Introduction to linear independence

WEBVTT
Kind: captions
Language: en

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Let's say I had the set of
vectors-- I don't want to do

00:00:06.670 --> 00:00:08.039
it that thick.

00:00:08.039 --> 00:00:14.350
Let's say one of the vectors is
the vector 2, 3, and then

00:00:14.350 --> 00:00:20.910
the other vector is
the vector 4, 6.

00:00:20.910 --> 00:00:24.050
And I just want to answer the
question: what is the span of

00:00:24.050 --> 00:00:24.685
these vectors?

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And let's assume that these
are position vectors.

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What are all of the
vectors that these

00:00:28.330 --> 00:00:30.150
two vectors can represent?

00:00:30.150 --> 00:00:34.000
Well, if you just look at it,
and remember, the span is just

00:00:34.000 --> 00:00:36.150
all of the vectors that can
be represented by linear

00:00:36.150 --> 00:00:37.320
combinations of these.

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So it's the set of all the
vectors that if I have some

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constant times 2 times that
vector plus some other

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constant times this vector,
it's all the possibilities

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that I can represent when I just
put a bunch of different

00:00:52.970 --> 00:00:55.640
real numbers for c1 and c2.

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Now, the first thing you might
realize is that, look, this

00:00:58.640 --> 00:01:02.250
vector 2, this is just
the same thing as

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2 times this vector.

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So I could just rewrite
it like this.

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I could just rewrite it as c1
times the vector 2, 3 plus c2

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times the vector-- and here,
instead of writing the vector

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4, 6, I'm going to write 2 times
the vector 2, 3, because

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this vector is just a multiple
of that vector.

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So I could write c2 times
2 times 2, 3.

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I think you see that this is
equivalent to the 4, 6.

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2 times 2 is 4.

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2 times 3 is 6.

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Well, then we can simplify
this a little bit.

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We can rewrite this as just c1
plus 2c2, all of that, times

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2, 3, times our vector 2, 3.

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And this is just some
arbitrary constant.

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It's some arbitrary constant
plus 2 times some other

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arbitrary constant.

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So we can just call this c3
times my vector 2, 3.

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So in this situation, even
though we started with two

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vectors, and I said, well, you
know, the span of these two

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vectors is equal to all of
the vectors that can be

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constructed with some linear
combination of these, any

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linear combination of these,
if I just use this

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substitution right here, can
be reduced to just a scalar

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multiple of my first vector.

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And I could have gone the
other way around.

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I could have substituted this
vector as being 1/2 times

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this, and just made any
combination of scalar multiple

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of the second vector.

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But the fact is, that instead
of talking about linear

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combinations of two vectors,
I can reduce this to just a

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scalar combination
of one vector.

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And we've seen in R2 a scalar
combination of one vector,

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especially if they're
position vectors.

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For example, this vector 2, 3.

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It's 2, 3.

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It looks like this.

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All the scalar combinations of
that vector are just going to

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lie along this line.

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So 2, 3, it's going
to be right there.

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They're all just going to lie
along that line right there,

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so along this line going in
both directions forever.

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And if I take negative
values of 2, 3, I'm

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going to go down here.

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If I take positive values,
I'm going to go here.

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If I get really large positive
values, it's

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going to go up here.

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But I can just represent the
vectors, and when you put them

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in standard form, their arrows
essentially would

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trace out this line.

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So you could say that the span
of my set of vectors-- let me

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put it over here.

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The span of the set of vectors
2, 3 and 4, 6 is

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just this line here.

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Even though we have
two vectors,

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they're essentially collinear.

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They're multiples
of each other.

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I mean, if this is 2, 3, 4,
6 is just this right here.

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It's just that longer
one right there.

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They're collinear.

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These two things
are collinear.

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Now, in this case, when we have
two collinear vectors in

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R2, essentially their span just
reduces to that line.

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You can't represent
some vector like--

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let me do a new color.

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You can't represent this vector
right there with some

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combination of those
two vectors.

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There's no way to kind of
break out of this line.

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So there's no way that you can
represent everything in R2.

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So the span is just
that line there.

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Now, a related idea to this,
and notice, you had two

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vectors, but it kind of reduced
to one vector when you

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took its linear combinations.

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The related idea here is that we
call this set-- we call it

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linearly dependent.

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Let me write that down:
linearly dependent.

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This is a linearly
dependent set.

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And linearly dependent just
means that one of the vectors

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in the set can be represented
by some combination of the

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other vectors in the set.

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A way to think about it is
whichever vector you pick that

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can be represented by the
others, it's not adding any

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new directionality or any
new information, right?

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In this case, we already had
a vector that went in this

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direction, and when you throw
this 4, 6 on there, you're

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going in the same direction,
just scaled up.

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So it's not giving us any new
dimension, letting us break

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out of this line, right?

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And you can imagine in three
space, if you have one vector

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that looks like this and another
vector that looks like

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this, two vectors that aren't
collinear, they're going to

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define a kind of two-dimensional
space.

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They can define a
two-dimensional space.

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Let's say that this is
the plane defined

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by those two vectors.

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In order to define R3, a third
vector in that set can't be

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coplanar with those
two, right?

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If this third vector is coplanar
with these, it's not

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adding any more directionality.

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So this set of three
vectors will

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also be linearly dependent.

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And another way to think about
it is that these two purple

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vectors span this plane, span
the plane that they define,

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essentially, right?

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Anything in this plane going in
any direction can be-- any

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vector in this plane, when we
say span it, that means that

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any vector can be represented
by a linear combination of

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this vector and this vector,
which means that if this

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vector is on that plane, it can
be represented as a linear

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combination of that vector
and that vector.

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So this green vector I added
isn't going to add anything to

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the span of our set of vectors
and that's because this is a

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linearly dependent set.

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This one can be represented by a
sum of that one and that one

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because this one and this
one span this plane.

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In order for the span of these
three vectors to kind of get

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more dimensionality or start
representing R3, the third

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vector will have to break
out of that plane.

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It would have to break
out of that plane.

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And if a vector is breaking out
of that plane, that means

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it's a vector that can't be
represented anywhere on that

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plane, so it's outside of the
span of those two vectors.

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Where it's outside, it can't
be represented by a linear

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combination of this
one and this one.

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So if you had a vector of this
one, this one, and this one,

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and just those three, none of
these other things that I

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drew, that would be linearly
independent.

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Let me draw a couple more
examples for you.

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That one might have been
a little too abstract.

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So, for example, if I have the
vectors 2, 3 and I have the

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vector 7, 2, and I have the
vector 9, 5, and I were to ask

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you, are these linearly
dependent or independent?

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So at first you say, well, you
know, it's not trivial.

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Let's see, this isn't a scalar
multiple of that.

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That doesn't look like a scalar
multiple of either of

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the other two.

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Maybe they're linearly
independent.

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But then, if you kind of inspect
them, you kind of see

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that v, if we call this v1,
vector 1, plus vector 2, if we

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call this vector 2, is
equal to vector 3.

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So vector 3 is a linear
combination of

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these other two vectors.

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So this is a linearly
dependent set.

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And if we were to show it, draw
it in kind of two space,

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and it's just a general idea
that-- well, let me see.

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Let me draw it in R2.

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There's a general idea that if
you have three two-dimensional

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vectors, one of them is
going to be redundant.

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Well, one of them definitely
will be redundant.

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For example, if we do 2, 3, if
we do the vector 2, 3, that's

00:08:49.030 --> 00:08:50.872
the first one right there.

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I draw it in the standard
position.

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And I draw the vector 7, 2 right
there, I could show you

00:09:06.010 --> 00:09:13.090
that any point in R2 can be
represented by some linear

00:09:13.090 --> 00:09:14.450
combination of these
two vectors.

00:09:14.450 --> 00:09:17.570
We can even do a kind of a
graphical representation.

00:09:17.570 --> 00:09:20.750
I've done that in the previous
video, so I could write that

00:09:20.750 --> 00:09:30.630
the span of v1 and v2
is equal to R2.

00:09:30.630 --> 00:09:35.100
That means that every vector,
every position here can be

00:09:35.100 --> 00:09:39.530
represented by some linear
combination of these two guys.

00:09:39.530 --> 00:09:49.830
Now, the vector 9,
5, it is in R2.

00:09:49.830 --> 00:09:51.730
It is in R2, right?

00:09:51.730 --> 00:09:52.060
Clearly.

00:09:52.060 --> 00:09:53.490
I just graphed it
on this plane.

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It's in our two-dimensional,
real number space.

00:09:56.870 --> 00:10:02.100
Or I guess we could call it
a space or in our set R2.

00:10:02.100 --> 00:10:03.280
It's there.

00:10:03.280 --> 00:10:04.720
It's right there.

00:10:04.720 --> 00:10:07.930
So we just said that anything in
R2 can be represented by a

00:10:07.930 --> 00:10:09.760
linear combination of
those two guys.

00:10:09.760 --> 00:10:12.720
So clearly, this is in R2, so
it can be represented as a

00:10:12.720 --> 00:10:13.850
linear combination.

00:10:13.850 --> 00:10:15.940
So hopefully, you're starting
to see the relationship

00:10:15.940 --> 00:10:20.810
between span and linear
independence or linear

00:10:20.810 --> 00:10:21.370
dependence.

00:10:21.370 --> 00:10:24.100
Let me do another example.

00:10:24.100 --> 00:10:30.263
Let's say I have the vectors--
let me do a new color.

00:10:30.263 --> 00:10:32.620
Let's say I have the vector--
and this one will be a little

00:10:32.620 --> 00:10:38.140
bit obvious-- 7, 0, so that's
my v1, and then I have my

00:10:38.140 --> 00:10:42.060
second vector, which
is 0, minus 1.

00:10:42.060 --> 00:10:43.850
That's v2.

00:10:43.850 --> 00:10:49.170
Now, is this set linearly
independent?

00:10:49.170 --> 00:10:50.840
Is it linearly independent?

00:10:50.840 --> 00:10:53.370
Well, can I represent
either of these as a

00:10:53.370 --> 00:10:54.400
combination of the other?

00:10:54.400 --> 00:10:56.020
And really when I say as a
combination, you'd have to

00:10:56.020 --> 00:10:58.270
scale up one to get the other,
because there's only two

00:10:58.270 --> 00:10:59.820
vectors here.

00:10:59.820 --> 00:11:02.310
If I am trying to add up to this
vector, the only thing I

00:11:02.310 --> 00:11:03.730
have to deal with is this
one, so all I can

00:11:03.730 --> 00:11:05.030
do is scale it up.

00:11:05.030 --> 00:11:06.850
Well, there's nothing
I can do.

00:11:06.850 --> 00:11:10.190
No matter what I multiply this
vector by, you know, some

00:11:10.190 --> 00:11:14.520
constant and add it to itself
or scale it up, this term

00:11:14.520 --> 00:11:16.400
right here is always
going to be zero.

00:11:16.400 --> 00:11:17.560
It's always going to be zero.

00:11:17.560 --> 00:11:20.300
So nothing I can multiply
this by is going to

00:11:20.300 --> 00:11:21.970
get me to this vector.

00:11:21.970 --> 00:11:25.820
Likewise, no matter what I
multiply this vector by, the

00:11:25.820 --> 00:11:27.590
top term is always
going to be zero.

00:11:27.590 --> 00:11:29.920
So there's no way I could
get to this vector.

00:11:29.920 --> 00:11:32.020
So both of these vectors,
there's no way that you can

00:11:32.020 --> 00:11:34.720
represent one as a combination
of the other.

00:11:34.720 --> 00:11:37.970
So these two are linearly
independent.

00:11:37.970 --> 00:11:41.220
And you can even see
it if we graph it.

00:11:41.220 --> 00:11:44.245
One is 7, 0, which
is like that.

00:11:44.245 --> 00:11:46.195
Let me do it in a non-yellow
color.

00:11:48.990 --> 00:11:50.140
7, 0.

00:11:50.140 --> 00:11:54.250
And one is 0, minus 1.

00:11:54.250 --> 00:11:57.030
And I think you can clearly see
that if you take a linear

00:11:57.030 --> 00:11:59.780
combination of any of these
two, you can represent

00:11:59.780 --> 00:12:00.780
anything in R2.

00:12:00.780 --> 00:12:04.320
So the span of these, just to
kind of get used to our notion

00:12:04.320 --> 00:12:09.550
of span of v1 and v2,
is equal to R2.

00:12:09.550 --> 00:12:12.790
Now, this is another interesting
point to make.

00:12:12.790 --> 00:12:15.640
I said the span of
v1 and v2 is R2.

00:12:15.640 --> 00:12:22.850
Now what is the span of
v1, v2, and v3 in

00:12:22.850 --> 00:12:24.810
this example up here?

00:12:24.810 --> 00:12:25.550
I already told you.

00:12:25.550 --> 00:12:28.170
I already showed you that
this third vector can be

00:12:28.170 --> 00:12:30.030
represented as a linear
combination of these two.

00:12:30.030 --> 00:12:32.050
It's actually just these
two summed up.

00:12:32.050 --> 00:12:33.210
I can even draw it right here.

00:12:33.210 --> 00:12:34.620
It's just those two
vectors summed up.

00:12:34.620 --> 00:12:37.280
So it clearly can be represented
as a linear

00:12:37.280 --> 00:12:38.860
combination of those two.

00:12:38.860 --> 00:12:40.670
So what's its span?

00:12:40.670 --> 00:12:43.270
Well, the fact that this is
redundant means that it

00:12:43.270 --> 00:12:44.200
doesn't change its span.

00:12:44.200 --> 00:12:47.000
It doesn't change all of the
possible linear combinations.

00:12:47.000 --> 00:12:50.500
So its span is also
going to be R2.

00:12:50.500 --> 00:12:52.540
It's just that this was
more vectors than you

00:12:52.540 --> 00:12:54.210
needed to span R2.

00:12:54.210 --> 00:12:57.210
R2 is a two-dimensional space,
and you needed two vectors.

00:12:57.210 --> 00:13:02.180
So this was kind of a more
efficient way of providing a

00:13:02.180 --> 00:13:06.830
basis, and I haven't defined
basis formally, yet, but I

00:13:06.830 --> 00:13:09.720
just want to use it a little
conversationally, and then

00:13:09.720 --> 00:13:11.780
it'll make sense to you when
I define it formally.

00:13:11.780 --> 00:13:15.060
This provides a better basis, or
this provides a basis, kind

00:13:15.060 --> 00:13:19.460
of a non-redundant set of
vectors that can represent R2.

00:13:19.460 --> 00:13:22.910
While this one, right
here, is redundant.

00:13:22.910 --> 00:13:27.620
So it's not a good
basis for R2.

00:13:27.620 --> 00:13:30.060
Let me give you one more example
in three dimensions.

00:13:30.060 --> 00:13:31.940
And then in the next video,
I'm going to make a more

00:13:31.940 --> 00:13:36.400
formal definition of linear
dependence or independence.

00:13:36.400 --> 00:13:42.205
So let's say that I had
the vector 2, 0, 0.

00:13:42.205 --> 00:13:45.470
Let me make a similar argument
that I made up there: the

00:13:45.470 --> 00:13:53.330
vector 2, 0, 0, the vector 0, 1,
0, and the vector 0, 0, 7.

00:13:56.560 --> 00:13:58.050
We are now in R3, right?

00:13:58.050 --> 00:14:00.550
Each of these are
three-dimensional vectors.

00:14:00.550 --> 00:14:03.290
Now, are these linear dependent
or linearly

00:14:03.290 --> 00:14:04.010
independent?

00:14:04.010 --> 00:14:07.080
Sorry, are they linearly
dependent or independent?

00:14:07.080 --> 00:14:10.960
Well, there's no way with some
combination of these two

00:14:10.960 --> 00:14:14.880
vectors that I can end up with
a non-zero term right here to

00:14:14.880 --> 00:14:16.160
make this third vector, right?

00:14:16.160 --> 00:14:18.160
Because no matter what I
multiply this one by and this

00:14:18.160 --> 00:14:20.400
one by, this last term
is going to be zero.

00:14:20.400 --> 00:14:22.690
So this is kind of adding
a new direction

00:14:22.690 --> 00:14:24.640
to our set of vectors.

00:14:24.640 --> 00:14:26.810
Likewise, there's nothing
I can do-- there's no

00:14:26.810 --> 00:14:30.720
combination of this guy and
this guy that I can get a

00:14:30.720 --> 00:14:32.430
non-zero term here.

00:14:32.430 --> 00:14:34.930
And finally, no combination of
this guy and this guy that I

00:14:34.930 --> 00:14:36.680
can get a non-zero term here.

00:14:36.680 --> 00:14:39.035
So this set is linearly
independent.

00:14:43.940 --> 00:14:46.550
And if you were to graph these
in three dimensions, you would

00:14:46.550 --> 00:14:50.610
see that none of these--
these three do not

00:14:50.610 --> 00:14:51.690
lie on the same plane.

00:14:51.690 --> 00:14:54.580
Obviously, any two of them lie
on the same plane, but if you

00:14:54.580 --> 00:14:56.730
were to actually graph
it, you get 2, 0.

00:14:56.730 --> 00:14:58.350
Let me say that that's x-axis.

00:14:58.350 --> 00:15:00.690
That's 2, 0, 0.

00:15:00.690 --> 00:15:02.510
Then you have this, 0, 1, 0.

00:15:02.510 --> 00:15:03.700
Maybe that's the y-axis.

00:15:03.700 --> 00:15:05.370
And then you have 0, 0, 7.

00:15:05.370 --> 00:15:07.070
It would look something
like this.

00:15:07.070 --> 00:15:09.300
So it almost looks like, your
three-dimensional axes, it

00:15:09.300 --> 00:15:10.860
almost looks like the
vectors i, j, k.

00:15:10.860 --> 00:15:12.390
They're just scaled
up a little bit.

00:15:12.390 --> 00:15:15.410
But you can always correct
that by just scaling them

00:15:15.410 --> 00:15:16.160
down, right?

00:15:16.160 --> 00:15:19.240
Because we care about any linear
combination of these.

00:15:19.240 --> 00:15:25.090
So the span of these three
vectors right here, because

00:15:25.090 --> 00:15:28.580
they're all adding new
directionality, is R3.

00:15:31.490 --> 00:15:33.690
Anyway, I thought I would leave
you there in this video.

00:15:33.690 --> 00:15:35.950
I realize I've been making
longer and longer videos, and

00:15:35.950 --> 00:15:38.230
I want to get back in the habit
of making shorter ones.

00:15:38.230 --> 00:15:40.270
In the next video, I'm going
to make a more formal

00:15:40.270 --> 00:15:43.450
definition of linear dependence,
and we'll do a

00:15:43.450 --> 00:15:45.200
bunch more examples.


Linear combinations and span

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.720
One term you are going to hear
a lot of in these videos, and

00:00:03.720 --> 00:00:06.080
in linear algebra in general,
is the idea of a linear

00:00:06.080 --> 00:00:07.330
combination.

00:00:11.620 --> 00:00:16.020
And all a linear combination of
vectors are, they're just a

00:00:16.020 --> 00:00:16.809
linear combination.

00:00:16.809 --> 00:00:18.910
Let me show you what
that means.

00:00:18.910 --> 00:00:26.160
So let's say I have a couple
of vectors, v1, v2, and it

00:00:26.160 --> 00:00:27.970
goes all the way to vn.

00:00:27.970 --> 00:00:30.580
And they're all in, you know,
it can be in R2 or Rn.

00:00:30.580 --> 00:00:33.510
Let's say that they're
all in Rn.

00:00:33.510 --> 00:00:37.600
They're in some dimension of
real space, I guess you could

00:00:37.600 --> 00:00:39.800
call it, but the idea
is fairly simple.

00:00:39.800 --> 00:00:42.910
A linear combination of these
vectors means you just add up

00:00:42.910 --> 00:00:44.640
the vectors.

00:00:44.640 --> 00:00:50.420
It's some combination of a sum
of the vectors, so v1 plus v2

00:00:50.420 --> 00:00:53.990
plus all the way to vn,
but you scale them

00:00:53.990 --> 00:00:56.110
by arbitrary constants.

00:00:56.110 --> 00:01:01.560
So you scale them by c1, c2,
all the way to cn, where

00:01:01.560 --> 00:01:04.720
everything from c1
to cn are all a

00:01:04.720 --> 00:01:06.970
member of the real numbers.

00:01:06.970 --> 00:01:08.870
That's all a linear
combination is.

00:01:08.870 --> 00:01:11.070
Let me show you a concrete
example of linear

00:01:11.070 --> 00:01:11.810
combinations.

00:01:11.810 --> 00:01:14.960
Let me make the vector.

00:01:14.960 --> 00:01:18.320
Let me define the vector a to
be equal to-- and these are

00:01:18.320 --> 00:01:19.180
all bolded.

00:01:19.180 --> 00:01:21.140
These purple, these are all
bolded, just because those are

00:01:21.140 --> 00:01:23.490
vectors, but sometimes it's
kind of onerous to keep

00:01:23.490 --> 00:01:24.160
bolding things.

00:01:24.160 --> 00:01:30.890
So let's just say I define the
vector a to be equal to 1, 2.

00:01:30.890 --> 00:01:36.820
And I define the vector
b to be equal to 0, 3.

00:01:36.820 --> 00:01:39.780
What is the linear combination
of a and b?

00:01:39.780 --> 00:01:43.490
Well, it could be any constant
times a plus any

00:01:43.490 --> 00:01:44.550
constant times b.

00:01:44.550 --> 00:01:50.870
So it could be 0 times a plus--
well, it could be 0

00:01:50.870 --> 00:01:54.690
times a plus 0 times b, which,
of course, would be what?

00:01:54.690 --> 00:01:57.650
That would be 0 times 0,
that would be 0, 0.

00:01:57.650 --> 00:02:00.810
That would be the 0 vector, but
this is a completely valid

00:02:00.810 --> 00:02:01.860
linear combination.

00:02:01.860 --> 00:02:04.980
And we can denote the
0 vector by just a

00:02:04.980 --> 00:02:06.945
big bold 0 like that.

00:02:09.979 --> 00:02:11.580
I could do 3 times a.

00:02:11.580 --> 00:02:13.880
I'm just picking these
numbers at random.

00:02:13.880 --> 00:02:18.960
3 times a plus-- let me do a
negative number just for fun.

00:02:18.960 --> 00:02:23.590
So I'm going to do plus
minus 2 times b.

00:02:23.590 --> 00:02:24.820
What is that equal to?

00:02:24.820 --> 00:02:27.230
Let's figure it out.

00:02:27.230 --> 00:02:28.190
Let me write it out.

00:02:28.190 --> 00:02:34.140
It's 3 minus 2 times 0,
so minus 0, and it's

00:02:34.140 --> 00:02:36.490
3 times 2 is 6.

00:02:36.490 --> 00:02:45.270
6 minus 2 times 3, so minus 6,
so it's the vector 3, 0.

00:02:45.270 --> 00:02:48.590
This is a linear combination
of a and b.

00:02:48.590 --> 00:02:52.170
I can keep putting in a bunch
of random real numbers here

00:02:52.170 --> 00:02:55.070
and here, and I'll just get a
bunch of different linear

00:02:55.070 --> 00:02:57.740
combinations of my
vectors a and b.

00:02:57.740 --> 00:03:01.720
If I had a third vector here,
if I had vector c, and maybe

00:03:01.720 --> 00:03:06.200
that was just, you know, 7, 2,
then I could add that to the

00:03:06.200 --> 00:03:10.640
mix and I could throw in
plus 8 times vector c.

00:03:10.640 --> 00:03:13.150
These are all just linear
combinations.

00:03:13.150 --> 00:03:16.680
Now why do we just call
them combinations?

00:03:16.680 --> 00:03:19.960
Why do you have to add that
little linear prefix there?

00:03:19.960 --> 00:03:22.130
Because we're just
scaling them up.

00:03:22.130 --> 00:03:25.360
We're not multiplying the
vectors times each other.

00:03:25.360 --> 00:03:28.280
We haven't even defined what it
means to multiply a vector,

00:03:28.280 --> 00:03:30.010
and there's actually several
ways to do it.

00:03:30.010 --> 00:03:33.350
But, you know, we can't square
a vector, and we haven't even

00:03:33.350 --> 00:03:36.260
defined what this means yet, but
this would all of a sudden

00:03:36.260 --> 00:03:38.270
make it nonlinear
in some form.

00:03:38.270 --> 00:03:40.400
So all we're doing is we're
adding the vectors, and we're

00:03:40.400 --> 00:03:43.610
just scaling them up by some
scaling factor, so that's why

00:03:43.610 --> 00:03:46.780
it's called a linear
combination.

00:03:46.780 --> 00:03:48.920
Now you might say, hey Sal, why
are you even introducing

00:03:48.920 --> 00:03:51.880
this idea of a linear
combination?

00:03:51.880 --> 00:03:55.610
Because I want to introduce the
idea, and this is an idea

00:03:55.610 --> 00:03:58.270
that confounds most students
when it's first taught.

00:03:58.270 --> 00:04:00.730
I think it's just the very
nature that it's taught.

00:04:00.730 --> 00:04:07.080
Over here, I just kept putting
different numbers for the

00:04:07.080 --> 00:04:10.110
weights, I guess we could call
them, for c1 and c2 in this

00:04:10.110 --> 00:04:12.350
combination of a and b, right?

00:04:12.350 --> 00:04:13.860
Let's ignore c for
a little bit.

00:04:13.860 --> 00:04:17.320
I just put in a bunch of
different numbers there.

00:04:17.320 --> 00:04:23.470
But it begs the question: what
is the set of all of the

00:04:23.470 --> 00:04:25.280
vectors I could have created?

00:04:25.280 --> 00:04:27.530
And this is just one
member of that set.

00:04:27.530 --> 00:04:31.390
But what is the set of all of
the vectors I could've created

00:04:31.390 --> 00:04:34.500
by taking linear combinations
of a and b?

00:04:34.500 --> 00:04:35.990
So let me draw a and b here.

00:04:35.990 --> 00:04:38.280
Maybe we can think about it
visually, and then maybe we

00:04:38.280 --> 00:04:41.190
can think about it
mathematically.

00:04:41.190 --> 00:04:42.850
So let's say a and b.

00:04:42.850 --> 00:04:44.160
So a is 1, 2.

00:04:44.160 --> 00:04:46.650
So 1, 2 looks like that.

00:04:46.650 --> 00:04:49.060
That's vector a.

00:04:49.060 --> 00:04:50.760
Let me do vector b in
a different color.

00:04:50.760 --> 00:04:51.720
We're going to do
it in yellow.

00:04:51.720 --> 00:04:53.330
Vector b is 0, 3.

00:04:53.330 --> 00:04:57.400
So vector b looks
like that: 0, 3.

00:04:57.400 --> 00:05:00.320
So what's the set of all of
the vectors that I can

00:05:00.320 --> 00:05:03.120
represent by adding and
subtracting these vectors?

00:05:03.120 --> 00:05:06.120
And we said, if we multiply them
both by zero and add them

00:05:06.120 --> 00:05:08.450
to each other, we
end up there.

00:05:08.450 --> 00:05:12.820
If we take 3 times a, that's
the equivalent of

00:05:12.820 --> 00:05:14.030
scaling up a by 3.

00:05:14.030 --> 00:05:16.880
So you go 1a, 2a, 3a.

00:05:16.880 --> 00:05:21.410
So that's 3a, 3 times a
will look like that.

00:05:21.410 --> 00:05:26.750
So this vector is 3a, and then
we added to that 2b, right?

00:05:26.750 --> 00:05:30.400
Oh no, we subtracted 2b
from that, so minus

00:05:30.400 --> 00:05:31.680
b looks like this.

00:05:34.860 --> 00:05:39.120
Minus 2b looks like this.

00:05:39.120 --> 00:05:43.710
This is minus 2b, all the way,
in standard form, standard

00:05:43.710 --> 00:05:45.930
position, minus 2b.

00:05:45.930 --> 00:05:51.410
So if you add 3a to minus 2b,
we get to this vector.

00:05:51.410 --> 00:05:57.860
3a to minus 2b, you get this
vector right here, and that's

00:05:57.860 --> 00:06:00.060
exactly what we did when we
solved it mathematically.

00:06:00.060 --> 00:06:02.430
You get the vector 3, 0.

00:06:02.430 --> 00:06:06.500
You get this vector
right here, 3, 0.

00:06:06.500 --> 00:06:09.060
But this is just one
combination, one linear

00:06:09.060 --> 00:06:10.620
combination of a and b.

00:06:10.620 --> 00:06:14.560
Instead of multiplying a times
3, I could have multiplied a

00:06:14.560 --> 00:06:17.970
times 1 and 1/2 and just
gotten right here.

00:06:17.970 --> 00:06:24.840
So 1 and 1/2 a minus 2b would
still look the same.

00:06:24.840 --> 00:06:26.830
It would look like something
like this.

00:06:26.830 --> 00:06:30.550
It would look something like--
let me make sure I'm doing

00:06:30.550 --> 00:06:33.060
this-- it would look something
like this.

00:06:33.060 --> 00:06:36.160
And so our new vector that
we would find would be

00:06:36.160 --> 00:06:38.910
something like this.

00:06:38.910 --> 00:06:41.060
So I just showed you, I can find
this vector with a linear

00:06:41.060 --> 00:06:41.420
combination.

00:06:41.420 --> 00:06:43.430
I can find this vector with
a linear combination.

00:06:43.430 --> 00:06:46.190
And actually, it turns out that
you can represent any

00:06:46.190 --> 00:06:52.530
vector in R2 with some linear
combination of these vectors

00:06:52.530 --> 00:06:54.270
right here, a and b.

00:06:54.270 --> 00:07:00.020
Now, let's just think of an
example, or maybe just try a

00:07:00.020 --> 00:07:01.400
mental visual example.

00:07:01.400 --> 00:07:06.350
Wherever we want to go, we
could go arbitrarily-- we

00:07:06.350 --> 00:07:12.840
could scale a up by some
arbitrary value.

00:07:12.840 --> 00:07:15.650
So this is some weight on a,
and then we can add up

00:07:15.650 --> 00:07:17.060
arbitrary multiples of b.

00:07:17.060 --> 00:07:19.510
B goes straight up and down,
so we can add up arbitrary

00:07:19.510 --> 00:07:21.060
multiples of b to that.

00:07:21.060 --> 00:07:24.740
So we could get any point on
this line right there.

00:07:24.740 --> 00:07:29.070
Now, if we scaled a up a little
bit more, and then

00:07:29.070 --> 00:07:32.400
added any multiple b, we'd get
anything on that line.

00:07:32.400 --> 00:07:36.100
If we multiplied a times a
negative number and then added

00:07:36.100 --> 00:07:38.260
a b in either direction, we'll
get anything on that line.

00:07:38.260 --> 00:07:39.110
We can keep doing that.

00:07:39.110 --> 00:07:42.360
And there's no reason why we
can't pick an arbitrary a that

00:07:42.360 --> 00:07:43.800
can fill in any of these gaps.

00:07:43.800 --> 00:07:47.765
If we want a point here, we just
take a little smaller a,

00:07:47.765 --> 00:07:49.540
and then we can add all
the b's that fill

00:07:49.540 --> 00:07:50.390
up all of that line.

00:07:50.390 --> 00:07:54.060
So we can fill up any
point in R2 with the

00:07:54.060 --> 00:07:55.700
combinations of a and b.

00:07:55.700 --> 00:08:01.990
So what we can write here is
that the span-- let me write

00:08:01.990 --> 00:08:03.230
this word down.

00:08:03.230 --> 00:08:10.600
The span of the vectors a and
b-- so let me write that

00:08:10.600 --> 00:08:18.140
down-- it equals R2 or it equals
all the vectors in R2,

00:08:18.140 --> 00:08:20.120
which is, you know, it's
all the tuples.

00:08:20.120 --> 00:08:23.300
R2 is all the tuples
made of two ordered

00:08:23.300 --> 00:08:24.550
tuples of two real numbers.

00:08:24.550 --> 00:08:25.780
So it equals all of R2.

00:08:25.780 --> 00:08:28.450
This just means that I can
represent any vector in R2

00:08:28.450 --> 00:08:32.220
with some linear combination
of a and b.

00:08:32.220 --> 00:08:34.710
And you're like, hey, can't I do
that with any two vectors?

00:08:34.710 --> 00:08:40.799
Well, what if a and b were the
vector-- let's say the vector

00:08:40.799 --> 00:08:48.490
2, 2 was a, so a is equal to 2,
2, and let's say that b is

00:08:48.490 --> 00:08:54.590
the vector minus 2, minus
2, so b is that vector.

00:08:54.590 --> 00:08:59.320
So b is the vector
minus 2, minus 2.

00:08:59.320 --> 00:09:01.420
Now, can I represent any
vector with these?

00:09:01.420 --> 00:09:05.200
Well, I can scale a up and down,
so I can scale a up and

00:09:05.200 --> 00:09:09.670
down to get anywhere on this
line, and then I can add b

00:09:09.670 --> 00:09:11.940
anywhere to it, and
b is essentially

00:09:11.940 --> 00:09:13.120
going in the same direction.

00:09:13.120 --> 00:09:14.590
It's just in the opposite
direction, but I can multiply

00:09:14.590 --> 00:09:16.750
it by a negative and go
anywhere on the line.

00:09:16.750 --> 00:09:19.590
So any combination of a and b
will just end up on this line

00:09:19.590 --> 00:09:21.320
right here, if I draw
it in standard form.

00:09:21.320 --> 00:09:24.790
It'll be a vector with the same
slope as either a or b,

00:09:24.790 --> 00:09:27.300
or same inclination, whatever
you want to call it.

00:09:27.300 --> 00:09:31.050
I could never-- there's no
combination of a and b that I

00:09:31.050 --> 00:09:33.330
could represent this vector,
that I could

00:09:33.330 --> 00:09:34.550
represent vector c.

00:09:34.550 --> 00:09:35.440
I just can't do it.

00:09:35.440 --> 00:09:38.150
I can add in standard form.

00:09:38.150 --> 00:09:41.020
I could just keep adding scale
up a, scale up b, put them

00:09:41.020 --> 00:09:43.520
heads to tails, I'll just get
the stuff on this line.

00:09:43.520 --> 00:09:45.230
I'll never get to this.

00:09:45.230 --> 00:09:50.530
So in this case, the span--
and I want to be clear.

00:09:50.530 --> 00:09:53.640
This is for this particular a
and b, not for the a and b--

00:09:53.640 --> 00:09:58.590
for this blue a and this yellow
b, the span here is

00:09:58.590 --> 00:10:00.820
just this line.

00:10:00.820 --> 00:10:01.820
It's just this line.

00:10:01.820 --> 00:10:03.830
It's not all of R2.

00:10:03.830 --> 00:10:07.980
So this isn't just some kind of
statement when I first did

00:10:07.980 --> 00:10:08.720
it with that example.

00:10:08.720 --> 00:10:12.040
It's like, OK, can
any two vectors

00:10:12.040 --> 00:10:13.000
represent anything in R2?

00:10:13.000 --> 00:10:13.730
Well, no.

00:10:13.730 --> 00:10:17.220
I just showed you two vectors
that can't represent that.

00:10:17.220 --> 00:10:23.550
What is the span of
the 0 vector?

00:10:23.550 --> 00:10:26.900
I'll put a cap over it, the 0
vector, make it really bold.

00:10:26.900 --> 00:10:31.790
Well, the 0 vector is just 0,
0, so I don't care what

00:10:31.790 --> 00:10:33.760
multiple I put on it.

00:10:33.760 --> 00:10:36.490
The span of it is all of the
linear combinations of this,

00:10:36.490 --> 00:10:39.650
so essentially, I could put
arbitrary real numbers here,

00:10:39.650 --> 00:10:42.430
but I'm just going to end
up with a 0, 0 vector.

00:10:42.430 --> 00:10:46.700
So the span of the 0 vector
is just the 0 vector.

00:10:46.700 --> 00:10:49.830
The only vector I can get with
a linear combination of this,

00:10:49.830 --> 00:10:54.830
the 0 vector by itself, is
just the 0 vector itself.

00:10:54.830 --> 00:10:59.720
Likewise, if I take the span of
just, you know, let's say I

00:10:59.720 --> 00:11:01.620
go back to this example
right here.

00:11:01.620 --> 00:11:04.540
My a vector was right
like that.

00:11:04.540 --> 00:11:06.400
Let me draw it in
a better color.

00:11:06.400 --> 00:11:08.010
My a vector looked like that.

00:11:08.010 --> 00:11:13.350
If I were to ask just what the
span of a is, it's all the

00:11:13.350 --> 00:11:15.280
vectors you can get by
creating a linear

00:11:15.280 --> 00:11:16.690
combination of just a.

00:11:16.690 --> 00:11:18.220
So it's really just scaling.

00:11:18.220 --> 00:11:20.450
You can't even talk about
combinations, really.

00:11:20.450 --> 00:11:23.370
So it's just c times a,
all of those vectors.

00:11:23.370 --> 00:11:30.230
And we saw in the video where
I parametrized or showed a

00:11:30.230 --> 00:11:34.430
parametric representation of a
line, that this, the span of

00:11:34.430 --> 00:11:37.750
just this vector a, is the line
that's formed when you

00:11:37.750 --> 00:11:40.050
just scale a up and down.

00:11:40.050 --> 00:11:42.340
So span of a is just a line.

00:11:42.340 --> 00:11:45.620
You have to have two vectors,
and they can't be collinear,

00:11:45.620 --> 00:11:47.500
in order span all of R2.

00:11:47.500 --> 00:11:49.620
And I haven't proven that to you
yet, but we saw with this

00:11:49.620 --> 00:11:52.780
example, if you pick this a and
this b, you can represent

00:11:52.780 --> 00:11:56.080
all of R2 with just
these two vectors.

00:11:56.080 --> 00:11:58.640
Now, the two vectors that you're
most familiar with to

00:11:58.640 --> 00:12:02.550
that span R2 are, if you take
a little physics class, you

00:12:02.550 --> 00:12:04.320
have your i and j
unit vectors.

00:12:09.580 --> 00:12:12.990
And in our notation, i, the unit
vector i that you learned

00:12:12.990 --> 00:12:16.560
in physics class, would
be the vector 1, 0.

00:12:16.560 --> 00:12:21.710
So this is i, that's the vector
i, and then the vector

00:12:21.710 --> 00:12:25.320
j is the unit vector 0, 1.

00:12:25.320 --> 00:12:27.110
This is what you learned
in physics class.

00:12:27.110 --> 00:12:28.310
Let me do it in a
different color.

00:12:28.310 --> 00:12:29.560
This is j.

00:12:29.560 --> 00:12:31.420
j is that.

00:12:31.420 --> 00:12:33.270
And you learned that they're
orthogonal, and we're going to

00:12:33.270 --> 00:12:35.840
talk a lot more about what
orthogonality means, but in

00:12:35.840 --> 00:12:38.690
our traditional sense that we
learned in high school, it

00:12:38.690 --> 00:12:39.840
means that they're 90 degrees.

00:12:39.840 --> 00:12:45.060
But you can clearly represent
any angle, or any vector, in

00:12:45.060 --> 00:12:47.660
R2, by these two vectors.

00:12:47.660 --> 00:12:49.610
And the fact that they're
orthogonal makes them extra

00:12:49.610 --> 00:12:51.816
nice, and that's why these
form-- and I'm going to throw

00:12:51.816 --> 00:12:53.500
out a word here that I
haven't defined yet.

00:12:53.500 --> 00:12:54.970
These form the basis.

00:12:54.970 --> 00:12:57.290
These form a basis for R2.

00:12:57.290 --> 00:13:00.070
In fact, you can represent
anything in R2 by these two

00:13:00.070 --> 00:13:00.600
vectors. line.

00:13:00.600 --> 00:13:02.130
I'm not going to even define
what basis is.

00:13:02.130 --> 00:13:03.670
That's going to be
a future video.

00:13:03.670 --> 00:13:06.940
But let me just write the formal
math-y definition of

00:13:06.940 --> 00:13:09.000
span, just so you're
satisfied.

00:13:09.000 --> 00:13:17.430
So if I were to write the span
of a set of vectors, v1, v2,

00:13:17.430 --> 00:13:24.020
all the way to vn, that just
means the set of all of the

00:13:24.020 --> 00:13:31.340
vectors, where I have c1 times
v1 plus c2 times v2 all the

00:13:31.340 --> 00:13:37.780
way to cn-- let me scroll over--
all the way to cn vn.

00:13:37.780 --> 00:13:42.670
So this is a set of vectors
because I can pick my ci's to

00:13:42.670 --> 00:13:47.640
be any member of the real
numbers, and that's true for

00:13:47.640 --> 00:13:53.830
i-- so I should write for i to
be anywhere between 1 and n.

00:13:53.830 --> 00:13:58.310
All I'm saying is that look, I
can multiply each of these

00:13:58.310 --> 00:14:02.350
vectors by any value, any
arbitrary value, real value,

00:14:02.350 --> 00:14:03.430
and then I can add them up.

00:14:03.430 --> 00:14:06.340
And now the set of all of the
combinations, scaled-up

00:14:06.340 --> 00:14:12.280
combinations I can get, that's
the span of these vectors.

00:14:12.280 --> 00:14:15.850
You can kind of view it as the
space of all of the vectors

00:14:15.850 --> 00:14:21.140
that can be represented by a
combination of these vectors

00:14:21.140 --> 00:14:22.040
right there.

00:14:22.040 --> 00:14:23.930
And so the word span,
I think it does have

00:14:23.930 --> 00:14:24.800
an intuitive sense.

00:14:24.800 --> 00:14:29.720
I mean, if I say that, you know,
in my first example, I

00:14:29.720 --> 00:14:34.770
showed you those two vectors
span, or a and b spans R2.

00:14:34.770 --> 00:14:36.720
I wrote it right here.

00:14:36.720 --> 00:14:41.900
That tells me that any vector in
R2 can be represented by a

00:14:41.900 --> 00:14:44.930
linear combination of a and b.

00:14:44.930 --> 00:14:47.980
And actually, just in case
that visual kind of

00:14:47.980 --> 00:14:50.960
pseudo-proof doesn't do you
justice, let me prove it to

00:14:50.960 --> 00:14:52.300
you algebraically.

00:14:52.300 --> 00:14:55.910
I'm telling you that I can
take-- let's say I want to

00:14:55.910 --> 00:15:00.670
represent, you know, I have
some-- let me rewrite my a's

00:15:00.670 --> 00:15:02.740
and b's again.

00:15:02.740 --> 00:15:04.060
So this was my vector a.

00:15:04.060 --> 00:15:06.260
It was 1, 2, and b was 0, 3.

00:15:06.260 --> 00:15:08.250
Let me remember that.

00:15:08.250 --> 00:15:17.540
So my vector a is 1, 2, and
my vector b was 0, 3.

00:15:17.540 --> 00:15:20.410
Now my claim was that I can
represent any point.

00:15:20.410 --> 00:15:25.270
Let's say I want to represent
some arbitrary point x in R2,

00:15:25.270 --> 00:15:28.640
so its coordinates
are x1 and x2.

00:15:28.640 --> 00:15:31.210
I need to be able to prove to
you that I can get to any x1

00:15:31.210 --> 00:15:34.760
and any x2 with some combination
of these guys.

00:15:34.760 --> 00:15:41.240
So let's say that my
combination, I say c1 times a

00:15:41.240 --> 00:15:48.460
plus c2 times b has to be
equal to my vector x.

00:15:48.460 --> 00:15:52.530
Let me show you that I can
always find a c1 or c2 given

00:15:52.530 --> 00:15:54.405
that you give me some x's.

00:15:57.190 --> 00:16:01.870
So let's just write this right
here with the actual vectors

00:16:01.870 --> 00:16:04.010
being represented in their
kind of column form.

00:16:04.010 --> 00:16:11.420
So we have c1 times this vector
plus c2 times the b

00:16:11.420 --> 00:16:18.020
vector 0, 3 should be able to
be equal to my x vector,

00:16:18.020 --> 00:16:21.630
should be able to be equal to my
x1 and x2, where these are

00:16:21.630 --> 00:16:22.940
just arbitrary.

00:16:22.940 --> 00:16:24.600
So let's see if I can
set that to be true.

00:16:24.600 --> 00:16:29.720
So if this is true, then the
following must be true. c1

00:16:29.720 --> 00:16:36.410
times 1 plus 0 times c2
must be equal to x1.

00:16:36.410 --> 00:16:39.250
We just get that from our
definition of multiplying

00:16:39.250 --> 00:16:41.820
vectors times scalars
and adding vectors.

00:16:41.820 --> 00:16:48.550
And then we also know that
2 times c2-- sorry.

00:16:48.550 --> 00:16:58.090
c1 times 2 plus c2 times 3, 3c2,
should be equal to x2.

00:16:58.090 --> 00:17:01.760
Now, if I can show you that I
can always find c1's and c2's

00:17:01.760 --> 00:17:07.130
given any x1's and x2's, then
I've proven that I can get to

00:17:07.130 --> 00:17:10.450
any point in R2 using just
these two vectors.

00:17:10.450 --> 00:17:12.839
So let me see if
I can do that.

00:17:12.839 --> 00:17:17.440
So this is just a system
of two unknowns.

00:17:17.440 --> 00:17:18.349
This is just 0.

00:17:18.349 --> 00:17:19.599
We can ignore it.

00:17:19.599 --> 00:17:23.780
So let's multiply this equation
up here by minus 2

00:17:23.780 --> 00:17:24.550
and put it here.

00:17:24.550 --> 00:17:28.670
So we get minus 2, c1--
I'm just multiplying

00:17:28.670 --> 00:17:30.040
this times minus 2.

00:17:30.040 --> 00:17:35.070
We get a 0 here, plus 0
is equal to minus 2x1.

00:17:35.070 --> 00:17:37.590
And then you add these two.

00:17:37.590 --> 00:17:40.500
You get 3c2, right?

00:17:40.500 --> 00:17:41.400
These cancel out.

00:17:41.400 --> 00:17:44.950
You get 3-- let me write it
in a different color.

00:17:44.950 --> 00:17:51.860
You get 3c2 is equal
to x2 minus 2x1.

00:17:51.860 --> 00:17:55.420
Or divide both sides by 3,
you get c2 is equal to

00:17:55.420 --> 00:18:00.180
1/3 x2 minus x1.

00:18:00.180 --> 00:18:04.780
Now we'd have to go substitute
back in for c1.

00:18:04.780 --> 00:18:08.890
But we have this first equation
right here, that c1,

00:18:08.890 --> 00:18:11.670
this first equation that says
c1 plus 0 is equal to x1, so

00:18:11.670 --> 00:18:14.530
c1 is equal to x1.

00:18:14.530 --> 00:18:17.380
So that one just
gets us there.

00:18:17.380 --> 00:18:20.260
So c1 is equal to x1.

00:18:20.260 --> 00:18:23.750
So you give me any point in R2--
these are just two real

00:18:23.750 --> 00:18:26.730
numbers-- and I can just perform
this operation, and

00:18:26.730 --> 00:18:31.100
I'll tell you what weights to
apply to a and b to get to

00:18:31.100 --> 00:18:31.890
that point.

00:18:31.890 --> 00:18:35.190
If you say, OK, what combination
of a and b can get

00:18:35.190 --> 00:18:38.520
me to the point-- let's say I
want to get to the point-- let

00:18:38.520 --> 00:18:39.930
me go back up here.

00:18:39.930 --> 00:18:41.430
Oh, it's way up there.

00:18:41.430 --> 00:18:46.130
Let's say I'm looking to
get to the point 2, 2.

00:18:46.130 --> 00:18:47.860
So x1 is 2.

00:18:47.860 --> 00:18:49.400
Let me write it down here.

00:18:49.400 --> 00:18:53.510
Say I'm trying to get to the
point the vector 2, 2.

00:18:53.510 --> 00:18:55.830
What combinations of a
and b can be there?

00:18:55.830 --> 00:18:59.930
Well, I know that c1 is equal
to x1, so that's equal to 2,

00:18:59.930 --> 00:19:07.760
and c2 is equal to 1/3
times 2 minus 2.

00:19:07.760 --> 00:19:13.000
So 2 minus 2 is 0, so
c2 is equal to 0.

00:19:13.000 --> 00:19:17.230
So if I want to just get to
the point 2, 2, I just

00:19:17.230 --> 00:19:20.120
multiply-- oh, I
just realized.

00:19:20.120 --> 00:19:21.350
This was looking suspicious.

00:19:21.350 --> 00:19:23.430
I made a slight error here,
and this was good that I

00:19:23.430 --> 00:19:25.550
actually tried it out
with real numbers.

00:19:25.550 --> 00:19:29.740
Over here, when I had 3c2 is
equal to x2 minus 2x1, I got

00:19:29.740 --> 00:19:30.920
rid of this 2 over here.

00:19:30.920 --> 00:19:31.970
There's a 2 over here.

00:19:31.970 --> 00:19:33.150
I divide both sides by 3.

00:19:33.150 --> 00:19:36.200
I get 1/3 times x2 minus 2x1.

00:19:36.200 --> 00:19:39.260
And that's why I was like, wait,
this is looking strange.

00:19:39.260 --> 00:19:40.540
So I had to take a
moment of pause.

00:19:40.540 --> 00:19:44.720
So let's go to my corrected
definition of c2.

00:19:44.720 --> 00:19:49.630
C2 is equal to 1/3 times x2.

00:19:49.630 --> 00:19:55.830
So 2 minus 2 times x1,
so minus 2 times 2.

00:19:55.830 --> 00:20:01.880
So it's equal to 1/3 times 2
minus 4, which is equal to

00:20:01.880 --> 00:20:04.670
minus 2, so it's equal
to minus 2/3.

00:20:04.670 --> 00:20:14.410
So if I multiply 2 times my
vector a minus 2/3 times my

00:20:14.410 --> 00:20:18.600
vector b, I will get
to the vector 2, 2.

00:20:18.600 --> 00:20:20.480
And you can verify
it for yourself.

00:20:20.480 --> 00:20:29.470
2 times my vector a 1, 2, minus
2/3 times my vector b 0,

00:20:29.470 --> 00:20:34.000
3, should equal 2, 2.


Parametric representations of lines

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.520
Everything we've been doing in
linear algebra so far, you

00:00:03.520 --> 00:00:06.930
might be thinking, it's kind of
a more painful way of doing

00:00:06.930 --> 00:00:08.470
things that you already
knew how to do.

00:00:08.470 --> 00:00:10.850
You've already dealt
with vectors.

00:00:10.850 --> 00:00:12.970
I'm guessing that some of you
all have already dealt with

00:00:12.970 --> 00:00:15.220
vectors in your calculus or
your pre-calculus or your

00:00:15.220 --> 00:00:16.260
physics classes.

00:00:16.260 --> 00:00:18.880
But in this video I hope to show
you something that you're

00:00:18.880 --> 00:00:21.340
going to do in linear algebra
that you've never done before,

00:00:21.340 --> 00:00:24.170
and that it would have been very
hard to do had you not

00:00:24.170 --> 00:00:25.840
been exposed to these videos.

00:00:25.840 --> 00:00:28.810
Well I'm going to start with,
once again, a different way of

00:00:28.810 --> 00:00:30.970
doing something you already
know how to do.

00:00:30.970 --> 00:00:35.190
So let me just define some
vector here, instead of making

00:00:35.190 --> 00:00:37.550
them bold, I'll just draw it
with the arrow on top.

00:00:37.550 --> 00:00:39.620
I'm going to define my vector
to be-- I can do with the

00:00:39.620 --> 00:00:41.920
arrow on top or I can just
make it super bold.

00:00:41.920 --> 00:00:43.660
I'm just going to define my
vectors, it's going to be a

00:00:43.660 --> 00:00:46.280
vector in R2.

00:00:46.280 --> 00:00:52.230
Let's just say that my vector
is the vector 2, 1.

00:00:52.230 --> 00:00:54.200
If I were to draw it in
standard position,

00:00:54.200 --> 00:00:55.100
it looks like this.

00:00:55.100 --> 00:00:58.665
You go two to the right,
up one, like that.

00:00:58.665 --> 00:01:04.400
That's my, right there,
that is my vector v.

00:01:04.400 --> 00:01:07.850
Now, if I were to ask you, what
are all of the possible

00:01:07.850 --> 00:01:08.970
vectors I can create?

00:01:08.970 --> 00:01:10.300
So let me define a set.

00:01:10.300 --> 00:01:15.960
Let me define a set, s, and
it's equal to-- all of the

00:01:15.960 --> 00:01:19.460
vectors I can create, if I were
to multiply v times some

00:01:19.460 --> 00:01:25.280
constant, so I multiply some
constant, some scalar, times

00:01:25.280 --> 00:01:28.880
my vector v, and just to maybe
be a little bit formal, I'll

00:01:28.880 --> 00:01:36.995
say such that c is a member
of the real numbers.

00:01:40.510 --> 00:01:44.800
Now what would be a graphical
representation of this set?

00:01:44.800 --> 00:01:47.370
Well, if we draw them all in
standard position, c could be

00:01:47.370 --> 00:01:48.280
any real number.

00:01:48.280 --> 00:01:51.270
So if I were to multiply,
c could be 2.

00:01:51.270 --> 00:01:55.390
If c is 2, let me
do it this way.

00:01:55.390 --> 00:01:58.460
If I do 2 times our vector,
I'm going to get

00:01:58.460 --> 00:02:01.080
the vector 4, 2.

00:02:01.080 --> 00:02:03.550
Let me draw that in standard
position, 4, 2.

00:02:03.550 --> 00:02:04.390
It's right there.

00:02:04.390 --> 00:02:08.240
It's this vector right there.

00:02:08.240 --> 00:02:10.350
It's collinear with
this first vector.

00:02:10.350 --> 00:02:13.640
It's along the same line, but
just goes out further 2.

00:02:13.640 --> 00:02:14.540
Now I could've done another.

00:02:14.540 --> 00:02:17.660
I could have done 1.5
times our vector v.

00:02:17.660 --> 00:02:19.660
Let me do that in a
different color.

00:02:19.660 --> 00:02:21.890
And maybe that would be,
that would be what?

00:02:21.890 --> 00:02:26.240
That'd be 1.5 times 2,
which is 3, 1.5.

00:02:26.240 --> 00:02:27.630
Where would that
vector get me?

00:02:27.630 --> 00:02:32.160
I'd go one 1.5 and then I'd
go 3, and then 1.5,

00:02:32.160 --> 00:02:34.380
I'd get right there.

00:02:34.380 --> 00:02:35.680
And I can multiply
by anything.

00:02:35.680 --> 00:02:39.390
I can multiply 1.4999
times vector v, and

00:02:39.390 --> 00:02:40.750
get right over here.

00:02:40.750 --> 00:02:44.220
I could do minus 0.0001
times vector v.

00:02:44.220 --> 00:02:45.210
Let me write that down.

00:02:45.210 --> 00:02:51.550
I could do 0.001 times
our vector v.

00:02:51.550 --> 00:02:52.810
And where were that put me?

00:02:52.810 --> 00:02:55.680
It would put me little super
small vector right there.

00:02:55.680 --> 00:02:59.430
If I did minus 0.01, it would
make a super small vector

00:02:59.430 --> 00:03:00.810
right there pointing
in that direction.

00:03:00.810 --> 00:03:03.126
If I were to do minus 10, I
would get a vector going in

00:03:03.126 --> 00:03:07.000
this direction that goes
way like that.

00:03:07.000 --> 00:03:09.760
But you can imagine that if
I were to plot all of the

00:03:09.760 --> 00:03:13.800
vectors in standard position,
all of them that could be

00:03:13.800 --> 00:03:16.250
represented by any c in real
numbers, I'll essentially

00:03:16.250 --> 00:03:19.550
get-- I'll end up drawing a
bunch of vectors where their

00:03:19.550 --> 00:03:23.880
arrows are all lined up along
this line right there, and all

00:03:23.880 --> 00:03:26.830
lined up in even in negative
direction-- let me make sure I

00:03:26.830 --> 00:03:31.440
draw it properly-- along
that line, like that.

00:03:31.440 --> 00:03:32.670
I think you get the idea.

00:03:32.670 --> 00:03:35.256
So it's a set of collinear
vectors.

00:03:35.256 --> 00:03:36.560
So let me write that down.

00:03:44.220 --> 00:03:49.850
And if we view these vectors as
position vectors, that this

00:03:49.850 --> 00:03:56.710
vector represents a point in
space in R2-- this R2 is just

00:03:56.710 --> 00:03:59.540
our Cartesian coordinate plane
right here in every

00:03:59.540 --> 00:04:03.840
direction-- if we view this
vector as a position vector--

00:04:03.840 --> 00:04:07.800
let me write that down-- if
we view it as kind of a

00:04:07.800 --> 00:04:11.495
coordinate in R2, then this set,
if we visually represent

00:04:11.495 --> 00:04:14.280
it as a bunch of position
vectors, it'll be represented

00:04:14.280 --> 00:04:15.900
by this whole line over here.

00:04:18.899 --> 00:04:22.750
And I want to make that point
clear because it's essentially

00:04:22.750 --> 00:04:24.600
a line, of slope 2.

00:04:24.600 --> 00:04:26.150
Right?

00:04:26.150 --> 00:04:27.410
Sorry, slope 1/2.

00:04:27.410 --> 00:04:29.080
Your rise is 1.

00:04:29.080 --> 00:04:31.620
Your rise is 1 for
going over 2.

00:04:31.620 --> 00:04:34.060
But I don't want to go
back to our Algebra 1

00:04:34.060 --> 00:04:34.840
notation too much.

00:04:34.840 --> 00:04:39.730
But I want to make this point
that this line of slope 2 that

00:04:39.730 --> 00:04:42.780
goes through the origin, this
is if we draw all of the

00:04:42.780 --> 00:04:46.110
vectors in the set as in their
standard form, or if we draw

00:04:46.110 --> 00:04:47.710
them all as position vectors.

00:04:47.710 --> 00:04:50.610
If I didn't make that
clarification, or that

00:04:50.610 --> 00:04:53.090
qualification, I could have
drawn these vectors anywhere.

00:04:53.090 --> 00:04:53.370
Right?

00:04:53.370 --> 00:05:00.190
Because this 4, 2 vector, I
could have drawn over here.

00:05:00.190 --> 00:05:02.720
And then, to say that it's
collinear probably wouldn't

00:05:02.720 --> 00:05:04.510
have made as much visual
sense to you.

00:05:04.510 --> 00:05:08.330
But I think this collinearity of
it makes more sense to you

00:05:08.330 --> 00:05:11.130
if you say, oh, let's draw them
all in standard form.

00:05:11.130 --> 00:05:15.030
All of them start at the origin,
and then their tails

00:05:15.030 --> 00:05:17.270
are at the origin, and their
heads go essentially to the

00:05:17.270 --> 00:05:18.370
coordinate they represent.

00:05:18.370 --> 00:05:20.390
That's what I mean by their
position vectors.

00:05:20.390 --> 00:05:23.200
They don't necessarily have to
be position vectors, but for

00:05:23.200 --> 00:05:27.830
the visualization in this video,
let's stick to that.

00:05:27.830 --> 00:05:30.530
Now I was only able to represent
something that goes

00:05:30.530 --> 00:05:33.020
through the origin
with this slope.

00:05:33.020 --> 00:05:35.580
So you can almost view that
this vector kind of

00:05:35.580 --> 00:05:38.570
represented its slope.

00:05:38.570 --> 00:05:40.820
You almost want to view it as a
slope vector, if you wanted

00:05:40.820 --> 00:05:43.280
to tie it in to what you
learned in Algebra 1.

00:05:43.280 --> 00:05:45.480
What if we wanted to represent
other lines

00:05:45.480 --> 00:05:46.190
that had that slope?

00:05:46.190 --> 00:05:53.470
What if we wanted to represent
the the same line, or I guess

00:05:53.470 --> 00:05:56.350
a parallel line-- that goes
through that point over there,

00:05:56.350 --> 00:06:01.120
the point 2 comma 4?

00:06:01.120 --> 00:06:03.250
Or if we're thinking in position
vectors, we could say

00:06:03.250 --> 00:06:19.105
that point is represented
by the vector, and we

00:06:19.105 --> 00:06:20.900
will call that x.

00:06:20.900 --> 00:06:23.130
It's represented by
the vector x.

00:06:23.130 --> 00:06:27.330
And the vector x is
equal to 2, 4.

00:06:27.330 --> 00:06:28.040
That point right there.

00:06:28.040 --> 00:06:30.600
What if I want to represent the
line that's parallel to

00:06:30.600 --> 00:06:34.030
this that goes through
that point 2, 4?

00:06:34.030 --> 00:06:36.040
So I want to represent
this line right here.

00:06:39.465 --> 00:06:43.470
I'll draw it as parallel
to this as I can.

00:06:43.470 --> 00:06:47.080
I think you get the idea, and it
just keeps going like that

00:06:47.080 --> 00:06:48.130
in every direction.

00:06:48.130 --> 00:06:50.120
These two lines are parallel.

00:06:50.120 --> 00:06:55.280
How can I represent the set of
all of these vectors, drawn in

00:06:55.280 --> 00:06:57.620
standard form, or all of the
vectors, that if I were to

00:06:57.620 --> 00:07:01.400
draw them in standard form,
would show this line?

00:07:01.400 --> 00:07:03.490
Well, you can think
about it this way.

00:07:03.490 --> 00:07:07.650
If every one of the vectors that
represented this line, if

00:07:07.650 --> 00:07:11.480
I start with any vector that was
on this line, and I add my

00:07:11.480 --> 00:07:20.070
x vector to it, I'll show up
at a corresponding point on

00:07:20.070 --> 00:07:22.250
this line that I
want to be at.

00:07:22.250 --> 00:07:23.500
Right?

00:07:29.130 --> 00:07:33.960
Let's say I do negative 2 times
my original, so minus 2

00:07:33.960 --> 00:07:38.160
times my vector v, that
equaled what?

00:07:38.160 --> 00:07:41.870
Minus 4, minus 2, so that's
that vector there.

00:07:41.870 --> 00:07:47.160
But if I were to add x to it, if
I were to add my x vector.

00:07:47.160 --> 00:07:51.040
So if I were to do minus 2 times
my vector v, but I were

00:07:51.040 --> 00:07:54.670
to add x to it, so plus x.

00:07:54.670 --> 00:07:58.360
I'm adding this vector 2 comma
4 to it, so from here I'd go

00:07:58.360 --> 00:08:00.860
right 2 and up 4,
so I'd go here.

00:08:00.860 --> 00:08:03.130
Or visually you could just
say, heads to tails, so I

00:08:03.130 --> 00:08:04.550
would go right there.

00:08:04.550 --> 00:08:05.610
So I would end up at a

00:08:05.610 --> 00:08:07.990
corresponding point over there.

00:08:13.020 --> 00:08:16.020
So when I define my set, s, as
the set of all points where I

00:08:16.020 --> 00:08:18.220
just multiply v times the
scalar, I got this thing that

00:08:18.220 --> 00:08:19.650
went through the origin.

00:08:19.650 --> 00:08:22.120
But now let me define
another set.

00:08:22.120 --> 00:08:28.580
Let me define a set l, maybe l
for line, that's equal to the

00:08:28.580 --> 00:08:34.669
set of all of vectors where the
vector x, I could do it

00:08:34.669 --> 00:08:39.510
bold or I'll just draw an
arrow on it, plus some

00:08:39.510 --> 00:08:42.230
scalar-- I could use c, but
let me use t, because I'm

00:08:42.230 --> 00:08:46.540
going to call this a
parametrization of the line--

00:08:46.540 --> 00:08:59.070
so plus some scalar, t times my
vector v such that t could

00:08:59.070 --> 00:09:03.380
be any member of the
real numbers.

00:09:03.380 --> 00:09:04.310
So what is this going to be?

00:09:04.310 --> 00:09:06.120
This is going to be
this blue line.

00:09:06.120 --> 00:09:08.820
If I were to draw all of these
vectors in standard position,

00:09:08.820 --> 00:09:10.460
I'm going to get my blue line.

00:09:10.460 --> 00:09:14.670
For example, if I do minus 2,
this is minus 2, times my

00:09:14.670 --> 00:09:16.160
vector v, I get here.

00:09:16.160 --> 00:09:18.730
Then if I add x, I go there.

00:09:18.730 --> 00:09:25.770
So this vector right here that
has its endpoint right there--

00:09:25.770 --> 00:09:27.680
its endpoint sits
on that line.

00:09:27.680 --> 00:09:29.330
I can do that with anything.

00:09:29.330 --> 00:09:33.980
If I take this vector, this is
some scalar times my vector v,

00:09:33.980 --> 00:09:38.990
and I add x to it, I end up
with this vector, whose

00:09:38.990 --> 00:09:41.700
endpoint, if I view it as a
position vector, it's endpoint

00:09:41.700 --> 00:09:43.950
dictates some coordinate
in the xy plane.

00:09:43.950 --> 00:09:44.740
So it will [UNINTELLIGIBLE]

00:09:44.740 --> 00:09:45.560
that point.

00:09:45.560 --> 00:09:48.330
So I can get to any
of these vectors.

00:09:48.330 --> 00:09:51.720
This is a set of vectors right
here, and all of these vectors

00:09:51.720 --> 00:09:54.430
are going to point-- they're
essentially going to point to

00:09:54.430 --> 00:09:57.446
something-- when I draw them
in standard form, if I draw

00:09:57.446 --> 00:09:59.650
them in standard form-- they're
going to point to a

00:09:59.650 --> 00:10:02.110
point on that blue line.

00:10:02.110 --> 00:10:05.630
Now you might say, hey Sal, this
was a really obtuse way

00:10:05.630 --> 00:10:07.210
of defining a line.

00:10:07.210 --> 00:10:09.320
I mean we do it in Algebra 1,
where we just say, hey you

00:10:09.320 --> 00:10:12.540
know, y is equal to mx plus b.

00:10:12.540 --> 00:10:15.200
And we figure out the slope by
figuring out the difference of

00:10:15.200 --> 00:10:17.360
two points, and then we do
a little substitution.

00:10:17.360 --> 00:10:19.670
And this is stuff you learned
in seventh or eighth grade.

00:10:19.670 --> 00:10:21.270
This was really straightforward.

00:10:21.270 --> 00:10:27.170
Why am I defining this obtuse
set here and making you think

00:10:27.170 --> 00:10:30.170
in terms of sets and vectors
and adding vectors?

00:10:30.170 --> 00:10:32.495
And the reason is, is because
this is very general.

00:10:35.570 --> 00:10:37.440
This worked well in R2.

00:10:37.440 --> 00:10:40.380
So in R2, this was great.

00:10:40.380 --> 00:10:42.860
I mean, we just have to worry
about x's and y's.

00:10:42.860 --> 00:10:45.620
But what about the situation, I
mean notice, in your algebra

00:10:45.620 --> 00:10:49.470
class, your teacher never really
told you much, at least

00:10:49.470 --> 00:10:52.430
in the ones I took, about how do
you present lines in three

00:10:52.430 --> 00:10:54.050
dimensions?

00:10:54.050 --> 00:10:56.310
Maybe some classes go there,
but they definitely didn't

00:10:56.310 --> 00:10:58.635
tell you how do you represent
lines in four dimensions, or a

00:10:58.635 --> 00:10:59.720
hundred dimensions.

00:10:59.720 --> 00:11:03.690
And that's what this is
going to do for us.

00:11:03.690 --> 00:11:08.750
Right here, I defined x and
v as vectors in R2.

00:11:08.750 --> 00:11:11.320
They're two-dimensional vectors,
but we can extend it

00:11:11.320 --> 00:11:14.950
to an arbitrary number
of dimensions.

00:11:14.950 --> 00:11:17.610
So just to kind of hit the point
home, let's do one more

00:11:17.610 --> 00:11:21.670
example in R2, where, it's kind
of the classic algebra

00:11:21.670 --> 00:11:24.650
problem where you need to find
the equation for the line.

00:11:24.650 --> 00:11:26.170
But here, we're going
to call it the set

00:11:26.170 --> 00:11:28.090
definition for the line.

00:11:28.090 --> 00:11:30.060
Let's say we have two vectors.

00:11:30.060 --> 00:11:39.410
Let's say we have the vector a,
which I'll define as-- let

00:11:39.410 --> 00:11:42.630
me just says it's 2, 1.

00:11:42.630 --> 00:11:47.550
So if I were draw it in standard
form, it's 2, 1.

00:11:47.550 --> 00:11:50.620
That's my vector
a, right there.

00:11:50.620 --> 00:11:56.530
And let's say I have vector
b, let me define vector b.

00:11:56.530 --> 00:11:59.890
I'm going to define it as,
I don't know, let me

00:11:59.890 --> 00:12:05.110
define it as 0, 3.

00:12:05.110 --> 00:12:07.563
So my vector b, 0-- I don't
move to the right at

00:12:07.563 --> 00:12:08.300
all and I go up.

00:12:08.300 --> 00:12:12.575
So my vector b will
look like that.

00:12:12.575 --> 00:12:14.760
Now I'm going to say that these
are position vectors,

00:12:14.760 --> 00:12:16.930
that we draw them in
standard form.

00:12:16.930 --> 00:12:19.670
When you draw them in standard
form, their endpoints

00:12:19.670 --> 00:12:21.030
represent some position.

00:12:21.030 --> 00:12:24.150
So you can almost view these
as coordinate points in R2.

00:12:24.150 --> 00:12:25.880
This is R2.

00:12:25.880 --> 00:12:28.950
All of these coordinate axes
I draw are going be R2.

00:12:28.950 --> 00:12:33.270
Now what if I asked you, give
me a parametrization of the

00:12:33.270 --> 00:12:35.710
line that goes through
these two points.

00:12:35.710 --> 00:12:38.380
So essentially, I want the
equation-- if you're thinking

00:12:38.380 --> 00:12:42.180
in Algebra 1 terms-- I want the
equation for the line that

00:12:42.180 --> 00:12:44.565
goes through these two points.

00:12:48.660 --> 00:12:50.910
So the classic way, you would
have figured out the slope and

00:12:50.910 --> 00:12:51.830
all of that, and then
you would have

00:12:51.830 --> 00:12:53.360
substituted back in.

00:12:53.360 --> 00:12:57.300
But instead, what we can do is,
we can say, hey look, this

00:12:57.300 --> 00:13:01.760
line that goes through both of
those points-- you could

00:13:01.760 --> 00:13:05.080
almost say that both of those
vectors lie on-- I guess

00:13:05.080 --> 00:13:06.320
that's a better-- Both of these

00:13:06.320 --> 00:13:09.220
vectors lie on this line.

00:13:09.220 --> 00:13:13.430
Now, what vector can be
represented by that line?

00:13:13.430 --> 00:13:18.630
Or even better, what vector,
if I take any arbitrary

00:13:18.630 --> 00:13:24.040
scalar-- can represent any other
vector on that line?

00:13:24.040 --> 00:13:25.950
Now let me do it this way.

00:13:25.950 --> 00:13:29.280
What if I were to take-- so this
is vector b here-- what

00:13:29.280 --> 00:13:31.520
if I were to take b minus a?

00:13:31.520 --> 00:13:33.880
We learned in, I think it was
the previous video, that b

00:13:33.880 --> 00:13:36.800
minus a, you'll get this
vector right here.

00:13:36.800 --> 00:13:38.620
You'll get the difference
in the two vectors.

00:13:38.620 --> 00:13:43.340
This is the vector b
minus the vector a.

00:13:43.340 --> 00:13:44.400
And you just think about it.

00:13:44.400 --> 00:13:46.500
What do I have to add
to a to get to b?

00:13:46.500 --> 00:13:48.860
I have to add b minus a.

00:13:48.860 --> 00:13:52.460
So if I can get the vector b
minus a-- right, we know how

00:13:52.460 --> 00:13:52.940
to do that.

00:13:52.940 --> 00:13:56.210
We just subtract the vectors,
and then multiply it by any

00:13:56.210 --> 00:14:01.380
scalar, then we're going to get
any point along that line.

00:14:01.380 --> 00:14:02.490
We have to be careful.

00:14:02.490 --> 00:14:07.220
So what happens if we take t,
so some scalar, times our

00:14:07.220 --> 00:14:10.570
vector, times the vectors
b minus a?

00:14:14.430 --> 00:14:15.740
What will we get then?

00:14:15.740 --> 00:14:17.120
So b minus a looks like that.

00:14:17.120 --> 00:14:19.610
But if we were to draw it in
standard form-- remember, in

00:14:19.610 --> 00:14:22.240
standard form b minus a would
look something like this.

00:14:26.330 --> 00:14:26.500
Right?

00:14:26.500 --> 00:14:28.340
It would start at 0, it would be
parallel to this, and then

00:14:28.340 --> 00:14:30.060
from 0 we would draw
its endpoint.

00:14:30.060 --> 00:14:33.830
So if we just multiplied some
scalar times b minus a, we

00:14:33.830 --> 00:14:38.600
would actually just get
points or vectors

00:14:38.600 --> 00:14:40.320
that lie on this line.

00:14:40.320 --> 00:14:43.520
Vectors that lie on that
line right there.

00:14:43.520 --> 00:14:45.250
Now, that's not what
we set out to do.

00:14:45.250 --> 00:14:49.300
We wanted to figure out an
equation, or parametrization,

00:14:49.300 --> 00:14:52.360
if you will, of this
line, or this set.

00:14:52.360 --> 00:14:53.810
Let's call this set l.

00:14:53.810 --> 00:14:56.780
So we want to know what
that set is equal to.

00:14:56.780 --> 00:15:03.020
So in order to get there, we
have to start with this, which

00:15:03.020 --> 00:15:05.630
is this line here, and
we have to shift it.

00:15:05.630 --> 00:15:08.370
And we could shift it either by
shifting it straight up, we

00:15:08.370 --> 00:15:10.960
could add vector b to it.

00:15:10.960 --> 00:15:13.550
So we could take this
line right here, and

00:15:13.550 --> 00:15:15.150
add vector b to it.

00:15:15.150 --> 00:15:17.520
And so any point on here
would have its

00:15:17.520 --> 00:15:18.640
corresponding point there.

00:15:18.640 --> 00:15:20.950
So when you add vector b, it
essentially shifts it up.

00:15:20.950 --> 00:15:21.660
That would work.

00:15:21.660 --> 00:15:26.720
So we could, say, we could
add vector b to it.

00:15:26.720 --> 00:15:31.100
And now all of these points
for any arbitrary-- t is a

00:15:31.100 --> 00:15:35.050
member of the real numbers, will
lie on this green line.

00:15:35.050 --> 00:15:37.430
Or the other option we could
have done is we could have

00:15:37.430 --> 00:15:38.250
added vector a.

00:15:38.250 --> 00:15:41.230
Vector a would have taken any
arbitrary point here and

00:15:41.230 --> 00:15:43.380
shifted it that way.

00:15:43.380 --> 00:15:43.600
Right?

00:15:43.600 --> 00:15:45.160
You would be adding
vector a to it.

00:15:45.160 --> 00:15:47.360
But either way, you're going to
get to the green line that

00:15:47.360 --> 00:15:50.100
we cared about, so you could
have also defined it as the

00:15:50.100 --> 00:15:55.390
set of vector a plus this line,
essentially, t times

00:15:55.390 --> 00:16:01.860
vector b minus a, where t is
a member of the reals.

00:16:01.860 --> 00:16:04.290
So my definition of
my line could be

00:16:04.290 --> 00:16:06.130
either of these things.

00:16:06.130 --> 00:16:11.510
The definition of my line could
be this set, or it could

00:16:11.510 --> 00:16:12.850
be this set.

00:16:12.850 --> 00:16:15.240
And all of this seems all very
abstract, but when you

00:16:15.240 --> 00:16:16.940
actually deal with the numbers,
it actually becomes

00:16:16.940 --> 00:16:17.670
very simple.

00:16:17.670 --> 00:16:22.230
It becomes arguably simpler than
what we did in Algebra 1.

00:16:22.230 --> 00:16:26.210
So this l, for these particular
case of a and b,

00:16:26.210 --> 00:16:27.480
let's figure it out.

00:16:27.480 --> 00:16:30.680
My line is equal to-- let me
just use the first example.

00:16:30.680 --> 00:16:37.740
It's vector b, so it's the
vector 0, 3 plus t, times the

00:16:37.740 --> 00:16:39.940
vector b minus a.

00:16:39.940 --> 00:16:42.260
Well what's b minus a?

00:16:42.260 --> 00:16:51.670
0 minus 2 is minus 2, 3, minus
1 is 2, for t is a

00:16:51.670 --> 00:16:54.090
member of the reals.

00:16:54.090 --> 00:16:57.110
Now, if this still seems kind
of like a convoluted set

00:16:57.110 --> 00:16:59.530
definition for you, I could
write it in terms that you

00:16:59.530 --> 00:17:01.190
might recognize better.

00:17:01.190 --> 00:17:05.290
If we want to plot points, if
we call this the y-axis, and

00:17:05.290 --> 00:17:09.949
we call this the x-axis,
and if we call this the

00:17:09.949 --> 00:17:13.339
x-coordinate, or maybe more
properly that the x-coordinate

00:17:13.339 --> 00:17:17.109
and call this the y-coordinate,
then we can set

00:17:17.109 --> 00:17:17.940
up an equation here.

00:17:17.940 --> 00:17:19.199
This actually is the x-slope.

00:17:21.960 --> 00:17:24.369
This is the x-coordinate,
that's the y-coordinate.

00:17:24.369 --> 00:17:27.510
Or actually, even better,
whatever-- actually, let me be

00:17:27.510 --> 00:17:29.620
very careful there.

00:17:29.620 --> 00:17:36.090
This is always going to end up
becoming some vector, l1, l2.

00:17:36.090 --> 00:17:36.490
Right?

00:17:36.490 --> 00:17:39.790
This is a set of vectors, and
any member of this set is

00:17:39.790 --> 00:17:41.950
going to look something
like this.

00:17:41.950 --> 00:17:45.900
So this could be li.

00:17:45.900 --> 00:17:49.910
So, this is the x-coordinate,
and this is the y-coordinate.

00:17:54.710 --> 00:17:56.900
And just to get this in a form
that you recognize, so we're

00:17:56.900 --> 00:18:00.330
saying that l is the set of
this vector x plus t times

00:18:00.330 --> 00:18:05.110
this vector b minus a here.

00:18:05.110 --> 00:18:08.200
If we wanted to write it in kind
of a parametric form, we

00:18:08.200 --> 00:18:12.070
can say, since this is what
determines our x-coordinate,

00:18:12.070 --> 00:18:18.420
we would say that x is equal to
0 plus t times minus 2, or

00:18:18.420 --> 00:18:21.120
minus 2 times t.

00:18:21.120 --> 00:18:24.090
And then we can say that y,
since this is what determines

00:18:24.090 --> 00:18:34.800
our y-coordinate, y is equal to
3 plus t times 2 plus 2t.

00:18:34.800 --> 00:18:37.630
So we could have rewritten that
first equation as just x

00:18:37.630 --> 00:18:43.790
is equal to minus 2t, and
y is equal to 2t plus 3.

00:18:43.790 --> 00:18:46.570
So if you watch the videos on
parametric equations, this is

00:18:46.570 --> 00:18:49.390
just a traditional parametric
definition of

00:18:49.390 --> 00:18:53.020
this line right there.

00:18:53.020 --> 00:18:55.860
Now, you might have still viewed
this as, Sal, this was

00:18:55.860 --> 00:18:57.580
a waste of time, this
was convoluted.

00:18:57.580 --> 00:19:00.100
You have to define these
sets and all that.

00:19:00.100 --> 00:19:03.020
But now I'm going to show you
something that you probably--

00:19:03.020 --> 00:19:05.490
well, unless you have done
this before, but I guess

00:19:05.490 --> 00:19:06.320
that's true of anything.

00:19:06.320 --> 00:19:07.680
But you probably haven't
seen in your

00:19:07.680 --> 00:19:10.260
traditional algebra class.

00:19:10.260 --> 00:19:12.420
Let's say I have two points, and
now I'm going to deal in

00:19:12.420 --> 00:19:13.910
three dimensions.

00:19:13.910 --> 00:19:15.520
So let's say I have
one vector.

00:19:15.520 --> 00:19:18.025
I'll just call it point
1, because these

00:19:18.025 --> 00:19:19.080
are position vectors.

00:19:19.080 --> 00:19:21.530
We'll just call it position 1.

00:19:21.530 --> 00:19:23.350
This is in three dimensions.

00:19:23.350 --> 00:19:28.090
Just make up some numbers,
negative 1, 2, 7.

00:19:28.090 --> 00:19:29.850
Let's say I have Point 2.

00:19:29.850 --> 00:19:32.540
Once again, this is in three
dimensions, so you have to

00:19:32.540 --> 00:19:34.050
specify three coordinates.

00:19:34.050 --> 00:19:36.530
This could be the x, the y,
and the z coordinate.

00:19:36.530 --> 00:19:37.420
Point 2, I don't know.

00:19:37.420 --> 00:19:42.580
Let's make it 0, 3, and 4.

00:19:42.580 --> 00:19:46.120
Now, what if I wanted to find
the equation of the line that

00:19:46.120 --> 00:19:49.670
passes through these
two points in R3?

00:19:49.670 --> 00:19:50.920
So this is in R3.

00:19:53.410 --> 00:19:57.020
Well, I just said that the
equation of this line-- so

00:19:57.020 --> 00:20:00.540
I'll just call that, or the set
of this line, let me just

00:20:00.540 --> 00:20:02.510
call this l.

00:20:02.510 --> 00:20:05.660
It's going to be equal to-- we
could just pick one of these

00:20:05.660 --> 00:20:10.720
guys, it could be P1, the
vector P1, these are all

00:20:10.720 --> 00:20:13.370
vectors, be careful here.

00:20:13.370 --> 00:20:18.310
The vector P1 plus some random
parameter, t, this t could be

00:20:18.310 --> 00:20:20.870
time, like you learn when you
first learn parametric

00:20:20.870 --> 00:20:24.750
equations, times the difference
of the two vectors,

00:20:24.750 --> 00:20:28.860
times P1, and it doesn't matter
what order you take it.

00:20:28.860 --> 00:20:29.870
So that's a nice thing too.

00:20:29.870 --> 00:20:31.860
P1 minus P2.

00:20:31.860 --> 00:20:35.450
It could be P2 minus P1--
because this can take on any

00:20:35.450 --> 00:20:40.580
positive or negative value--
where t is a member of the

00:20:40.580 --> 00:20:42.120
real numbers.

00:20:42.120 --> 00:20:43.690
So let's apply it to
these numbers.

00:20:43.690 --> 00:20:44.780
Let's apply it right here.

00:20:44.780 --> 00:20:47.530
What is P1 minus P2?

00:20:47.530 --> 00:20:54.860
P1 minus P2 is equal to-- let
me get some space here.

00:20:54.860 --> 00:20:59.650
P1 minus P2 is equal, minus
1 minus 0 is minus 1.

00:20:59.650 --> 00:21:04.740
2 minus 3 is minus 1.

00:21:04.740 --> 00:21:07.540
7 minus 4 is 3.

00:21:07.540 --> 00:21:09.140
So that thing is that vector.

00:21:09.140 --> 00:21:13.450
And so, our line can be
described as a set of vectors,

00:21:13.450 --> 00:21:18.280
that if you were to plot it in
standard position, it would be

00:21:18.280 --> 00:21:20.440
this set of position vectors.

00:21:20.440 --> 00:21:23.700
It would be P1, it would be--
let me do that in green-- it

00:21:23.700 --> 00:21:28.520
would be minus 1, 2, 7.

00:21:28.520 --> 00:21:39.010
I could've put P2 there, just as
easily-- plus t times minus

00:21:39.010 --> 00:21:45.360
1, minus 1, 3, where, or such
that, t is a member of the

00:21:45.360 --> 00:21:46.690
real numbers.

00:21:46.690 --> 00:21:49.770
Now, this also might not
be satisfying for you.

00:21:49.770 --> 00:21:52.650
You're like, gee, how do I plot
this in three dimensions?

00:21:52.650 --> 00:21:54.860
Where's my x, y's, and z's?

00:21:54.860 --> 00:21:57.850
And if you want to care about
x, y's, and z's, let's say

00:21:57.850 --> 00:22:05.860
that this is the z-axis.

00:22:05.860 --> 00:22:09.360
This is the x-axis, and
let's say the y-axis.

00:22:09.360 --> 00:22:13.050
It kind of goes into our board
like this, so the y-axis comes

00:22:13.050 --> 00:22:14.300
out like that.

00:22:17.530 --> 00:22:19.750
So what you can do, and actually
I probably won't

00:22:19.750 --> 00:22:23.540
graph, so the determinate for
the x-coordinate, just our

00:22:23.540 --> 00:22:27.030
convention, is going to be
this term right here.

00:22:27.030 --> 00:22:29.520
So we can write that x--
let me write that down.

00:22:29.520 --> 00:22:31.190
So that term is going to
determine our x-coordinate.

00:22:31.190 --> 00:22:36.420
So we can write that x is equal
to minus 1-- be careful

00:22:36.420 --> 00:22:41.840
with the colors-- minus 1,
plus minus 1 times t.

00:22:46.000 --> 00:22:48.600
That's our x-coordinate.

00:22:48.600 --> 00:22:52.800
Now, our y-coordinate is going
to be determined by this part

00:22:52.800 --> 00:22:55.280
of our vector addition because
these are the y-coordinates.

00:22:55.280 --> 00:22:59.000
So we can say the y-coordinate
is equal to-- I'll just write

00:22:59.000 --> 00:23:04.800
it like this-- 2 plus
minus 1 times t.

00:23:04.800 --> 00:23:08.690
And then finally, our
z-coordinate is determined by

00:23:08.690 --> 00:23:12.290
that there, the t shows up
because t times 3-- or I could

00:23:12.290 --> 00:23:13.940
just put this t into
all of this.

00:23:13.940 --> 00:23:20.115
So that the z-coordinate is
equal to 7 plus t times 3, or

00:23:20.115 --> 00:23:22.680
I could say plus 3t.

00:23:22.680 --> 00:23:25.640
And just like that, we have
three parametric equations.

00:23:25.640 --> 00:23:29.110
And when we did it in R2, I did
a parametric equation, but

00:23:29.110 --> 00:23:30.550
we learned in Algebra 1,
you can just have a

00:23:30.550 --> 00:23:31.870
regular y in terms x.

00:23:31.870 --> 00:23:33.830
You don't have to have a
parametric equation.

00:23:33.830 --> 00:23:37.420
But when you're dealing in R3,
the only way to define a line

00:23:37.420 --> 00:23:38.810
is to have a parametric
equation.

00:23:38.810 --> 00:23:41.380
If you have just an equation
with x's, y's, and z's, if I

00:23:41.380 --> 00:23:46.580
just have x plus y plus z is
equal to some number, this is

00:23:46.580 --> 00:23:49.210
not a line.

00:23:49.210 --> 00:23:51.110
And we'll talk more
about this in R3.

00:23:51.110 --> 00:23:52.360
This is a plane.

00:23:54.590 --> 00:23:57.910
The only way to define a line
or a curve in three

00:23:57.910 --> 00:24:01.020
dimensions, if I wanted to
describe the path of a fly in

00:24:01.020 --> 00:24:03.950
three dimensions, it has to
be a parametric equation.

00:24:03.950 --> 00:24:06.890
Or if I shoot a bullet in three
dimensions and it goes

00:24:06.890 --> 00:24:09.850
in a straight line, it has to
be a parametric equation.

00:24:09.850 --> 00:24:11.650
So these-- I guess you could
call it-- these are the

00:24:11.650 --> 00:24:15.600
equations of a line in
three dimensions.

00:24:15.600 --> 00:24:17.050
So hopefully you found
that interesting.

00:24:17.050 --> 00:24:20.340
And I think this will be the
first video where you have an

00:24:20.340 --> 00:24:23.450
appreciation that linear algebra
can solve problems or

00:24:23.450 --> 00:24:25.480
address issues that you
never saw before.

00:24:25.480 --> 00:24:27.930
And there's no reason why we
have to just stop at three,

00:24:27.930 --> 00:24:29.410
three coordinates, right here.

00:24:29.410 --> 00:24:31.410
We could have done this
with fifty dimensions.

00:24:31.410 --> 00:24:35.200
We could have defined a line in
fifty dimensions-- or the

00:24:35.200 --> 00:24:40.450
set of vectors that define a
line, that two points sit on,

00:24:40.450 --> 00:24:43.060
in fifty dimensions-- which is
very hard to visualize, but we

00:24:43.060 --> 00:24:45.340
can actually deal with
it mathematically.