at
least in the context of--

00:00:14.900 --> 00:00:17.470
there may be other convolution
theorems-- but we're talking

00:00:17.470 --> 00:00:20.050
about differential equations
and Laplace transforms. So

00:00:20.050 --> 00:00:23.440
this is the convolution theorem
as applies to Laplace

00:00:23.440 --> 00:00:27.400
transforms. And it tells us that
if I have a function f of

00:00:27.400 --> 00:00:32.259
t-- and I can define its Laplace
transform as, let's

00:00:32.259 --> 00:00:37.130
see, the Laplace transform of
f of t is capital F of s.

00:00:37.130 --> 00:00:38.660
We've done that before.

00:00:38.660 --> 00:00:43.580
And if I have another function,
g of t, and I take

00:00:43.580 --> 00:00:50.010
its Laplace transform, that of
course is capital G of s.

00:00:50.010 --> 00:00:53.220
Then if we were to convolute
these two functions, so if I

00:00:53.220 --> 00:00:58.670
were to take f and I were to
convolute it with g, which is

00:00:58.670 --> 00:01:00.990
going to be another function
of t-- and

00:01:00.990 --> 00:01:02.040
we already saw this.

00:01:02.040 --> 00:01:02.970
We saw that in the last video.

00:01:02.970 --> 00:01:06.860
I convoluted sine and cosine.

00:01:06.860 --> 00:01:09.460
So this is going to be
a function of t.

00:01:09.460 --> 00:01:13.350
That the Laplace transform of
this thing, and this the crux

00:01:13.350 --> 00:01:18.770
of the theorem, the Laplace
transform of the convolution

00:01:18.770 --> 00:01:22.860
of these two functions is equal
to the products of their

00:01:22.860 --> 00:01:29.030
Laplace transforms. It equals
F of s, big capital F of s,

00:01:29.030 --> 00:01:32.970
times big capital G of s.

00:01:32.970 --> 00:01:36.050
Now, this might seem very
abstract and very, you know,

00:01:36.050 --> 00:01:38.170
hard to kind of handle
for you right now.

00:01:38.170 --> 00:01:39.830
So let's do an actual example.

00:01:39.830 --> 00:01:42.260
And actually, even better, let's
do an inverse Laplace

00:01:42.260 --> 00:01:44.870
transform with an example.

00:01:44.870 --> 00:01:46.040
And actually, let me write
one more thing.

00:01:46.040 --> 00:01:47.950
If this is true, then we could
also do it the other way.

00:01:47.950 --> 00:01:51.580
We could also say that f-- and
I'll just do it all in yellow;

00:01:51.580 --> 00:01:54.200
it takes me too much time to
keep switching colors-- that

00:01:54.200 --> 00:01:58.800
the convolution of f and g, this
is just a function of t,

00:01:58.800 --> 00:02:01.790
I can just say it's the inverse
Laplace transform.

00:02:01.790 --> 00:02:06.480
It's just the inverse Laplace
transform of F of

00:02:06.480 --> 00:02:07.540
s times G of s.

00:02:07.540 --> 00:02:08.680
Although I couldn't resist it.

00:02:08.680 --> 00:02:09.930
Let me switch colors.

00:02:15.020 --> 00:02:16.420
There you go.

00:02:16.420 --> 00:02:18.570
Now, what good does
all of this do?

00:02:18.570 --> 00:02:21.510
Well, we can take inverse
Laplace transforms. Let's just

00:02:21.510 --> 00:02:25.760
say that I had-- let me write
it down here-- let's say I

00:02:25.760 --> 00:02:29.080
told you that the following
expression or function, let's

00:02:29.080 --> 00:02:34.430
say H of s-- let me write it
this way-- H of s is equal to

00:02:34.430 --> 00:02:38.510
2s over s squared plus 1.

00:02:38.510 --> 00:02:40.880
Now, we did this long
differential equations at the

00:02:40.880 --> 00:02:42.880
end, we end up with this thing
and we have to take the

00:02:42.880 --> 00:02:45.170
inverse Laplace transform
of it.

00:02:45.170 --> 00:02:50.580
So we want to figure out the
inverse Laplace transform of H

00:02:50.580 --> 00:02:53.960
of s, or the inverse Laplace
transform of

00:02:53.960 --> 00:02:55.830
this thing right there.

00:02:55.830 --> 00:03:02.090
So we want to figure out the
inverse Laplace transform of

00:03:02.090 --> 00:03:09.285
this expression right here, 2s
over s squared plus 1 squared.

00:03:09.285 --> 00:03:11.040
I don't want to lose that.

00:03:11.040 --> 00:03:12.170
Right there.

00:03:12.170 --> 00:03:15.760
Now, can we write this as the
product of two Laplace

00:03:15.760 --> 00:03:18.000
transforms that we do know?

00:03:18.000 --> 00:03:19.510
Let's try to do it.

00:03:19.510 --> 00:03:21.790
So we can rewrite this.

00:03:21.790 --> 00:03:23.965
And so this is the inverse
Laplace transform.

00:03:23.965 --> 00:03:26.880
So let me rewrite this
expression down here.

00:03:26.880 --> 00:03:34.920
So I can rewrite 2s over s
squared plus 1 squared.

00:03:34.920 --> 00:03:40.180
This is the same thing as-- let
me write it this way-- 2

00:03:40.180 --> 00:03:51.420
times 1 over s squared plus 1,
times s over s squared plus 1.

00:03:51.420 --> 00:03:52.490
I just kind of broke it up.

00:03:52.490 --> 00:03:54.590
If you multiply the numerators
here, you get 2 times

00:03:54.590 --> 00:03:56.580
1, times s, or 2s.

00:03:56.580 --> 00:03:59.200
If you multiply the denominators
here, s squared

00:03:59.200 --> 00:04:02.920
plus 1, times s squared plus 1,
well, that's just s squared

00:04:02.920 --> 00:04:04.130
plus 1 squared.

00:04:04.130 --> 00:04:05.360
So this is the same thing.

00:04:05.360 --> 00:04:07.640
So if we want to take the
inverse Laplace transform of

00:04:07.640 --> 00:04:11.530
this, it's the same thing as
taking the inverse Laplace

00:04:11.530 --> 00:04:14.650
transform of this right here.

00:04:14.650 --> 00:04:17.649
Now, something should
hopefully start

00:04:17.649 --> 00:04:18.870
popping out at you.

00:04:18.870 --> 00:04:22.360
If these were separate
transforms, if they were on

00:04:22.360 --> 00:04:24.470
their own, we know
what this is.

00:04:24.470 --> 00:04:28.380
If we call this F of s, if we
said this is the Laplace

00:04:28.380 --> 00:04:31.230
transform of some function, we
know what that function is.

00:04:31.230 --> 00:04:32.980
This is this piece right here.

00:04:32.980 --> 00:04:35.020
I'm just doing a little
dotted line around it.

00:04:35.020 --> 00:04:37.810
This is the Laplace transform
of sine of t.

00:04:37.810 --> 00:04:41.890
And then if we draw a little
box around this one right

00:04:41.890 --> 00:04:44.560
here, this is the Laplace
transform of

00:04:44.560 --> 00:04:47.480
cosine of t, G of s.

00:04:47.480 --> 00:04:51.110
So this is the Laplace transform
of sine of t, or we

00:04:51.110 --> 00:04:55.510
could write that this implies
that f of t is

00:04:55.510 --> 00:04:58.170
equal to sine of t.

00:04:58.170 --> 00:05:00.600
You should recognize
that one by now.

00:05:00.600 --> 00:05:04.650
And this implies that g of t,
if we define this as the

00:05:04.650 --> 00:05:10.650
Laplace transform of g, this
means that g of t is equal to

00:05:10.650 --> 00:05:13.770
cosine of t.

00:05:13.770 --> 00:05:15.170
And, of course, when you take
the inverse Laplace

00:05:15.170 --> 00:05:19.150
transforms, you could
take the 2's out.

00:05:19.150 --> 00:05:21.440
So now what can we say?

00:05:21.440 --> 00:05:26.170
We can now say that the-- let
me write it this way-- the

00:05:26.170 --> 00:05:28.110
inverse-- so actually, let
me write it this way.

00:05:28.110 --> 00:05:30.890
Or, actually, a better thing to
do, instead of taking the 2

00:05:30.890 --> 00:05:33.990
out, so I can leave it nice
and clean, we could, if we

00:05:33.990 --> 00:05:40.220
were to draw a box around this
whole thing, and define this

00:05:40.220 --> 00:05:46.470
whole thing as F of s, then F of
s is the Laplace transform

00:05:46.470 --> 00:05:48.130
of 2 sine of t.

00:05:48.130 --> 00:05:49.500
I just wanted to
include that 2.

00:05:49.500 --> 00:05:51.620
I didn't want to leave that
out and confuse the issue.

00:05:51.620 --> 00:05:55.150
I wanted a very pure F
of s times G of s.

00:05:55.150 --> 00:05:59.430
So this expression right here is
the product of the Laplace

00:05:59.430 --> 00:06:04.200
transform of 2 sine of t, and
the Laplace transform of

00:06:04.200 --> 00:06:05.610
cosine of t.

00:06:05.610 --> 00:06:11.400
Now, our convolution theorem
told us this right here.

00:06:11.400 --> 00:06:14.510
That if we want to take the
inverse Laplace transform of

00:06:14.510 --> 00:06:17.470
the Laplace transforms of two
functions-- I know that sounds

00:06:17.470 --> 00:06:19.530
very confusing --but you just
kind of pattern match.

00:06:19.530 --> 00:06:21.920
You say, OK look, this thing
that I had here, I could

00:06:21.920 --> 00:06:24.690
rewrite it as a product
of two Laplace

00:06:24.690 --> 00:06:26.460
transforms I can recognize.

00:06:26.460 --> 00:06:30.670
This right here is the Laplace
transform of 2 sine of t.

00:06:30.670 --> 00:06:33.050
This is the Laplace transform
of cosine of t.

00:06:33.050 --> 00:06:36.180
And we just wrote that as
G of s, and F of s.

00:06:36.180 --> 00:06:40.040
So if I have an expression
written like this, I can take

00:06:40.040 --> 00:06:42.440
the inverse Laplace transform
and it'll be equal to the

00:06:42.440 --> 00:06:48.220
convolution of the original
functions.

00:06:48.220 --> 00:06:51.830
It'll be equal to the
convolution of the inverse of

00:06:51.830 --> 00:06:53.630
g or the inverse of f.

00:06:53.630 --> 00:06:54.750
Let me write it this way.

00:06:54.750 --> 00:06:56.810
I could write it like this.

00:06:56.810 --> 00:07:03.410
We know that f of t is equal
to the inverse Laplace

00:07:03.410 --> 00:07:06.080
transform of F of s.

00:07:06.080 --> 00:07:09.830
And we know that g-- I should
have done it in a different

00:07:09.830 --> 00:07:14.520
color, but I'll do g in green--
we know that g of t is

00:07:14.520 --> 00:07:21.190
equal to the inverse Laplace
transform of G of s.

00:07:21.190 --> 00:07:24.450
So we can rewrite the
convolution theorem as the

00:07:24.450 --> 00:07:27.960
inverse-- and this might maybe
confuse you more than help,

00:07:27.960 --> 00:07:30.060
but I'll give my best shot.

00:07:30.060 --> 00:07:34.550
The inverse Laplace transform
of-- and I'll try to stay true

00:07:34.550 --> 00:07:43.480
to the colors-- of F of s times
G of s is equal to-- I'm

00:07:43.480 --> 00:07:45.220
just restating this convolution

00:07:45.220 --> 00:07:45.690
theorem right here.

00:07:45.690 --> 00:07:49.910
This is equal to the convolution
of the inverse

00:07:49.910 --> 00:07:52.340
Laplace transform of F of s.

00:07:52.340 --> 00:07:55.680
So it's equal to the convolution
of the inverse

00:07:55.680 --> 00:08:04.620
Laplace transform of F of s
with the inverse Laplace

00:08:04.620 --> 00:08:06.480
transform of G of s.

00:08:06.480 --> 00:08:10.380
With the inverse Laplace
transform of

00:08:10.380 --> 00:08:15.410
capital G, of G of s.

00:08:15.410 --> 00:08:17.740
I'm not sure if that helps you
or not, but if you go back to

00:08:17.740 --> 00:08:19.160
this example it might.

00:08:19.160 --> 00:08:22.260
This is F of s, this is
F of s right here.

00:08:22.260 --> 00:08:28.130
2 times-- I'll do it in the
light blue-- this is 2 over s

00:08:28.130 --> 00:08:29.320
squared plus 1.

00:08:29.320 --> 00:08:32.460
That's F of s in our example.

00:08:32.460 --> 00:08:39.820
And the G of s was s over
s squared plus 1.

00:08:39.820 --> 00:08:42.590
And all I got that from is I
just broke this up into two

00:08:42.590 --> 00:08:43.570
things that I recognize.

00:08:43.570 --> 00:08:46.980
If I multiply this together, I
get back to my original thing

00:08:46.980 --> 00:08:49.540
that I was trying to take the
inverse Laplace transform of.

00:08:49.540 --> 00:08:52.930
And so the convolution theorem
just says that, OK, well, the

00:08:52.930 --> 00:08:57.170
inverse Laplace transform of
this is equal to the inverse

00:08:57.170 --> 00:09:05.090
Laplace transform of 2 over s
squared plus 1, convoluted

00:09:05.090 --> 00:09:14.960
with the inverse Laplace
transform of our G of s, of s

00:09:14.960 --> 00:09:18.430
over s squared plus 1.

00:09:18.430 --> 00:09:20.430
And we know what these
things are.

00:09:20.430 --> 00:09:22.900
I already told them to you, but
they should be somewhat

00:09:22.900 --> 00:09:23.700
second nature now.

00:09:23.700 --> 00:09:25.650
This is 2 times sine of t.

00:09:28.720 --> 00:09:31.110
You take the Laplace transform
of sine of t, you get 1 over s

00:09:31.110 --> 00:09:33.190
squared plus 1, and then you
multiply it by 2, you get the

00:09:33.190 --> 00:09:34.260
2 up there.

00:09:34.260 --> 00:09:37.870
And you're going to have to
convolute that with the

00:09:37.870 --> 00:09:39.660
inverse Laplace transform
of this thing here.

00:09:39.660 --> 00:09:40.580
And we already went over this.

00:09:40.580 --> 00:09:41.900
This is cosine of t.

00:09:44.670 --> 00:09:49.480
So our result so far--
let me be very clear.

00:09:49.480 --> 00:09:52.130
It's always good to take a step
back and just think about

00:09:52.130 --> 00:09:55.270
what we're doing, much less
why we're doing it.

00:09:55.270 --> 00:10:00.960
But let's see, the inverse
Laplace transform of this

00:10:00.960 --> 00:10:06.650
thing up in this top left
corner, 2s over s squared plus

00:10:06.650 --> 00:10:11.220
1 squared, which before we did
what we're doing now was very

00:10:11.220 --> 00:10:13.740
hard to figure out-- actually,
this would be a curly bracket

00:10:13.740 --> 00:10:17.620
right here, but you get the
idea-- is equal to this.

00:10:17.620 --> 00:10:30.710
It's equal to 2 sine of t,
convoluted with cosine of t.

00:10:30.710 --> 00:10:32.830
And you're like, Sal, throughout
this whole process

00:10:32.830 --> 00:10:34.990
I've already forgotten what
it means to convolute two

00:10:34.990 --> 00:10:37.330
functions, so let's
convolute them.

00:10:37.330 --> 00:10:39.570
And I'll just write the
definition, or the definition

00:10:39.570 --> 00:10:41.310
we're using of the
convolution.

00:10:41.310 --> 00:10:44.370
That f convoluted with
g-- it's going to be

00:10:44.370 --> 00:10:45.040
a function of g.

00:10:45.040 --> 00:10:46.350
I'll just write this
short-hand-- is equal to the

00:10:46.350 --> 00:10:51.570
integral from 0 to t,
of f of t minus tau,

00:10:51.570 --> 00:10:54.910
times g of tau, dtau.

00:10:54.910 --> 00:11:03.280
So 2 sine of t convoluted with
cosine of t is equal to-- let

00:11:03.280 --> 00:11:09.575
me do a neutral color-- the
integral from 0 to t, of 2

00:11:09.575 --> 00:11:22.290
sine of t, minus tau, times
the cosine of tau, dtau.

00:11:22.290 --> 00:11:25.800
Now if you watched the very last
video I made, I actually

00:11:25.800 --> 00:11:29.515
solved this, or I solved a very
similar thing to this.

00:11:29.515 --> 00:11:34.400
If we take the 2 out we get 2,
times the integral from 0 to

00:11:34.400 --> 00:11:43.090
t, of sine of t minus tau,
times the cosine of tau.

00:11:43.090 --> 00:11:45.950
I actually solved this in
the previous video.

00:11:45.950 --> 00:11:48.490
This right here, this is the
convolution of sine of t and

00:11:48.490 --> 00:11:49.170
cosine of t.

00:11:49.170 --> 00:11:54.000
It's sine of t convoluted
with cosine of t.

00:11:54.000 --> 00:11:56.870
And I show you in the previous
video, just watch that video,

00:11:56.870 --> 00:11:59.490
where I introduce a convolution,
that this thing

00:11:59.490 --> 00:12:05.870
right here is equal
to 1/2t sine of t.

00:12:05.870 --> 00:12:08.460
Now, if this thing is equal to
1/2 t sine of t, and I have to

00:12:08.460 --> 00:12:14.750
multiply it by 2, then we get,
our big result, that the

00:12:14.750 --> 00:12:21.550
inverse Laplace transform of
2s over s squared plus 1

00:12:21.550 --> 00:12:26.170
squared is equal to the
convolution of 2 sine of t

00:12:26.170 --> 00:12:27.480
with cosine of t.

00:12:27.480 --> 00:12:31.550
Which is just 2 times this thing
here, which is 2 times

00:12:31.550 --> 00:12:37.650
1/2-- those cancel out-- so
it equals t sine of t.

00:12:37.650 --> 00:12:39.470
And once you get the hang of
it, you won't have to go

00:12:39.470 --> 00:12:40.820
through all of these steps.

00:12:40.820 --> 00:12:43.760
But the key is to recognize that
this could be broken down

00:12:43.760 --> 00:12:47.800
as the products of two Laplace
transforms that you recognize.

00:12:47.800 --> 00:12:50.160
This could be broken down as
the product of two Laplace

00:12:50.160 --> 00:12:51.840
transforms we recognized.

00:12:51.840 --> 00:12:53.630
This is the Laplace transform
of 2 sine of t.

00:12:53.630 --> 00:12:56.010
This was the Laplace transform
of cosine of t.

00:12:56.010 --> 00:12:59.320
So the inverse Laplace transform
of our original

00:12:59.320 --> 00:13:02.460
thing, or original expression,
is just the convolution of

00:13:02.460 --> 00:13:04.760
that with that.

00:13:04.760 --> 00:13:07.190
And if you watched the previous
video, you'd realize

00:13:07.190 --> 00:13:10.160
that actually calculating that
convolution was no simple

00:13:10.160 --> 00:13:11.890
task, but it can be done.

00:13:11.890 --> 00:13:13.840
So you actually can get
an integral form.

00:13:13.840 --> 00:13:16.200
Even if it can't be done, you
can get your answer, at least,

00:13:16.200 --> 00:13:19.510
in terms of some integral.

00:13:19.510 --> 00:13:22.920
So I haven't proven
the convolution

00:13:22.920 --> 00:13:23.730
theorem to you just yet.

00:13:23.730 --> 00:13:25.260
I'll do that in a
future video.

00:13:25.260 --> 00:13:30.540
But hopefully, this gave you a
little bit of a sense of how

00:13:30.540 --> 00:13:33.040
you can use it to actually
take inverse Laplace

00:13:33.040 --> 00:13:34.840
transforms. And remember, the
reason why we're learning to

00:13:34.840 --> 00:13:37.040
take inverse Laplace transforms,
and we have all of

00:13:37.040 --> 00:13:40.280
these tools to do it, is because
that's always that

00:13:40.280 --> 00:13:42.950
last step when you're solving
these differential equations,

00:13:42.950 --> 00:13:45.490
using your Laplace transforms.