Visualizations of left nullspace and rowspace

WEBVTT
Kind: captions
Language: en

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In the last video I had this 2
by 3 matrix A right here, and

00:00:05.070 --> 00:00:08.820
we figured out all of the
subspaces that are associated

00:00:08.820 --> 00:00:09.455
with this matrix.

00:00:09.455 --> 00:00:13.370
We figured out its null space,
its column space, we figured

00:00:13.370 --> 00:00:16.260
out the null space and column
space of its transpose, which

00:00:16.260 --> 00:00:19.930
you could also call the left
null space, and the row space,

00:00:19.930 --> 00:00:23.580
or what's essentially the space
spanned by A's rows.

00:00:23.580 --> 00:00:25.840
Let's write it all in one place,
because I realize it

00:00:25.840 --> 00:00:29.050
got a little disjointed, and see
if we can visualize what

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all of these look
like, especially

00:00:30.490 --> 00:00:31.620
relative to each other.

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So let me copy and paste
my original matrix.

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Copy, and then let me scroll
down here and paste it over

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here, and hit paste.

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Let me see if I can find
our key takeaways

00:00:49.190 --> 00:00:50.580
from the last video.

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So our column space right here,
of A, was this thing

00:00:55.410 --> 00:00:55.900
right here.

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Let me write this.

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This was our column space.

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It was the span of the
R2 vector 2, 4.

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Let me copy that.

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Copy that and bring it down.

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Hit paste.

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This was our column space.

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Let me write that.

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This is the column space of A.

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It was equal to that
right there.

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And now what other things
do we know?

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Well, we know that the left null
space was a span of 2, 1.

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Let me write that.

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So our left null space, or the
null space of our transpose,

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either way, it was equal to the
span of the R2 vector 2,

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1, just like that.

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And then what was
our null space?

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Our null space we figured
out in the last video.

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Here it is.

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It's the span of these
two R3 vectors.

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Let me copy and paste that.

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Hit copy.

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Let me go down here.

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Let me paste it.

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So that was our null
space right there.

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And then finally, what
was our row space?

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What was our row space or the
column space of our transpose?

00:02:23.520 --> 00:02:27.850
So the column space of our
transpose was the span of this

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R3 vector right there, So it
was this one right here.

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So let me copy and paste it.

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Copy and scroll down, and we can
paste it just like that.

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OK, let's see if we can
visualize this now, now that

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we have them all in one place.

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So first of all, if we imagine
a transformation, x, that is

00:02:57.170 --> 00:03:02.160
equal to A times x, our
transformation is going to be

00:03:02.160 --> 00:03:06.630
a mapping from what? x would be
a member of R3, so R3 would

00:03:06.630 --> 00:03:07.670
be our domain.

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So it would be a mapping from
R3 and then it would be a

00:03:10.340 --> 00:03:13.390
mapping to R2 because we have
two rows here, right?

00:03:13.390 --> 00:03:21.300
You multiply a 2-by-3 matrix
times a 3-by-1 vector, and

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you're going to get a 2-by-1
vector, so it's going to be a

00:03:23.330 --> 00:03:26.090
mapping to R2.

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So that's our codomain.

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So let's draw our domains
and our codomains.

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I'll just write them very
generally right here.

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So you could imagine
R3 is our domain.

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And then our codomain is going
to be R2 just like that.

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And our T is a mapping, or you
could even imagine A is a

00:03:51.020 --> 00:03:53.750
mapping between any vector there
and any vector there

00:03:53.750 --> 00:03:55.340
when you multiply them.

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Now, what is our column
space of A?

00:04:00.150 --> 00:04:04.260
Our column space of A is the
span of the vector 2 minus 4.

00:04:04.260 --> 00:04:05.470
It's an R2 vector.

00:04:05.470 --> 00:04:09.200
This is a subspace of R2.

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We could write this.

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So let me write this.

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So our column space of A, these
are just all of the

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vectors that are spanned
by this.

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We figured out that these guys
are just multiples of this

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first guy, or we could have
done it the other way.

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We could have said this guy and
that guy are multiples of

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that guy, either way.

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But the basis is just one
of these vectors.

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We just have to have one of
these vectors, and so it was

00:04:31.590 --> 00:04:33.660
equal to this right here.

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So the column space
is a subset of R2.

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And what else is
a subset of R2?

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Well, our left null space.

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Our left null space is
also a subset of R2.

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So let's graph them, actually.

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So I won't be too exact,
but you can imagine.

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Let's see, if we draw the
vector 2, 4-- let me

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draw some axes here.

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Let me scroll down
a little bit.

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So if you have some vector--
let me draw my-- do this as

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neatly as possible.

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That's my vertical axis.

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That is my horizontal axis.

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And then, what does the span of
our column space look like?

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So you draw the vector 2, minus
4, so you're going to go

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out one, two, and then you're
going to go down one, two,

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three, four.

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So that's what that
vector looks like.

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And the span of this vector
is essentially all of the

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multiples of this vector, where
you could say linear

00:05:44.970 --> 00:05:47.565
combinations of it, but you're
taking a combination of just

00:05:47.565 --> 00:05:49.530
one vector, so it's just going
to be all of the multiples of

00:05:49.530 --> 00:05:50.700
this vector.

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So if I were to graph it, it
would just be a line that is

00:05:55.750 --> 00:05:59.030
specified by all of the linear
combinations of that vector

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right there.

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This right here is a graphical
representation of the column

00:06:04.665 --> 00:06:07.160
space of A.

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Now, let's look at the left null
space of A, or you could

00:06:10.140 --> 00:06:12.130
imagine, the null space
of the transpose.

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They are the same thing.

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You saw why in the last video.

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What does this look like?

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So the left null space
is a span of 2, 1.

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So if you graph 2, and then you
go up 1, it's the graph of

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2, 1, and it looks like this.

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Let me do it in a
different color.

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So that's what the vector
looks like.

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The vector looks like that,
but of course, we want the

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span of that vector, so it's
going to be all of the

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combinations.

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All you can do when you combine
one vector is just

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multiply it by a bunch of
scalars, so it's going to be

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all of the scalar multiples
of that vector.

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So let me draw it like that.

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It's going to be like that.

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And the first thing you might
notice, let me write this.

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This is our left null space
of A or the null

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space of our transpose.

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This is equal to the left
null space of A.

00:07:07.870 --> 00:07:09.670
And actually, since we're
writing, we wrote this in

00:07:09.670 --> 00:07:10.660
terms of A transpose.

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It's the null space of A
transpose, which is the left

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null space of A.

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Let's write the column
space of A also

00:07:15.900 --> 00:07:17.640
in terms of A transpose.

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This is equal to the row space
of A transpose, right?

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If you're looking at the columns
of A, everything it

00:07:26.400 --> 00:07:29.110
spans, the columns of A are the
same things as the rows of

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A transpose.

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But the first thing that you
see, when I just at least

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visually drew it like this, is
that these two spaces look to

00:07:36.710 --> 00:07:38.400
be orthogonal to each other.

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It looks like I drew it in R2.

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It looks like there's a
90-degree angle there.

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And if we wanted to verify it,
all we have to do is take the

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dot product.

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Well, any vector that is in our
column space, you could

00:07:55.400 --> 00:07:58.110
take an arbitrary vector that's
in our column space,

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it's going to be equal
to c times 2 minus 4.

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So let me write that down.

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I want this stuff up here.

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I'll scroll down a little bit.

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Let's say v1 is a member
of our column space.

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And that means that v1 is going
to be equal to some

00:08:23.170 --> 00:08:27.270
scalar multiple times the
spanning vector of our column

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space, so some scale or
multiple of this.

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So we could say it's equal
to c1 times 2 minus 4.

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That's some member of
our column space.

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Now, if we want some member of
our left null space-- let's

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write it here.

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So let's say that v2 is some
member of our left null space,

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or the null space of
the transpose, then

00:08:57.430 --> 00:08:58.050
what does that mean?

00:08:58.050 --> 00:09:01.820
That means v2 is going to be
equal to some scalar multiple

00:09:01.820 --> 00:09:05.130
of the spanning vector of our
left null space of 2, 1.

00:09:08.480 --> 00:09:10.540
So any vector that's in our
column space could be

00:09:10.540 --> 00:09:11.770
represented this way.

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Any vector in our left
null space can be

00:09:13.410 --> 00:09:14.620
represented this way.

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Now, what happens if you take
the dot product of these two

00:09:16.720 --> 00:09:17.695
characters?

00:09:17.695 --> 00:09:20.570
So let me do it down here.

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I want to save some space for
what we're going to do in R3,

00:09:23.850 --> 00:09:25.880
but let me take the dot
product of these two

00:09:25.880 --> 00:09:26.910
characters.

00:09:26.910 --> 00:09:33.460
So v1 dot v2 is equal to--
I'll arbitrarily switch

00:09:33.460 --> 00:09:42.840
colors-- c1 times 2 minus
4 dot c2 times 2, 1.

00:09:42.840 --> 00:09:45.630
And then the scalars, we've
seen this before.

00:09:45.630 --> 00:09:51.420
You can just say that this is
the same thing as c1, c2 times

00:09:51.420 --> 00:09:56.620
the dot product of 2
minus 4 dot 2, 1.

00:09:56.620 --> 00:09:58.600
And then what is
this equal to?

00:09:58.600 --> 00:10:05.130
This is going to be equal to
c1, c2 times 2 times 2 is 4

00:10:05.130 --> 00:10:09.090
plus minus 4 times 1: minus 4.

00:10:09.090 --> 00:10:11.280
Well, this is going to be equal
to 0, so this whole

00:10:11.280 --> 00:10:12.805
expression is going
to be equal to 0.

00:10:12.805 --> 00:10:15.350
And this was for any two vectors
that are members of

00:10:15.350 --> 00:10:17.090
our column space and our
left null space.

00:10:17.090 --> 00:10:18.950
They're orthogonal
to each other.

00:10:18.950 --> 00:10:22.160
So every member of our column
space is going to be

00:10:22.160 --> 00:10:26.660
orthogonal to every member of
our left null space, or every

00:10:26.660 --> 00:10:29.980
member of the null space of our
transpose, and that was

00:10:29.980 --> 00:10:31.270
the case in this example.

00:10:31.270 --> 00:10:35.360
It actually turns out this is
always going to be the case,

00:10:35.360 --> 00:10:39.530
that your column space of
a matrix, its orthogonal

00:10:39.530 --> 00:10:43.280
complement is the left null
space, or the null space of

00:10:43.280 --> 00:10:44.300
its transpose.

00:10:44.300 --> 00:10:46.640
I'll prove that probably in the
next video, either in the

00:10:46.640 --> 00:10:48.770
next video or the video after
that, but you can see it

00:10:48.770 --> 00:10:50.680
visually for this example.

00:10:50.680 --> 00:10:55.620
Now let's draw the other two
characters that we're

00:10:55.620 --> 00:10:57.310
dealing with here.

00:10:57.310 --> 00:11:03.930
So we have our null space, which
is the span of these two

00:11:03.930 --> 00:11:04.950
vectors in R3.

00:11:04.950 --> 00:11:08.330
It's a little bit more difficult
to draw it, these

00:11:08.330 --> 00:11:10.430
two vectors in R3 right there.

00:11:10.430 --> 00:11:13.510
But what is the span of
two vectors in R3?

00:11:13.510 --> 00:11:16.135
All of the linear combinations
of two vectors in R3 is going

00:11:16.135 --> 00:11:18.000
to be a plane in R3.

00:11:18.000 --> 00:11:21.045
So I'll draw it in just very
general terms right here.

00:11:21.045 --> 00:11:25.860
If we draw it in just very
general terms, let me see.

00:11:25.860 --> 00:11:30.410
So it's a plane in R3 that
looks like that.

00:11:30.410 --> 00:11:33.300
Maybe I'll fill in the plane
a little bit, give you some

00:11:33.300 --> 00:11:34.370
sense of what it looks like.

00:11:34.370 --> 00:11:38.640
This is the null space of A.

00:11:38.640 --> 00:11:40.960
It's spanned by these
two vectors.

00:11:40.960 --> 00:11:42.950
Now, you could imagine these
two vectors look something

00:11:42.950 --> 00:11:45.990
like-- I'm drawing it very
general, but if you take any

00:11:45.990 --> 00:11:49.800
linear combinations of these two
guys, you're going to get

00:11:49.800 --> 00:11:54.040
any vector that's along this
plane that goes in infinite

00:11:54.040 --> 00:11:54.480
directions.

00:11:54.480 --> 00:11:56.000
And, of course, the origin
will be in these.

00:11:56.000 --> 00:11:58.090
All of these are valid
subspaces.

00:11:58.090 --> 00:12:05.930
Now, what does the row
space of A look like?

00:12:05.930 --> 00:12:08.820
Or you could say the column
space of A transpose?

00:12:08.820 --> 00:12:15.000
Well, it's the span of this
vector in R3, but let's see

00:12:15.000 --> 00:12:17.580
something interesting about
this vector in R3.

00:12:17.580 --> 00:12:20.090
How does it relate to
these two vectors?

00:12:20.090 --> 00:12:22.200
Well, you may not see it
immediately, although if you

00:12:22.200 --> 00:12:24.910
look at it closely, it might pop
out at you, that this guy

00:12:24.910 --> 00:12:28.670
is orthogonal to both
of these guys.

00:12:28.670 --> 00:12:37.170
Notice, if you take the dot
product of 2 minus 1 minus 3,

00:12:37.170 --> 00:12:43.360
and you dotted it with 1/2, 1,
0, what are you going to get?

00:12:43.360 --> 00:12:46.720
You're going to get 2 times 1/2,
which is 1, plus minus 1

00:12:46.720 --> 00:12:49.190
times 1, which is minus
1, plus minus 3

00:12:49.190 --> 00:12:51.390
times 0, which is 0.

00:12:51.390 --> 00:12:55.230
So that's when I dotted that guy
with that guy right there.

00:12:55.230 --> 00:12:57.490
And then, when I take the dot
of this guy with that guy,

00:12:57.490 --> 00:12:58.150
what do you get?

00:12:58.150 --> 00:13:04.580
You get 3/2, 0 and 1, dotted
with-- let me scroll down a

00:13:04.580 --> 00:13:05.210
little bit.

00:13:05.210 --> 00:13:11.650
I don't want to write too
small-- dotted with 1, dotted

00:13:11.650 --> 00:13:16.440
with 2 minus 1 minus 3.

00:13:16.440 --> 00:13:20.010
In the row space of A,
I wrote the spanning

00:13:20.010 --> 00:13:21.970
vector there this time.

00:13:21.970 --> 00:13:23.280
I probably shouldn't have
switched the order.

00:13:23.280 --> 00:13:26.230
But here, I'm dotting it with
this guy, and then here, I'm

00:13:26.230 --> 00:13:28.110
dotting it with this
guy right there.

00:13:28.110 --> 00:13:33.860
So if you take it, 3/2 times 2
is equal to 3 plus 0 times

00:13:33.860 --> 00:13:37.966
minus 1 is 0, plus 1 times minus
3 is minus 3, so it's

00:13:37.966 --> 00:13:39.490
equal to 0.

00:13:39.490 --> 00:13:44.350
So the fact that this guy is
orthogonal to both of these

00:13:44.350 --> 00:13:46.530
spanning vectors, it also means
that it's orthogonal to

00:13:46.530 --> 00:13:48.470
any linear combination
of those guys.

00:13:48.470 --> 00:13:51.410
Maybe it might be useful
for you to see that.

00:13:51.410 --> 00:13:58.640
So let's take some member
of our null space.

00:13:58.640 --> 00:14:04.020
So let's say the vector v3 is
a member of our null space.

00:14:07.320 --> 00:14:11.680
That means it's a linear
combination of that

00:14:11.680 --> 00:14:13.340
guy and that guy.

00:14:13.340 --> 00:14:14.850
Those are the two spanning
vectors.

00:14:14.850 --> 00:14:16.970
I'd written it up here.

00:14:16.970 --> 00:14:18.930
These are our two spanning
vectors.

00:14:18.930 --> 00:14:20.860
I need the space down here,
so let me scroll

00:14:20.860 --> 00:14:21.520
down a little bit.

00:14:21.520 --> 00:14:22.840
These are the two spanning
vectors.

00:14:22.840 --> 00:14:27.910
So that means that v3 can be
written as some linear

00:14:27.910 --> 00:14:31.780
combination of these two guys
that I squared off in pink.

00:14:31.780 --> 00:14:40.810
So let me just write it as maybe
A times 3/2, 0, 1 plus b

00:14:40.810 --> 00:14:48.520
times 1/2, 1, 0.

00:14:48.520 --> 00:14:56.900
Now, what happens if I take the
dot product of v3 and I

00:14:56.900 --> 00:15:03.110
dot it with any member of
my row space right here?

00:15:03.110 --> 00:15:08.650
So any member of my row space
is going to be a multiple of

00:15:08.650 --> 00:15:09.490
this guy right here.

00:15:09.490 --> 00:15:12.170
That is the spanning vector
of my row space.

00:15:12.170 --> 00:15:13.750
Just let me actually
create that.

00:15:13.750 --> 00:15:20.750
So let me say that v4 is a
member of my row space, which

00:15:20.750 --> 00:15:23.800
is the column space of
the transpose of A.

00:15:23.800 --> 00:15:28.500
And that means that v4 is equal
to, let's say, some

00:15:28.500 --> 00:15:29.300
scaling vector.

00:15:29.300 --> 00:15:30.860
I always use c a lot.

00:15:30.860 --> 00:15:32.200
Let me use d.

00:15:32.200 --> 00:15:34.490
Let's say it's d times
my spanning vector.

00:15:34.490 --> 00:15:38.520
d times 2 minus 1, 3.

00:15:38.520 --> 00:15:43.520
So what is v3, which is just
any member of my null space

00:15:43.520 --> 00:15:50.420
dotted with v4, which is any
member of my row space?

00:15:50.420 --> 00:15:52.740
So what is this going
to be equal to?

00:15:52.740 --> 00:15:57.890
This is going to be
equal to this guy.

00:15:57.890 --> 00:15:59.620
So let me write it like this.

00:15:59.620 --> 00:16:13.850
A times 3/2, 0, 1 plus v times
1/2, 1, 0 dotted with this

00:16:13.850 --> 00:16:18.830
guy, dot d times 2 minus 1, 3.

00:16:18.830 --> 00:16:20.500
Now, what is this going
to be equal to?

00:16:20.500 --> 00:16:22.310
Well, we know all of
the properties

00:16:22.310 --> 00:16:23.930
of vector dot products.

00:16:23.930 --> 00:16:26.330
We can distribute it and then
take the scalars out.

00:16:26.330 --> 00:16:28.940
So this is going to be equal
to-- I'll skip a few steps

00:16:28.940 --> 00:16:32.970
here, but it's going to be equal
to-- ad times the dot

00:16:32.970 --> 00:16:41.755
product of 3/2, 0, 1, dot 2
minus 1, 3-- just distribute

00:16:41.755 --> 00:16:50.790
it out to here-- plus bd times
the dot product of 1/2, 1, 0,

00:16:50.790 --> 00:16:54.410
dotted with 2 minus 1, 3.

00:16:54.410 --> 00:16:55.660
This is the dot product.

00:16:55.660 --> 00:16:57.400
I just distributed this
term along these

00:16:57.400 --> 00:16:58.670
two terms right here.

00:16:58.670 --> 00:17:01.040
And we already know what these
dot products are equal to.

00:17:01.040 --> 00:17:02.410
We did it right here.

00:17:02.410 --> 00:17:04.800
This dot product right here
is that dot product.

00:17:04.800 --> 00:17:07.420
I just switched the order,
so this is equal to 0.

00:17:07.420 --> 00:17:10.700
And this dot product is that dot
product, so this is also

00:17:10.700 --> 00:17:12.369
equal to 0.

00:17:12.369 --> 00:17:17.190
So you take any member of your
row space and you dot it with

00:17:17.190 --> 00:17:21.819
any member of your null space,
and you're going to get 0, or

00:17:21.819 --> 00:17:25.690
any member of your row space is
orthogonal to any member of

00:17:25.690 --> 00:17:27.170
your null space.

00:17:27.170 --> 00:17:29.720
And I did all of that to
help our visualization.

00:17:29.720 --> 00:17:32.420
So we just saw that any member
of our row space, which is the

00:17:32.420 --> 00:17:36.930
span of this vector, is
orthogonal to any member of or

00:17:36.930 --> 00:17:37.790
null space.

00:17:37.790 --> 00:17:40.200
So my row space, which is just
going to be a line in R3

00:17:40.200 --> 00:17:41.550
because it's just a multiple
of a vector.

00:17:41.550 --> 00:17:42.800
It's going to look like this.

00:17:45.780 --> 00:17:47.500
It's going to be a line,
and then it's going to

00:17:47.500 --> 00:17:48.170
maybe go behind it.

00:17:48.170 --> 00:17:49.100
You can't see it there.

00:17:49.100 --> 00:17:50.710
It's going to look like that,
but it's going to be

00:17:50.710 --> 00:17:51.605
orthogonal.

00:17:51.605 --> 00:17:52.690
So let me draw it.

00:17:52.690 --> 00:17:59.440
So this pink line right here in
R3, that is our row space

00:17:59.440 --> 00:18:03.540
of A, which is equal to the
column space of A transpose

00:18:03.540 --> 00:18:06.500
because the rows of A are the
same thing as the columns of A

00:18:06.500 --> 00:18:09.820
transposed, and the row space
is just the space spanned by

00:18:09.820 --> 00:18:11.030
your row vectors.

00:18:11.030 --> 00:18:16.850
And then this is the null space
of A, which is a plane.

00:18:16.850 --> 00:18:19.240
It's spanned by two
vectors in R3.

00:18:19.240 --> 00:18:22.750
Or we could also call
that the left

00:18:22.750 --> 00:18:29.800
null space of A transpose.

00:18:29.800 --> 00:18:31.590
And I never used this term in
the last video, but it's

00:18:31.590 --> 00:18:32.620
symmetric, right?

00:18:32.620 --> 00:18:34.880
If the null space of A transpose
is the left null

00:18:34.880 --> 00:18:37.710
space of A, then the null space
of A is the left null

00:18:37.710 --> 00:18:42.810
space of A transpose, which is
an interesting takeaway.

00:18:42.810 --> 00:18:50.570
Notice that you have here the
row space of A is orthogonal

00:18:50.570 --> 00:18:52.520
to the null space of A.

00:18:52.520 --> 00:18:56.640
And here, you have the row
space of A transpose is

00:18:56.640 --> 00:18:59.750
orthogonal to the null
space of A transpose.

00:18:59.750 --> 00:19:03.750
Or you could say the left null
space of A is orthogonal to

00:19:03.750 --> 00:19:05.350
the column space of A.

00:19:05.350 --> 00:19:08.220
Or you could say the left null
space of A transpose is

00:19:08.220 --> 00:19:11.780
orthogonal to the column
space of A transpose.

00:19:11.780 --> 00:19:13.280
So these are just
very interesting

00:19:13.280 --> 00:19:15.890
takeaways, in general.

00:19:15.890 --> 00:19:18.290
And just like I said here, that
look, these happen to be

00:19:18.290 --> 00:19:18.830
orthogonal.

00:19:18.830 --> 00:19:20.460
These also happen to
be orthogonal.

00:19:20.460 --> 00:19:22.590
And this isn't just some
strange coincidence.

00:19:22.590 --> 00:19:26.490
In the next video or two, I'll
show you that this space, this

00:19:26.490 --> 00:19:30.780
pink space, is the orthogonal
complement of the null space

00:19:30.780 --> 00:19:36.710
right here, which means it
represents all of the vectors

00:19:36.710 --> 00:19:39.100
that are orthogonal
to the null space.

00:19:39.100 --> 00:19:41.110
And these two guys
are orthogonal

00:19:41.110 --> 00:19:42.130
complements to each other.

00:19:42.130 --> 00:19:45.910
They each represent all of the
vectors that are orthogonal to

00:19:45.910 --> 00:19:49.210
the other guy in their
respective spaces.


Rowspace and left nullspace

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:03.420
I've got this matrix, A, here,
it's a 2 by 3 matrix.

00:00:03.420 --> 00:00:06.210
And just as a bit of review,
let's figure out its nullspace

00:00:06.210 --> 00:00:07.920
and its columnspace.

00:00:07.920 --> 00:00:13.300
So the nullspace of A is the set
of all vectors x that are

00:00:13.300 --> 00:00:16.550
member of-- let's see we have
3 columns here-- so a member

00:00:16.550 --> 00:00:22.580
of R3, such that A times the
vector are going to be equal

00:00:22.580 --> 00:00:25.210
to the 0 vector.

00:00:25.210 --> 00:00:26.820
So we can just set this up.

00:00:26.820 --> 00:00:29.800
Let me just-- we just need to
figure out all of the x's that

00:00:29.800 --> 00:00:31.990
satisfy this in R3.

00:00:31.990 --> 00:00:35.710
So we take our matrix
A 2, minus 1, minus

00:00:35.710 --> 00:00:39.120
3, minus 4, 2, 6.

00:00:39.120 --> 00:00:42.965
Multiply them times some
arbitrary vector in R3 here.

00:00:42.965 --> 00:00:47.060
So you get x1, x2, x3.

00:00:47.060 --> 00:00:49.210
And you set them equal
to the 0 vector.

00:00:49.210 --> 00:00:50.930
It's going to be the
0 vector in R2.

00:00:50.930 --> 00:00:52.120
Because we have 2 rows here.

00:00:52.120 --> 00:00:55.770
You multiply a 2 by 3 matrix
times a vector in R3, you're

00:00:55.770 --> 00:00:59.250
going to get a 2 by 1 vector
or 2 by 1 matrix.

00:00:59.250 --> 00:01:02.560
So you're going to get
the 0 vector in R3.

00:01:02.560 --> 00:01:04.590
And to solve what is essentially
a system of

00:01:04.590 --> 00:01:08.570
equations-- you get 2 x1 minus
x2 minus 3 x3 is equal to 0

00:01:08.570 --> 00:01:09.930
and so on and so forth.

00:01:09.930 --> 00:01:12.610
We can just set up an
augmented matrix.

00:01:12.610 --> 00:01:15.890
So we can just set up this
augmented matrix right here.

00:01:15.890 --> 00:01:21.240
2 minus 1 minus 3
minus 4, 2, 6.

00:01:21.240 --> 00:01:24.140
And then augment it with what
we're trying to set it equal

00:01:24.140 --> 00:01:26.420
to to solve the system.

00:01:26.420 --> 00:01:29.690
And you know we're going to
perform a bunch of row

00:01:29.690 --> 00:01:32.690
operations here to put this in
reduced row echelon form.

00:01:32.690 --> 00:01:35.610
And they're not going to change
the right-hand side of

00:01:35.610 --> 00:01:36.560
this augmented matrix.

00:01:36.560 --> 00:01:38.700
And that's essentially the
argument as to why the

00:01:38.700 --> 00:01:42.390
nullspace of the reduced row
echelon form of A is the same

00:01:42.390 --> 00:01:43.600
thing as the nullspace of A.

00:01:43.600 --> 00:01:45.430
But anyway, that's just
a bit of review.

00:01:45.430 --> 00:01:48.690
So let's perform some row
operations to solve this a

00:01:48.690 --> 00:01:50.680
little bit better.

00:01:50.680 --> 00:01:54.600
So, the first thing I might
want to do is divide the

00:01:54.600 --> 00:01:57.145
first row by 2.

00:01:57.145 --> 00:02:02.970
So if I divide the first row by
2 I get a 1 minus 1/2 minus

00:02:02.970 --> 00:02:07.220
3/2 and then of course
0 divided by 2 is 0.

00:02:07.220 --> 00:02:14.200
And let's just divide this row
right here by-- I don't know

00:02:14.200 --> 00:02:18.680
just to simplify things--
let's divide it by 4.

00:02:18.680 --> 00:02:20.890
So I'm doing two row operations
in one step.

00:02:20.890 --> 00:02:21.560
And you can do that.

00:02:21.560 --> 00:02:23.510
I could have done it in
two separate steps.

00:02:23.510 --> 00:02:29.100
So if we divide it by 4, this
becomes minus 1, 1/2 and then

00:02:29.100 --> 00:02:33.640
you get 3/2 and then
you get 0.

00:02:33.640 --> 00:02:36.570
And now, let's keep my
first row the same.

00:02:36.570 --> 00:02:38.150
I'm going to keep my
first row the same.

00:02:38.150 --> 00:02:43.960
It's 1 minus 1/2 minus 3/2 and
of course the 0 is the

00:02:43.960 --> 00:02:45.070
right-hand side.

00:02:45.070 --> 00:02:47.420
Now let's replace my second
row with my second row

00:02:47.420 --> 00:02:49.550
plus my first row.

00:02:49.550 --> 00:02:52.580
So these are just linear
operations on these guys.

00:02:52.580 --> 00:02:55.170
So negative 1 plus 1 is 0.

00:02:55.170 --> 00:02:58.190
1/2 plus minus 1/2 is 0.

00:02:58.190 --> 00:03:00.400
3/2 plus minus 3/2 is 0.

00:03:00.400 --> 00:03:04.190
And of course, 0 plus 0 is 0.

00:03:04.190 --> 00:03:05.800
So what are we left with?

00:03:05.800 --> 00:03:07.430
we're left with this
right here.

00:03:07.430 --> 00:03:13.310
This is another way of saying
that x1-- let me write it this

00:03:13.310 --> 00:03:19.940
way-- x1-- I guess the easiest
way to think about it is--

00:03:19.940 --> 00:03:23.070
you're multiplying the reduced
row echelon form of A now.

00:03:23.070 --> 00:03:26.750
1 minus 1/2 minus 3/2.

00:03:26.750 --> 00:03:28.950
You have a bunch of 0's here.

00:03:28.950 --> 00:03:37.310
Times x1, x2, x3 is equal
to the R2 0 vector.

00:03:37.310 --> 00:03:40.680
This is another interpretation
of this augmented matrix.

00:03:40.680 --> 00:03:43.150
So this is just saying,
this is useless.

00:03:43.150 --> 00:03:45.540
This is saying 0 times that plus
0 times that plus 0 times

00:03:45.540 --> 00:03:46.680
that is equal to 0.

00:03:46.680 --> 00:03:47.690
So it's giving us
no information.

00:03:47.690 --> 00:03:54.280
But this first row tells us
that-- let me switch colors--

00:03:54.280 --> 00:04:04.740
1 times x1 minus 1/2 times
x2 minus 3/2 times

00:04:04.740 --> 00:04:07.010
x3 is equal to 0.

00:04:07.010 --> 00:04:10.240
All of the vectors whose
components satisfy this are in

00:04:10.240 --> 00:04:11.580
my nullspace.

00:04:11.580 --> 00:04:13.950
If I want to write it a little
bit differently I could write

00:04:13.950 --> 00:04:22.079
as, x1 is equal to 1/2
x2 plus 3/2 x3.

00:04:22.079 --> 00:04:25.450
Or if I wanted to write my
solution set in vector form, I

00:04:25.450 --> 00:04:29.170
could write that my nullspace is
going to be the set of all

00:04:29.170 --> 00:04:34.960
the vectors x1, x2, x3 that
satisfy these conditions.

00:04:34.960 --> 00:04:37.150
That are equal to what?

00:04:37.150 --> 00:04:39.950
Well, x2 and x3 are
free variables.

00:04:39.950 --> 00:04:42.760
They're associated with the
non-pivot entries, or the

00:04:42.760 --> 00:04:45.670
non-pivot columns in our reduced
row echelon form.

00:04:45.670 --> 00:04:50.500
That is a pivot column
right there.

00:04:54.720 --> 00:04:55.595
So let me write it this way.

00:04:55.595 --> 00:04:59.670
It's going to be x2
times something

00:04:59.670 --> 00:05:03.100
plus x3 times something.

00:05:03.100 --> 00:05:05.180
Those are my two
free variables.

00:05:05.180 --> 00:05:08.270
And we have here,
x1 is 1/2 x2.

00:05:08.270 --> 00:05:11.690
1/2 times x2 plus
3/2 times x3.

00:05:14.850 --> 00:05:19.790
x2 is just going to be x2
times 1 plus 0 times x3.

00:05:19.790 --> 00:05:25.060
x3 is going to be 0 times
x2 plus 1 times x3.

00:05:25.060 --> 00:05:29.310
So, our nullspace, these
can be any real

00:05:29.310 --> 00:05:30.470
numbers right here.

00:05:30.470 --> 00:05:31.780
They're free variables.

00:05:31.780 --> 00:05:34.440
So our nullspace is essentially
all of the linear

00:05:34.440 --> 00:05:37.420
combinations of this
guy and that guy.

00:05:37.420 --> 00:05:45.430
Or another way to write it, the
nullspace of A is equal to

00:05:45.430 --> 00:05:48.730
the span, which is the same
thing as all the linear

00:05:48.730 --> 00:05:55.280
combinations of the
span of 1/2, 1, 0.

00:05:55.280 --> 00:05:57.050
Notice these are
vectors in R3.

00:05:57.050 --> 00:05:59.620
And that makes sense because the
nullspace is going to be a

00:05:59.620 --> 00:06:00.710
set of vectors in R3.

00:06:00.710 --> 00:06:02.310
So it's the span of that.

00:06:02.310 --> 00:06:03.560
And that right there.

00:06:06.080 --> 00:06:10.850
So 3/2, 0, and 1.

00:06:10.850 --> 00:06:12.460
Just like that.

00:06:12.460 --> 00:06:20.600
And what is the columnspace
of our original matrix, A?

00:06:20.600 --> 00:06:27.850
So the columnspace of A is equal
to just, the subspace

00:06:27.850 --> 00:06:30.430
created by all of the linear
combinations of these guys.

00:06:30.430 --> 00:06:32.830
Or essentially the span
of the column vectors.

00:06:32.830 --> 00:06:39.310
Is equal to the span
of 2 minus 4 minus

00:06:39.310 --> 00:06:42.550
1, 2, minus 3, 6.

00:06:42.550 --> 00:06:45.180
These are all each
separate vectors.

00:06:45.180 --> 00:06:48.360
So it's the span of
these 3 vectors.

00:06:48.360 --> 00:06:51.540
Now, these guys might not
be linearly independent.

00:06:51.540 --> 00:06:54.394
And actually, when you put
this guy in reduced row

00:06:54.394 --> 00:06:57.960
echelon form, you know that the
basis vectors for this are

00:06:57.960 --> 00:07:00.010
the vectors that are associated

00:07:00.010 --> 00:07:01.530
with our pivot columns.

00:07:01.530 --> 00:07:02.760
So we have one pivot
column here.

00:07:02.760 --> 00:07:03.740
It's our first column.

00:07:03.740 --> 00:07:07.900
So we could say that we could
use this as a basis vector.

00:07:07.900 --> 00:07:08.780
And it makes sense.

00:07:08.780 --> 00:07:11.280
Because this guy right here
is minus 2 times this guy.

00:07:11.280 --> 00:07:15.660
This guy right here is minus
3/2 times that guy.

00:07:15.660 --> 00:07:18.030
So these two guys can definitely
be represented as

00:07:18.030 --> 00:07:19.870
linear combinations
of that guy.

00:07:19.870 --> 00:07:30.670
So it's equal to the span of
just the vector, 2, minus 4.

00:07:30.670 --> 00:07:34.570
So if you were to ask me, and
this is the basis for our

00:07:34.570 --> 00:07:35.300
columnspace.

00:07:35.300 --> 00:07:37.160
So if you wanted to know the
rank-- and this is all a bit

00:07:37.160 --> 00:07:43.050
of review-- the rank of A is
equal to the number of vectors

00:07:43.050 --> 00:07:45.290
in our basis for our
columnspace.

00:07:45.290 --> 00:07:48.370
So it's going to
be equal to 1.

00:07:48.370 --> 00:07:51.580
Now, everything I just did
is a bit of review.

00:07:51.580 --> 00:07:53.740
But with the last couple of
videos, we've been dealing

00:07:53.740 --> 00:07:55.520
with transposes.

00:07:55.520 --> 00:07:58.540
So let's actually figure out
the same ideas for the

00:07:58.540 --> 00:07:59.790
transpose of A.

00:08:03.120 --> 00:08:08.160
So A transpose looks
like this.

00:08:08.160 --> 00:08:18.790
A transpose is equal to the
matrix 2, minus 1, minus 3 is

00:08:18.790 --> 00:08:20.470
the first column right there.

00:08:20.470 --> 00:08:25.910
And then the second column is
going to be minus 4, 2 and 6.

00:08:25.910 --> 00:08:28.240
That is our transpose.

00:08:28.240 --> 00:08:31.100
So let's figure out the
nullspace and the columnspace

00:08:31.100 --> 00:08:32.350
of our transpose.

00:08:37.080 --> 00:08:38.940
Let me put this in reduced
row echelon form so

00:08:38.940 --> 00:08:41.159
we can get the nullspace.

00:08:41.159 --> 00:08:42.919
Let me get the nullspace
of this guy.

00:08:42.919 --> 00:08:46.900
So we could do the exact
same exercise.

00:08:46.900 --> 00:08:47.660
Let me write it this way.

00:08:47.660 --> 00:08:51.090
The nullspace of
A transpose-- A

00:08:51.090 --> 00:08:53.480
transpose is a 3 by 2 matrix.

00:08:53.480 --> 00:08:57.350
So it's equal to all of
the vectors, x, that

00:08:57.350 --> 00:09:00.540
are members of R2.

00:09:00.540 --> 00:09:03.040
Not R3 anymore-- because now we
are taking the transpose's

00:09:03.040 --> 00:09:08.810
nullspace-- such that A
transpose times R vectors are

00:09:08.810 --> 00:09:10.880
equal to the 0 vector in R3.

00:09:15.210 --> 00:09:17.930
And we can do that the same
exact way we did before.

00:09:17.930 --> 00:09:20.230
We set up an augmented matrix.

00:09:20.230 --> 00:09:23.740
We could just put it in reduced
row echelon form and

00:09:23.740 --> 00:09:24.770
set them all equal to 0.

00:09:24.770 --> 00:09:25.980
So let's just do that.

00:09:25.980 --> 00:09:29.480
So if we-- let me just put it
in reduced row echelon form.

00:09:29.480 --> 00:09:35.460
So let me divide my
first row by 2.

00:09:35.460 --> 00:09:39.460
Let's divide the
first row by 2.

00:09:39.460 --> 00:09:40.770
I'm just going to put it in
reduced row echelon form.

00:09:40.770 --> 00:09:44.760
The first row divided
by 2 is 1 minus 2.

00:09:44.760 --> 00:09:48.090
And then the second row, let
me divide it-- let me just,

00:09:48.090 --> 00:09:51.900
I'll just keep it the same--
so minus 1, 2.

00:09:51.900 --> 00:09:54.460
And then this last row,
let me divide it by 3.

00:09:54.460 --> 00:09:57.490
So it becomes minus 1 and 2.

00:09:57.490 --> 00:10:00.420
And now, let me keep my
first row the same.

00:10:00.420 --> 00:10:02.910
1 minus 2.

00:10:02.910 --> 00:10:05.190
And now let me replace my second
row with my second row

00:10:05.190 --> 00:10:06.050
plus my first row.

00:10:06.050 --> 00:10:07.480
So minus 1 plus 1 is 0.

00:10:07.480 --> 00:10:09.440
2 plus minus 2 is 0.

00:10:09.440 --> 00:10:10.830
You get some 0's.

00:10:10.830 --> 00:10:12.160
I'm going to do the same thing
with the third row.

00:10:12.160 --> 00:10:14.420
Replace it with it plus
the first row.

00:10:14.420 --> 00:10:16.800
Once again you're going
to get some 0's.

00:10:16.800 --> 00:10:19.840
So this is the reduced row
echelon form of A transpose.

00:10:24.830 --> 00:10:28.190
And its nullspace is the same
as A transpose's nullspace.

00:10:30.990 --> 00:10:36.770
We could say, to find this
nullspace we can find all of

00:10:36.770 --> 00:10:42.740
the solutions to this equation
times the vectors x1 and x2 is

00:10:42.740 --> 00:10:50.060
equal to 0, 0, and 0.

00:10:50.060 --> 00:10:50.830
These aren't vectors.

00:10:50.830 --> 00:10:52.650
These are just entries
right here.

00:10:52.650 --> 00:10:54.830
0, 0, 0.

00:10:54.830 --> 00:10:57.190
So these two lines give
us no information, but

00:10:57.190 --> 00:10:58.140
this first one does.

00:10:58.140 --> 00:11:00.620
So we get 1 times x1-- and
notice, this is the pivot

00:11:00.620 --> 00:11:02.580
column right here.

00:11:02.580 --> 00:11:03.640
It's associated.

00:11:03.640 --> 00:11:06.930
So x1 is going to be
a pivot variable.

00:11:06.930 --> 00:11:08.530
x2 will be a free variable.

00:11:08.530 --> 00:11:10.390
And just to be clear that
the first column

00:11:10.390 --> 00:11:11.260
is our pivot column.

00:11:11.260 --> 00:11:14.420
So if we go back to A transpose,
it's this first

00:11:14.420 --> 00:11:17.470
column here that is associated
with the pivot column.

00:11:17.470 --> 00:11:21.770
So when we talk about its
columnspace, this by itself

00:11:21.770 --> 00:11:23.660
will span the columnspace.

00:11:23.660 --> 00:11:25.310
This is all a review of
what we did before.

00:11:25.310 --> 00:11:28.190
We're just applying it
to the transpose.

00:11:28.190 --> 00:11:30.600
Let's go back to
our nullspace.

00:11:30.600 --> 00:11:37.610
So this tells us that 1 times
x1, so x1, minus 2 times x2,

00:11:37.610 --> 00:11:41.030
is equal to 0.

00:11:41.030 --> 00:11:44.830
Or we could say that x1 is
equal to 2 times x2.

00:11:44.830 --> 00:11:48.620
So all of the vectors in R2 that
satisfy these conditions

00:11:48.620 --> 00:11:52.190
with these entries will be in
the nullspace of A transpose.

00:11:52.190 --> 00:11:53.440
Let me write it this way.

00:11:57.010 --> 00:12:02.700
So the nullspace of A is going
to be the set of all the

00:12:02.700 --> 00:12:05.930
vectors-- let me write it here--
the set of all the

00:12:05.930 --> 00:12:12.130
vectors, x1, x2, that are member
of R2, clearly, such

00:12:12.130 --> 00:12:19.670
that x1, x2 is going to be
equal to-- well, our free

00:12:19.670 --> 00:12:26.800
variable is x2-- so it's
x2 times the vector.

00:12:26.800 --> 00:12:28.860
So x1 has to be 2 times x2.

00:12:28.860 --> 00:12:34.140
And obviously x2-- that's a 2--
is going to be 1 times x2.

00:12:34.140 --> 00:12:35.710
So what is this going to be?

00:12:35.710 --> 00:12:38.100
Well this is all of the linear
combinations of this vector

00:12:38.100 --> 00:12:40.080
right here.

00:12:40.080 --> 00:12:50.460
So we could say it's equal to
the span of our vector 2, 1.

00:12:50.460 --> 00:12:51.540
Now, that's the nullspace.

00:12:51.540 --> 00:12:54.200
Sorry, this was the nullspace
of A transpose.

00:12:54.200 --> 00:12:56.090
I have to be very
careful there.

00:12:56.090 --> 00:12:57.970
Now what is the columnspace?

00:12:57.970 --> 00:13:01.210
The columnspace of A tranpose?

00:13:01.210 --> 00:13:05.690
Well, the columnspace of A
transpose is the set of all

00:13:05.690 --> 00:13:07.795
vectors spanned by
the columns of A.

00:13:07.795 --> 00:13:10.590
So you could just say the span
of this column vector and this

00:13:10.590 --> 00:13:11.570
column vector.

00:13:11.570 --> 00:13:14.020
But we know, when we put it into
reduced row echelon form,

00:13:14.020 --> 00:13:17.620
only this column vector
was associated

00:13:17.620 --> 00:13:20.210
with a pivot column.

00:13:20.210 --> 00:13:22.750
So this by itself, this
guy is a linear

00:13:22.750 --> 00:13:23.640
combination of this guy.

00:13:23.640 --> 00:13:26.690
If you multiply him by
minus 2, you get

00:13:26.690 --> 00:13:27.523
that guy right there.

00:13:27.523 --> 00:13:29.960
So it's consistent with
everything we've learned.

00:13:29.960 --> 00:13:37.610
So it equals the span of just
this guy right here.

00:13:37.610 --> 00:13:47.380
Of just the vector 2, minus
1, and minus 3.

00:13:47.380 --> 00:13:50.620
That's just a nice, neat
exercise that we did.

00:13:50.620 --> 00:13:54.260
Notice that your span here,
it's in R3, but it's just

00:13:54.260 --> 00:13:57.190
going to be a line in R3.

00:13:57.190 --> 00:14:00.490
Maybe in the next video I'll
do a more graphical

00:14:00.490 --> 00:14:02.130
representation of it.

00:14:02.130 --> 00:14:06.700
But I did this whole exercise to
introduce you to the ideas

00:14:06.700 --> 00:14:09.760
of the nullspace of your
transpose and the columnspace

00:14:09.760 --> 00:14:10.900
of your transpose.

00:14:10.900 --> 00:14:12.140
Think about what the
columnspace of

00:14:12.140 --> 00:14:13.580
your transpose is.

00:14:13.580 --> 00:14:19.390
It's the subspace spanned by
that vector-- sorry-- spanned

00:14:19.390 --> 00:14:21.810
by this vector and
that vector.

00:14:21.810 --> 00:14:24.880
And it turns out that this guy
is a multiple of that guy.

00:14:24.880 --> 00:14:26.890
So we could say just
by that guy.

00:14:26.890 --> 00:14:31.370
But these were the rows of
our original matrix, A.

00:14:31.370 --> 00:14:40.080
So we could also view this as
the span of the row vectors of

00:14:40.080 --> 00:14:41.420
our original guy.

00:14:41.420 --> 00:14:46.190
This is that column that is the
basis for the column span

00:14:46.190 --> 00:14:48.410
of the R transpose matrix.

00:14:48.410 --> 00:14:51.250
And of course this guy was a
linear combination of that.

00:14:51.250 --> 00:14:54.600
So we could also view
the column span of

00:14:54.600 --> 00:14:56.750
our transpose matrix.

00:14:56.750 --> 00:14:58.570
It's equivalent to the subspace

00:14:58.570 --> 00:15:01.010
spanned by these rows.

00:15:01.010 --> 00:15:04.320
Or we could call that
the row space of A.

00:15:04.320 --> 00:15:06.750
Let me write that down.

00:15:06.750 --> 00:15:09.110
So the columnspace of
A transpose-- and

00:15:09.110 --> 00:15:10.000
this is just general.

00:15:10.000 --> 00:15:10.840
Let me write this generally.

00:15:10.840 --> 00:15:13.870
It doesn't just apply
to this example.

00:15:13.870 --> 00:15:18.310
So the columnspace of the
transpose of any matrix, this

00:15:18.310 --> 00:15:24.030
is called the rowspace of A.

00:15:24.030 --> 00:15:26.370
And it's a very natural name.

00:15:26.370 --> 00:15:29.850
Because if A's got a bunch of
rows, we could call them the

00:15:29.850 --> 00:15:31.360
transpose of some vectors.

00:15:31.360 --> 00:15:33.150
So that's first row.

00:15:33.150 --> 00:15:34.390
You got the second row.

00:15:34.390 --> 00:15:38.430
All the way to, maybe,
the nth row.

00:15:38.430 --> 00:15:39.290
Just like that.

00:15:39.290 --> 00:15:40.530
These are vector transposes.

00:15:40.530 --> 00:15:42.890
They're really just rows.

00:15:42.890 --> 00:15:45.680
If you imagine the space
that's spanned by these

00:15:45.680 --> 00:15:48.830
vectors, by the different rows,
that's essentially the

00:15:48.830 --> 00:15:50.320
columnspace of the transpose.

00:15:50.320 --> 00:15:51.660
Because when you transpose
it, each of

00:15:51.660 --> 00:15:53.550
these guys become columns.

00:15:53.550 --> 00:15:56.210
So that's what the
rowspace is.

00:15:56.210 --> 00:16:01.100
Now, the nullspace of our
transpose-- let's write it

00:16:01.100 --> 00:16:04.430
like this-- it was all
of the vectors x that

00:16:04.430 --> 00:16:06.770
satisfied this equation.

00:16:06.770 --> 00:16:11.870
Equals the 0 vector
right there.

00:16:11.870 --> 00:16:18.460
Now, what happens if we take the
transpose of both sides of

00:16:18.460 --> 00:16:19.710
this equation?

00:16:24.360 --> 00:16:26.900
Well, we've learned from our
transpose properties, this is

00:16:26.900 --> 00:16:30.375
equal to the reverse product of
each of those transposes.

00:16:30.375 --> 00:16:33.950
So this is going to be equal
to, this is a vector, the

00:16:33.950 --> 00:16:38.890
vector x transpose.

00:16:38.890 --> 00:16:41.260
If this is a column vector
before, now it's going to

00:16:41.260 --> 00:16:42.830
become a row vector.

00:16:42.830 --> 00:16:47.610
And then, times A transpose
transpose.

00:16:47.610 --> 00:16:49.210
And that's going to
be equal to the

00:16:49.210 --> 00:16:52.300
transpose of the 0 vector.

00:16:52.300 --> 00:16:56.130
Or, we could just write
this like this.

00:16:56.130 --> 00:16:59.910
We could write this as some
matrix-- well let me just

00:16:59.910 --> 00:17:01.070
write it like this.

00:17:01.070 --> 00:17:06.130
Some column vector x-- what's
the transpose of A transpose?

00:17:06.130 --> 00:17:09.290
Well that's just equal to A.

00:17:09.290 --> 00:17:11.940
So you take the transpose
of this column vector.

00:17:11.940 --> 00:17:13.000
You now get a row vector.

00:17:13.000 --> 00:17:16.079
You could view it as a
matrix if you want.

00:17:16.079 --> 00:17:21.790
If this was a member of Rn, this
is now going to be an 1

00:17:21.790 --> 00:17:23.349
by n matrix.

00:17:23.349 --> 00:17:26.170
If this was a member of Rn.

00:17:31.550 --> 00:17:32.930
We kind of switched
the orders.

00:17:32.930 --> 00:17:35.990
And we multiply it times the
transpose of the transpose.

00:17:35.990 --> 00:17:37.580
We just get the matrix, A.

00:17:37.580 --> 00:17:41.230
And we set that equal to the
transpose of the 0 vector.

00:17:41.230 --> 00:17:44.080
Now this is interesting.

00:17:44.080 --> 00:17:47.290
We now have it in terms of
our original matrix, A.

00:17:47.290 --> 00:17:49.540
Now what did the nullspace of
our matrix, A look like?

00:17:49.540 --> 00:17:52.140
The nullspace where all of the
vectors x that satisfied this

00:17:52.140 --> 00:17:54.180
equation is equal to 0.

00:17:54.180 --> 00:17:56.170
So the x was on the right.

00:17:56.170 --> 00:17:59.230
So the nullspace is all the
x's that satisfy this.

00:17:59.230 --> 00:18:03.780
The nullspace of our transpose
is all of the x's that satisfy

00:18:03.780 --> 00:18:05.030
this equation.

00:18:07.610 --> 00:18:14.260
So let me say the set of all
of the x's such that A

00:18:14.260 --> 00:18:18.560
transpose times x
is equal to 0.

00:18:18.560 --> 00:18:20.730
That is the nullspace
of A transpose.

00:18:24.380 --> 00:18:28.700
Or we could also write this as
the set of all of the x's such

00:18:28.700 --> 00:18:33.620
that the transpose of our x
times A is equal to the

00:18:33.620 --> 00:18:35.970
transpose of the 0 vector.

00:18:35.970 --> 00:18:37.880
And we have another
name for this.

00:18:37.880 --> 00:18:48.390
This is called the left
nullspace of A.

00:18:48.390 --> 00:18:50.490
Why is it called the
left nullspace?

00:18:50.490 --> 00:18:52.270
Because now we have
x on our left.

00:18:52.270 --> 00:18:54.920
In just a regular nullspace
you have x on the right.

00:18:54.920 --> 00:18:59.660
But now, if you take the
nullspace of the transpose,

00:18:59.660 --> 00:19:02.690
using just our transpose
properties, that's equivalent

00:19:02.690 --> 00:19:04.890
to this transpose vector
right here.

00:19:04.890 --> 00:19:05.800
Actually let me write this.

00:19:05.800 --> 00:19:07.380
The transpose right there.

00:19:07.380 --> 00:19:10.140
This transpose vector
multiplying A from the

00:19:10.140 --> 00:19:11.080
left-hand side.

00:19:11.080 --> 00:19:14.360
So all of the x's that satisfy
this is the left nullspace.

00:19:14.360 --> 00:19:17.730
And it's going to be different
than your nullspace.

00:19:17.730 --> 00:19:21.400
Notice, your nullspace of A
transpose was the span of this

00:19:21.400 --> 00:19:22.750
right here.

00:19:22.750 --> 00:19:31.040
This is also the left
nullspace of A.

00:19:31.040 --> 00:19:33.690
Now what was just the regular
nullspace of A?

00:19:33.690 --> 00:19:38.500
The regular nullspace of A was
essentially a plane in R3.

00:19:38.500 --> 00:19:41.150
That's the nullspace of A.

00:19:41.150 --> 00:19:48.070
The left nullspace of A
is just a line in R2.

00:19:48.070 --> 00:19:51.320
Very different things.

00:19:51.320 --> 00:19:57.330
And if you go to the rowspace,
what is the rowspace of A?

00:19:57.330 --> 00:20:01.570
The rowspace of A
is a line in R3.

00:20:01.570 --> 00:20:03.980
The rowspace of A
is a line in R3.

00:20:03.980 --> 00:20:06.590
Well what is the columnspace
of A?

00:20:06.590 --> 00:20:12.700
The columnspace of A, right
here, where did I have it?

00:20:12.700 --> 00:20:16.100
Well, this is the only linearly
independent vector.

00:20:16.100 --> 00:20:18.600
It was essentially
a line in R2.

00:20:18.600 --> 00:20:20.290
So they're all very
different things.

00:20:20.290 --> 00:20:23.450
And we'll study a little bit
more how they're all related.

00:20:23.450 --> 00:20:26.330
Now there's one thing I
want to relate to you.

00:20:26.330 --> 00:20:29.980
We figured out that the
rank of this vector

00:20:29.980 --> 00:20:32.290
right here is 1.

00:20:32.290 --> 00:20:36.210
Because when you put it in
reduced row echelon form there

00:20:36.210 --> 00:20:40.130
was one pivot column.

00:20:40.130 --> 00:20:43.780
And the basis vectors are those
associated with that

00:20:43.780 --> 00:20:44.610
pivot column.

00:20:44.610 --> 00:20:47.400
And if you count your basis
vectors, that's your dimension

00:20:47.400 --> 00:20:48.180
of your space.

00:20:48.180 --> 00:20:50.870
So the dimension of your
columnspace is 1.

00:20:50.870 --> 00:20:52.760
And that's the same thing
as your rank.

00:20:52.760 --> 00:20:55.620
Now what is the rank
of A transpose?

00:20:55.620 --> 00:20:58.460
The rank of A transpose in the
example, when you put it in

00:20:58.460 --> 00:21:02.260
reduced row echelon form, you
got one linearly independent

00:21:02.260 --> 00:21:03.050
column vector.

00:21:03.050 --> 00:21:05.770
So the basis for our
columnspace was

00:21:05.770 --> 00:21:07.500
also equal to 1.

00:21:07.500 --> 00:21:09.840
And in general, that's always
going to be the case.

00:21:12.860 --> 00:21:17.340
That the rank of A, which
is the dimension of its

00:21:17.340 --> 00:21:22.590
columnspace, is equal to the
rank of A transpose.

00:21:22.590 --> 00:21:25.360
And if you think about it,
it makes a lot of sense.

00:21:25.360 --> 00:21:28.140
To figure out the rank of A, you
essentially figure out how

00:21:28.140 --> 00:21:34.200
many pivot columns they have.
Or another way to say it is

00:21:34.200 --> 00:21:45.030
how many pivot entries they
have. When you want to find

00:21:45.030 --> 00:21:51.760
the rank of your transpose
vector, you're essentially

00:21:51.760 --> 00:21:53.810
just saying-- and I know this is
maybe getting a little bit

00:21:53.810 --> 00:21:56.150
confusing-- but when you want
the rank of your transpose

00:21:56.150 --> 00:21:59.090
vector, you're saying, how
many of these columns are

00:21:59.090 --> 00:22:00.080
linearly independent?

00:22:00.080 --> 00:22:02.880
Or which of these are linearly
independent?

00:22:02.880 --> 00:22:05.730
And that's the same question
as saying, how many of your

00:22:05.730 --> 00:22:08.250
rows up here are linearly
independent?

00:22:08.250 --> 00:22:10.410
If you want to know how many
columns in your transpose are

00:22:10.410 --> 00:22:13.320
linearly independent, that's
equivalent to asking how many

00:22:13.320 --> 00:22:16.300
rows in your original matrix
are linearly independent.

00:22:16.300 --> 00:22:19.770
And when you put this matrix in
reduced row echelon form,

00:22:19.770 --> 00:22:22.180
everything in reduced row
echelon form are just row

00:22:22.180 --> 00:22:22.810
operations.

00:22:22.810 --> 00:22:24.240
So they are just linear
combinations of

00:22:24.240 --> 00:22:25.560
these things up here.

00:22:25.560 --> 00:22:27.130
Or you could go vice versa.

00:22:27.130 --> 00:22:29.680
Everything up here is just
linear combinations of your

00:22:29.680 --> 00:22:31.830
matrix in reduced row
echelon form.

00:22:31.830 --> 00:22:36.940
So if you only have one pivot
entry, then this guy right

00:22:36.940 --> 00:22:41.560
here, by himself, or one pivot
row, that guy by himself can

00:22:41.560 --> 00:22:46.780
represent a basis for
your rowspace.

00:22:46.780 --> 00:22:49.480
Or, all of your rows can be
represented by a linear

00:22:49.480 --> 00:22:51.570
combination of your
pivot rows.

00:22:51.570 --> 00:22:53.570
And because of that, you
just count that.

00:22:53.570 --> 00:22:55.530
You say, OK there's
one in this case.

00:22:55.530 --> 00:22:59.190
So the dimension of
my rowspace is 1.

00:22:59.190 --> 00:23:05.980
And that's the same thing
as the dimension of my

00:23:05.980 --> 00:23:07.430
transpose's columnspace.

00:23:07.430 --> 00:23:08.890
I know it's getting all
confusing and it's late in the

00:23:08.890 --> 00:23:10.270
day for me as well.

00:23:10.270 --> 00:23:13.820
So that, hopefully, will
convince you that the rank of

00:23:13.820 --> 00:23:16.280
our transpose is the
same as the rank of

00:23:16.280 --> 00:23:17.950
our original matrix.


Alternate Solution to Ratio Problem (HD Version)

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.930
It's always good to see the
same problem done more than one

00:00:03.930 --> 00:00:06.430
way, and it's especially true
with these ratio problems

00:00:06.430 --> 00:00:09.360
because they, in general, can
be done in many, many different

00:00:09.360 --> 00:00:12.810
ways, and different types of
ways gel with different people.

00:00:12.810 --> 00:00:14.900
So I just did a video on
this where I said that the

00:00:14.900 --> 00:00:19.720
ratio of apples to oranges
starts off at 5 to 8.

00:00:19.720 --> 00:00:21.580
And we take away 15.

00:00:21.580 --> 00:00:25.870
We take away 15 apples, and
the ratio becomes 1 to 4.

00:00:25.870 --> 00:00:28.220
And then the question that we
have to answer is how much

00:00:28.220 --> 00:00:32.060
total fruit do we have after
taking away the 15 apples?

00:00:32.060 --> 00:00:32.860
And we figured it out.

00:00:32.860 --> 00:00:34.990
We figured out that it's
50 pieces of fruit.

00:00:34.990 --> 00:00:36.490
But let's see if we can
do it a different way.

00:00:36.490 --> 00:00:38.960
The last time I kind of
talked through it and

00:00:38.960 --> 00:00:39.790
used a lot of words.

00:00:39.790 --> 00:00:42.620
Let see if we can do it a
little bit more algebraically.

00:00:42.620 --> 00:00:54.890
So let a is equal to the number
of apples we are starting with,

00:00:54.890 --> 00:00:59.390
and let o equal the
number of oranges.

00:00:59.390 --> 00:01:01.360
That doesn't change throughout
the whole problem.

00:01:05.610 --> 00:01:09.190
So we're starting off with
a apples and o oranges.

00:01:09.190 --> 00:01:15.470
So the ratio of a to
o is equal to what?

00:01:15.470 --> 00:01:19.160
The ratio of a-- let me
actually-- well, instead of

00:01:19.160 --> 00:01:21.100
trying to fit this whole
problem into this part of the

00:01:21.100 --> 00:01:23.650
screen, let me just copy and
paste this part, and do it at

00:01:23.650 --> 00:01:25.730
the bottom part of the screen.

00:01:25.730 --> 00:01:29.710
Let me copy it, and then let me
scroll down here where I have

00:01:29.710 --> 00:01:32.380
some extra real estate because
I think I'm going to need more

00:01:32.380 --> 00:01:35.020
than that little pocket
of area right there.

00:01:35.020 --> 00:01:37.483
So I copied and pasted
just like that.

00:01:37.483 --> 00:01:38.920
OK, so there we have it.

00:01:38.920 --> 00:01:41.370
a is the number of apples
to start with, and o is

00:01:41.370 --> 00:01:43.490
the number of oranges.

00:01:43.490 --> 00:01:45.930
So we could write-- let
me do it in a color that

00:01:45.930 --> 00:01:46.900
I haven't used before.

00:01:46.900 --> 00:01:48.420
I actually haven't used white.

00:01:48.420 --> 00:01:50.960
So we can say that the
ratio of apples to oranges

00:01:50.960 --> 00:01:54.180
is equal to 5 to 8.

00:01:54.180 --> 00:01:56.680
That's what they tell us, that
the number of apples we start

00:01:56.680 --> 00:01:59.370
with divided by the number of
oranges, or the ratio of them--

00:01:59.370 --> 00:02:02.260
i have to be a little bit
more particular-- is 5 to 8.

00:02:02.260 --> 00:02:04.400
Then we removed 15 apples.

00:02:04.400 --> 00:02:06.890
So if we remove 15 apples,
how many apples do we have?

00:02:06.890 --> 00:02:10.290
We have a minus 15 apples.

00:02:10.290 --> 00:02:13.530
And they tell us that the
ratio-- then the ratio of

00:02:13.530 --> 00:02:16.620
apples to oranges
becomes 1 to 4.

00:02:16.620 --> 00:02:19.540
So a minus 15, which is the
number of apples we start with

00:02:19.540 --> 00:02:25.680
minus 15, the ratio of that to
oranges is equal to 1 over 4.

00:02:25.680 --> 00:02:28.970
And let's see if we can
solve these two equations.

00:02:28.970 --> 00:02:32.050
I have two equations here,
and I have two unknowns.

00:02:32.050 --> 00:02:33.130
Let's see what we can do here.

00:02:33.130 --> 00:02:35.880
And I should probably shouldn't
have labeled-- there's a reason

00:02:35.880 --> 00:02:38.370
why o isn't used in algebra a
lot, because it looks just

00:02:38.370 --> 00:02:40.730
like a zero, but I'll
try to be careful.

00:02:40.730 --> 00:02:42.450
So this first-- let
me write down my two

00:02:42.450 --> 00:02:43.500
equations over here.

00:02:43.500 --> 00:02:45.760
It never hurts to rewrite it.

00:02:45.760 --> 00:02:53.250
The original apples to
oranges ratio is 5 to 8.

00:02:53.250 --> 00:02:57.730
Then when we remove 15 apples,
the ratio of our new number of

00:02:57.730 --> 00:03:00.970
apples to oranges is
equal to 1 over 4.

00:03:00.970 --> 00:03:06.070
This is just an algebraic
way of stating our problem.

00:03:06.070 --> 00:03:07.620
Let's see if we can solve this.

00:03:07.620 --> 00:03:10.230
If we cross-multiply both
sides of this equation,

00:03:10.230 --> 00:03:14.610
we get 8a is equal to 5o.

00:03:14.610 --> 00:03:15.670
That's not a zero.

00:03:15.670 --> 00:03:17.420
It's equal to 5o.

00:03:17.420 --> 00:03:21.480
Or if we subtract 5o
from both sides, get 8a

00:03:21.480 --> 00:03:25.600
minus 5o is equal to 0.

00:03:25.600 --> 00:03:28.590
So this is just a linear
equation with two unknowns.

00:03:28.590 --> 00:03:30.295
Now, what do we get on
this right-hand side?

00:03:30.295 --> 00:03:35.500
If we cross-multiply,
we get 4a minus 60.

00:03:35.500 --> 00:03:38.710
I'm just multiplying
4 times a minus 15.

00:03:38.710 --> 00:03:42.990
That is equal to 1 times o,
or we could just write--

00:03:42.990 --> 00:03:44.600
well, I'll write 1o.

00:03:44.600 --> 00:03:45.320
That's not a 10.

00:03:45.320 --> 00:03:47.990
That's 1o, or 1 orange.

00:03:47.990 --> 00:03:51.120
And if we rearrange this, if
we subtract the 1o from both

00:03:51.120 --> 00:03:57.470
sides, we get 4a minus 1o
minus 60 is equal to 0.

00:03:57.470 --> 00:04:01.570
Or if we add 60 to both
sides, we get 4a minus

00:04:01.570 --> 00:04:05.320
1o is equal to 60.

00:04:05.320 --> 00:04:07.570
And here, you might recognize
that these are just two

00:04:07.570 --> 00:04:08.880
linear equations right here.

00:04:08.880 --> 00:04:11.830
That's one of them, and
that's the second of them.

00:04:11.830 --> 00:04:13.250
And I have two unknowns.

00:04:13.250 --> 00:04:14.620
I have a and o.

00:04:14.620 --> 00:04:18.299
I probably should have called
them x and y to prevent this o

00:04:18.299 --> 00:04:20.540
from looking so confusing
because that's a zero

00:04:20.540 --> 00:04:21.260
and that's an o.

00:04:21.260 --> 00:04:23.890
But I think you get
what I'm saying.

00:04:23.890 --> 00:04:25.600
So how do we solve this?

00:04:25.600 --> 00:04:28.390
Well, what we can do is we can
multiply one of these times a

00:04:28.390 --> 00:04:31.520
convenient factor, and then
add them or subtract them to

00:04:31.520 --> 00:04:35.400
get rid of one of the two
variables, so let's do that.

00:04:35.400 --> 00:04:39.030
Let's multiply this equation
times minus 2, so it becomes

00:04:39.030 --> 00:04:45.520
minus 8a plus 2o is
equal to minus 120.

00:04:45.520 --> 00:04:50.002
I just multiplied this equation
right here times minus 2.

00:04:50.002 --> 00:04:53.215
If I do that and then if I add
it to this equation right here,

00:04:53.215 --> 00:04:59.920
this equation is 8a minus 5o
is equal to 0, what do I get?

00:04:59.920 --> 00:05:02.300
Let's just add these
two equations.

00:05:02.300 --> 00:05:04.130
This becomes 0a.

00:05:04.130 --> 00:05:11.530
Minus 5 plus 2 is minus 3o is
equal to minus 120, or o is

00:05:11.530 --> 00:05:15.040
equal to-- you divide both
sides-- you can ignore this.

00:05:15.040 --> 00:05:21.200
You divide both sides by minus
3, you get o is equal to 40.

00:05:21.200 --> 00:05:21.900
So let's see.

00:05:21.900 --> 00:05:26.110
So the oranges is equal to 40.

00:05:26.110 --> 00:05:27.570
And that's before and after.

00:05:27.570 --> 00:05:28.960
The o never changes.

00:05:28.960 --> 00:05:31.000
o is equal to 40.

00:05:31.000 --> 00:05:32.860
So what are our
apples equal to?

00:05:32.860 --> 00:05:34.385
We could go to either
of these equations.

00:05:34.385 --> 00:05:36.150
We could go to any of
these that we have.

00:05:36.150 --> 00:05:38.260
Maybe we can go to this one.

00:05:38.260 --> 00:05:44.280
So we have 8 times our number
of apples is equal to 5 times

00:05:44.280 --> 00:05:50.550
the number of oranges,
or 5 times 40.

00:05:50.550 --> 00:05:55.440
8 times the number of apples is
equal to 200, or the apples is

00:05:55.440 --> 00:06:01.800
equal to 200 over 8, which
is equal to 100 over 4,

00:06:01.800 --> 00:06:04.390
which is equal to 25.

00:06:04.390 --> 00:06:05.990
So remember what a and o are.

00:06:05.990 --> 00:06:09.320
O is the number of oranges we
have throughout the problem.

00:06:09.320 --> 00:06:11.930
a is the number of
apples we start with.

00:06:11.930 --> 00:06:13.190
That's what we start with.

00:06:15.870 --> 00:06:18.790
So you can say that we start
with 25 apples, 40 oranges,

00:06:18.790 --> 00:06:25.375
or we start with 65
pieces of fruit.

00:06:27.950 --> 00:06:32.480
Then if you remove 15 fruit,
you'll end up with 50 fruit,

00:06:32.480 --> 00:06:35.750
which is the exact same answer
we got the other way of doing

00:06:35.750 --> 00:06:38.415
this problem, and it's nice
to see it done two ways.

00:06:38.415 --> 00:06:40.700
If you want to know the number
of apples we end up with,

00:06:40.700 --> 00:06:43.000
we started with 25 apples.

00:06:43.000 --> 00:06:46.040
If you were to take 15 away
from them, you're going

00:06:46.040 --> 00:06:48.330
to get 10 apples.

00:06:48.330 --> 00:06:52.070
So you start with 25 apples,
40 oranges, 65 fruit.

00:06:52.070 --> 00:06:56.250
You end with 10 apples,
40 oranges, or 50 fruit.

00:06:56.250 --> 00:06:56.990
Either way.

00:06:56.990 --> 00:06:59.730
Hopefully, it was reasonably
useful to see this

00:06:59.730 --> 00:07:01.830
done another way.


More advanced ratio problem--with Algebra (HD version)

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:04.150
Let's do another ratio problem
that's a bit of a classic.

00:00:04.150 --> 00:00:08.300
So, let's say that I start off
with a certain number of fruit.

00:00:08.300 --> 00:00:10.430
So, I have some
number of fruit.

00:00:10.430 --> 00:00:11.800
But I'll get back to
that in a second.

00:00:11.800 --> 00:00:13.080
But we know that the ratio.

00:00:13.080 --> 00:00:19.070
We know that the ratio -- we
know that the ratio of apples

00:00:19.070 --> 00:00:23.080
-- I'll do apples in
this reddish color.

00:00:23.080 --> 00:00:28.840
The ratio of apples to oranges.

00:00:28.840 --> 00:00:31.930
To oranges, those colors
are very similar.

00:00:31.930 --> 00:00:37.570
Let's say, it is equal to --
let's say it is equal to 5:8.

00:00:37.570 --> 00:00:41.390
So it's equal to 5:8.

00:00:41.390 --> 00:00:42.100
5:8.

00:00:42.100 --> 00:00:45.260
Or I could also write it
as 5:8, just like that.

00:00:45.260 --> 00:00:48.670
That's the ratio of
apples to oranges.

00:00:48.670 --> 00:00:51.890
Now, the next thing that I know
about this problem is that when

00:00:51.890 --> 00:00:56.320
I take away 15 -- when
I take away 15 apples.

00:00:56.320 --> 00:01:13.810
So, take away 15 apples,
and the ratio becomes 1:4.

00:01:13.810 --> 00:01:19.320
The ratio of apples to
oranges becomes 1:4.

00:01:19.320 --> 00:01:24.080
How many apples -- let
me write it this way.

00:01:24.080 --> 00:01:45.320
How much total fruit do
I have after taking

00:01:45.320 --> 00:01:51.860
away the 15 apples?

00:01:51.860 --> 00:01:54.750
The 15 apples?

00:01:54.750 --> 00:01:57.790
How much total fruit do I have?

00:01:57.790 --> 00:02:00.280
So let's do a little
bit of algebra.

00:02:00.280 --> 00:02:07.660
Let's say that x is the number,
x is equal to the number of

00:02:07.660 --> 00:02:11.005
fruit that we start with.

00:02:15.110 --> 00:02:17.050
How much fruit are we going
to end up with at the end?

00:02:17.050 --> 00:02:18.220
We're taking away 15 apples.

00:02:18.220 --> 00:02:20.280
We're not doing anything
to the oranges.

00:02:20.280 --> 00:02:23.440
So we know that x minus 30,
I'll do that in a different

00:02:23.440 --> 00:02:31.600
color, x minus 15, we take away
15 apples, x minus 15 is equal

00:02:31.600 --> 00:02:38.595
to the number of fruit
we end up with.

00:02:41.690 --> 00:02:43.660
We know these two
things are true.

00:02:43.660 --> 00:02:47.790
And now, we know that we start
off with a ratio of 5:8.

00:02:47.790 --> 00:02:52.560
So we know that the ratio of
apples to oranges is equal to

00:02:52.560 --> 00:02:54.840
5:8 that we start off with.

00:02:54.840 --> 00:02:58.660
Or, that's just another way of
saying that for every group --

00:02:58.660 --> 00:03:05.020
assuming that we can divide our
group of fruit into even -- or

00:03:05.020 --> 00:03:07.270
we could divide it into a set
of groups, and every group

00:03:07.270 --> 00:03:09.350
has the same number of
apples and oranges.

00:03:09.350 --> 00:03:23.210
That means that for every group
of, what, 13 fruits, 13 fruits,

00:03:23.210 --> 00:03:33.240
there are 5 apples
and 8 oranges.

00:03:37.820 --> 00:03:39.030
That's what this tells me.

00:03:39.030 --> 00:03:41.130
For every group of 13
fruits, there are 5

00:03:41.130 --> 00:03:42.750
apples and 18 oranges.

00:03:42.750 --> 00:03:45.480
And how many groups of
13 fruit are there?

00:03:45.480 --> 00:03:47.630
Well, I have x total fruit.

00:03:47.630 --> 00:03:53.020
So if I divide x divided by 13,
if I take x divided by 13,

00:03:53.020 --> 00:03:56.580
that is equal to -- this
is 13 thirteen right here.

00:03:56.580 --> 00:03:58.240
This is the 3.

00:03:58.240 --> 00:04:00.180
I have 13 fruits per group.

00:04:00.180 --> 00:04:01.850
I have x total fruit.

00:04:01.850 --> 00:04:04.390
So this is the total
number of groups.

00:04:04.390 --> 00:04:09.390
So this is equal to
the number of groups.

00:04:09.390 --> 00:04:12.110
Now, for every group,
I have 5 apples.

00:04:12.110 --> 00:04:14.600
So how many apples am
I starting off with?

00:04:14.600 --> 00:04:15.850
This is the number of groups.

00:04:15.850 --> 00:04:17.720
For every group,
I have 5 apples.

00:04:17.720 --> 00:04:21.730
So I'm starting off with
5 times that apples.

00:04:21.730 --> 00:04:25.510
5x over 13 apples.

00:04:25.510 --> 00:04:27.160
5x over 13 apples.

00:04:27.160 --> 00:04:29.940
Now, for every one of these
groups, I also have 8 oranges.

00:04:29.940 --> 00:04:31.800
So, how many oranges do I have?

00:04:31.800 --> 00:04:34.200
I have 8 times that.

00:04:34.200 --> 00:04:35.840
Because for every one of
these, I have 8 oranges.

00:04:35.840 --> 00:04:40.320
So 8 times that is
8x over 13 oranges.

00:04:40.320 --> 00:04:40.800
Fair enough.

00:04:40.800 --> 00:04:42.500
That's what I start off with.

00:04:42.500 --> 00:04:47.350
Now, when I'm done we said
that the ratio of apples

00:04:47.350 --> 00:04:52.390
to oranges becomes 1:4.

00:04:52.390 --> 00:04:55.080
Becomes 1:4.

00:04:55.080 --> 00:05:01.190
Or, another way of saying that
is that, for every 5 -- right,

00:05:01.190 --> 00:05:05.700
if you add these two numbers
together, for every 5 fruit

00:05:05.700 --> 00:05:15.196
there are 1 apple
and 4 oranges.

00:05:19.500 --> 00:05:20.510
Fair enough.

00:05:20.510 --> 00:05:24.590
So, how many groups of
5 fruit do we have?

00:05:24.590 --> 00:05:26.660
Well, this is the total number
of fruit we're left with

00:05:26.660 --> 00:05:28.080
after we took 15 away.

00:05:28.080 --> 00:05:30.600
So how many groups
of 5 do we have?

00:05:30.600 --> 00:05:32.550
We take the total
number of fruit.

00:05:32.550 --> 00:05:34.250
x minus 15.

00:05:34.250 --> 00:05:37.780
Divide it by how many
fruit we have per group.

00:05:37.780 --> 00:05:40.810
And this is how many groups
of -- this is equal to

00:05:40.810 --> 00:05:45.720
the groups of 5 fruit.

00:05:45.720 --> 00:05:47.250
Of 5 fruit.

00:05:47.250 --> 00:05:49.550
Now, for each of these
groups, I'm going to have

00:05:49.550 --> 00:05:51.790
1 apple and 4 oranges.

00:05:51.790 --> 00:05:55.180
So how many apples and oranges
do I now have after I took

00:05:55.180 --> 00:05:56.730
away 15 of the apples?

00:05:56.730 --> 00:05:59.090
I took away 15 fruit?

00:05:59.090 --> 00:06:01.370
Let's think about
this a little bit.

00:06:01.370 --> 00:06:03.920
How many apples am
I going to have?

00:06:03.920 --> 00:06:06.270
I have 1 apple for every
1 of these groups.

00:06:06.270 --> 00:06:08.350
So, that's how many -- so
the number of groups is the

00:06:08.350 --> 00:06:09.350
number of apples I have.

00:06:09.350 --> 00:06:14.610
So, I'm going to have x
minus 15 over 5 apples.

00:06:14.610 --> 00:06:16.920
And then how many oranges
am I going to have?

00:06:16.920 --> 00:06:19.630
For every one of these groups,
I'm going to have 4 oranges.

00:06:19.630 --> 00:06:21.790
So, I'm going to have 4
times this many oranges.

00:06:21.790 --> 00:06:27.290
So we could write that as 4/5
times x minus 15 oranges.

00:06:30.050 --> 00:06:33.630
That's just 5 times the
number of groups of 5 fruit.

00:06:33.630 --> 00:06:35.390
Now, what doesn't change?

00:06:35.390 --> 00:06:38.340
I'm messing with the number of
apples, but I never changed

00:06:38.340 --> 00:06:40.290
the number of oranges.

00:06:40.290 --> 00:06:42.760
So, this number which is
the number of oranges we

00:06:42.760 --> 00:06:45.660
started off with, has
to equal this number.

00:06:45.660 --> 00:06:47.570
The number of oranges
we ended up with.

00:06:47.570 --> 00:06:49.770
There's many ways to do this
problem, but this is the way

00:06:49.770 --> 00:06:52.122
that's popping out of me that
might be straightforward.

00:06:52.122 --> 00:06:54.230
So let's see if we
can solve for x.

00:06:54.230 --> 00:06:55.960
This has to be equal to that.

00:06:55.960 --> 00:06:57.350
That never changed.

00:06:57.350 --> 00:07:09.790
So you could say 8x over 13 is
equal to 4/5 times x minus 15.

00:07:09.790 --> 00:07:12.590
And, maybe, the best way to do
this, let's multiply both sides

00:07:12.590 --> 00:07:15.530
of this equation times 5/4.

00:07:15.530 --> 00:07:20.690
So, times 5/4 times 5/4.

00:07:20.690 --> 00:07:21.760
These cancel out.

00:07:21.760 --> 00:07:22.660
That just becomes 1.

00:07:22.660 --> 00:07:24.070
What does this become?

00:07:24.070 --> 00:07:26.680
This becomes, you get
a 4, divide it by 4,

00:07:26.680 --> 00:07:27.520
you get a 2 there.

00:07:27.520 --> 00:07:30.860
This becomes 10x over 13.

00:07:30.860 --> 00:07:35.140
10x over 13, that's the
left-thand side, is

00:07:35.140 --> 00:07:39.550
equal to x minus 15.

00:07:39.550 --> 00:07:42.070
Now, let's subtract
x from both sides.

00:07:42.070 --> 00:07:50.400
And we get 10x over 13 minus
x is equal to minus 15.

00:07:50.400 --> 00:07:52.040
Let me scroll down
a little bit.

00:07:52.040 --> 00:07:53.300
And so, what is this?

00:07:53.300 --> 00:08:01.430
This is the same thing as 10x
over 13 minus 13x over 13.

00:08:01.430 --> 00:08:02.570
That's just x.

00:08:02.570 --> 00:08:04.470
Is equal to minus 14.

00:08:04.470 --> 00:08:10.310
And then, 10 minus 13, this is
equal to -- so this is minus 3x

00:08:10.310 --> 00:08:17.180
over 13, is equal to minus 15.

00:08:17.180 --> 00:08:20.420
Now, we can just multiply
both sides of this equation

00:08:20.420 --> 00:08:24.570
by minus 13 over 3.

00:08:24.570 --> 00:08:28.460
Multiply this side
by minus 13 over 3.

00:08:28.460 --> 00:08:31.300
I picked that number so that
this and that cancel out.

00:08:31.300 --> 00:08:34.080
And that and that cancel out,
and the negatives cancel out.

00:08:34.080 --> 00:08:36.740
So I'm just left with
x on that side.

00:08:36.740 --> 00:08:39.080
And then, the right-hand
side becomes what?

00:08:39.080 --> 00:08:41.010
The negatives cancel out.

00:08:41.010 --> 00:08:45.260
I can divide the 15 and the 3
by 3, so that becomes a 5.

00:08:45.260 --> 00:08:47.260
And then this becomes a 1.

00:08:47.260 --> 00:08:49.510
And I'm just left
with 5 times 13.

00:08:49.510 --> 00:08:52.380
This was just, times plus 13, I
wrote that just to make sure

00:08:52.380 --> 00:08:54.740
that you understood that the
negatives canceled out.

00:08:54.740 --> 00:08:56.390
So, 5 times 13 is what?

00:08:56.390 --> 00:09:00.320
5 times 12 is 60, so
x is equal to 65.

00:09:00.320 --> 00:09:02.320
But that is not our answer.

00:09:02.320 --> 00:09:03.955
Remember what x was.

00:09:03.955 --> 00:09:07.970
x was the number of
fruit we started with.

00:09:07.970 --> 00:09:11.820
We just figured out that
this is equal to 65.

00:09:11.820 --> 00:09:14.940
But what the question asked,
the question wanted to know is,

00:09:14.940 --> 00:09:17.700
how much total fruit do I have
after taking away

00:09:17.700 --> 00:09:18.805
the 15 apples.

00:09:18.805 --> 00:09:21.330
So the question wants to
know, how many after

00:09:21.330 --> 00:09:22.800
taking the 15 apples.

00:09:22.800 --> 00:09:26.050
If I started with 65, I give
away 15 of the fruit, all of

00:09:26.050 --> 00:09:29.620
them are apples, then I'm
left with 65 minus 15,

00:09:29.620 --> 00:09:32.590
which is equal to?

00:09:32.590 --> 00:09:37.550
50 pieces of fruit.

00:09:37.550 --> 00:09:40.960
50 pieces of fruit.

00:09:40.960 --> 00:09:42.950
That's what they're asking for.

00:09:42.950 --> 00:09:44.180
50 pieces of fruit.

00:09:44.180 --> 00:09:46.310
If you want, you can go on and
figure out how many apples you

00:09:46.310 --> 00:09:49.060
started with, you finished
with, the number of oranges, or

00:09:49.060 --> 00:09:50.970
whatever else, but this
is what they're asking.

00:09:50.970 --> 00:09:54.060
How much total fruit, not the
number of apples, total fruit,

00:09:54.060 --> 00:09:57.680
do I have after taking
away the 15 apples?


Ratio problem with basic algebra

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.010
Now that have a basic
understanding of what a ratio

00:00:03.010 --> 00:00:06.130
is, let's see if we can tackle
some more advanced problems.

00:00:06.130 --> 00:00:07.250
So, I have a classroom.

00:00:07.250 --> 00:00:09.960
It's got 55 students in it.

00:00:09.960 --> 00:00:12.580
55 students.

00:00:12.580 --> 00:00:18.970
And I also know that the ratio
of boys to girls, the ratio of

00:00:18.970 --> 00:00:25.395
I'll write it this way,
boys to girls is 4:7.

00:00:28.380 --> 00:00:32.510
What I want to know is, how
many girls do I need to add.

00:00:32.510 --> 00:00:44.510
So, how many girls need to add,
or do I need to add, do I need

00:00:44.510 --> 00:00:57.360
to add, for the ratio, for
the ratio to become 1:2.

00:00:57.360 --> 00:00:59.180
1:2.

00:00:59.180 --> 00:01:01.210
And this is the ratio
of boys to girls.

00:01:01.210 --> 00:01:02.805
I'm not switching
the ratio on you.

00:01:02.805 --> 00:01:07.120
So, right now for every
4 boys, I have 7 girls.

00:01:07.120 --> 00:01:09.480
I want to add some girls so
that the ratio becomes for

00:01:09.480 --> 00:01:11.540
every boy, I have 2 girls.

00:01:11.540 --> 00:01:12.880
So how can you do this?

00:01:12.880 --> 00:01:15.620
Well, just going back to the
basic ratio video, the first

00:01:15.620 --> 00:01:18.140
thing we can do we, know the
total number of students.

00:01:18.140 --> 00:01:19.720
And we know the ratio.

00:01:19.720 --> 00:01:22.490
So maybe it'll help us to
figure out the number of boys

00:01:22.490 --> 00:01:24.650
and girls we have right now.

00:01:24.650 --> 00:01:27.200
So our current ratio is 4:7.

00:01:27.200 --> 00:01:33.660
So boys, boys, to girls,
right now is equal to 4:7.

00:01:33.660 --> 00:01:35.590
That means -- what
does that mean?

00:01:35.590 --> 00:01:38.820
That means for every, if
you add up these two.

00:01:38.820 --> 00:01:52.000
For every 11 students there
are 4 boys and 7 girls.

00:01:52.000 --> 00:01:58.945
There are 4 boys, 4
boys and 7 girls.

00:02:01.840 --> 00:02:03.380
That's what this
ratio tells us.

00:02:03.380 --> 00:02:08.170
You give me 11 students of
this -- you could split

00:02:08.170 --> 00:02:10.050
this classroom into
a bunch of groups.

00:02:10.050 --> 00:02:12.550
And for every group of 11,
you're going to have 4 boys

00:02:12.550 --> 00:02:15.440
and 7 girls that's
all that tells me.

00:02:15.440 --> 00:02:17.780
For every 11 students,
4 boys and 7 girls.

00:02:17.780 --> 00:02:20.830
If you split them, every
group has the same number

00:02:20.830 --> 00:02:22.260
of boys and girls.

00:02:22.260 --> 00:02:25.680
Now, how many groups of
11 students there are?

00:02:25.680 --> 00:02:27.510
So there are 55 students.

00:02:27.510 --> 00:02:31.770
We have 55 students total.

00:02:31.770 --> 00:02:34.130
And we're breaking them
into groups of 11.

00:02:34.130 --> 00:02:36.120
And every group will
have the same number

00:02:36.120 --> 00:02:37.660
of boys and of girls.

00:02:37.660 --> 00:02:39.380
So we have 11 groups.

00:02:41.970 --> 00:02:46.540
Sorry, if we divide by
11 students per group.

00:02:46.540 --> 00:02:50.400
Students per group.

00:02:50.400 --> 00:02:52.780
So how many groups of 11
students do we have?

00:02:52.780 --> 00:02:56.700
That means we have 5
groups of 11 students.

00:02:56.700 --> 00:03:04.250
5 groups of 11 students.

00:03:04.250 --> 00:03:06.010
And we've done this
drill before.

00:03:06.010 --> 00:03:08.920
Every group has 4 boys
and it has 7 girls.

00:03:08.920 --> 00:03:10.600
So we have 5 groups.

00:03:10.600 --> 00:03:15.160
5 groups each of them has 4
boys, that means we have 5

00:03:15.160 --> 00:03:19.460
times 4, which is
equal to 20 boys.

00:03:19.460 --> 00:03:24.700
And we have 5 times 7, we
have 5 groups, 5 times 7,

00:03:24.700 --> 00:03:26.390
each group has 7 girls.

00:03:26.390 --> 00:03:27.190
Of girls.

00:03:27.190 --> 00:03:30.740
So that's equal to 35 girls.

00:03:30.740 --> 00:03:31.840
And the numbers add up.

00:03:31.840 --> 00:03:34.000
20 plus 35 is 55.

00:03:34.000 --> 00:03:40.190
And 20 to 35, 20 to
35, is equal to 4:7.

00:03:40.190 --> 00:03:41.420
So it all works out.

00:03:41.420 --> 00:03:43.490
So that's what we
have right now.

00:03:43.490 --> 00:03:45.710
Now, we're going to add to
girls to the classroom

00:03:45.710 --> 00:03:47.475
to change the ratio.

00:03:47.475 --> 00:03:49.470
So let's do this.

00:03:49.470 --> 00:03:53.910
So right now we have 20 boys.

00:03:53.910 --> 00:03:55.360
We have 20 boys.

00:03:55.360 --> 00:03:56.850
And we have 35 girls.

00:03:56.850 --> 00:04:00.130
We just figure that out.

00:04:00.130 --> 00:04:02.590
And I'm going to add some
girls to the classroom.

00:04:02.590 --> 00:04:05.760
So I'm going to add some girls
to the classroom And my new

00:04:05.760 --> 00:04:10.110
ratio, after I add these g
girls to the classroom, my new

00:04:10.110 --> 00:04:13.360
ratio is going to be 1:2.

00:04:13.360 --> 00:04:14.830
1:2.

00:04:14.830 --> 00:04:17.050
So I have an equation
I have one unknown, I

00:04:17.050 --> 00:04:18.150
can just solve for it.

00:04:18.150 --> 00:04:20.185
You can almost do this
one in your head.

00:04:20.185 --> 00:04:24.450
20 is to what, 1 is to
2 as 20 as to what?

00:04:24.450 --> 00:04:28.240
You could say, oh, 20 is -- 1
is 1/2 of 20, 20 is 1/2 of 40.

00:04:28.240 --> 00:04:29.740
So you could say, oh, it's 40.

00:04:29.740 --> 00:04:31.400
So g would be equal to 5.

00:04:31.400 --> 00:04:32.620
That's how you would
solve it in your head.

00:04:32.620 --> 00:04:34.950
If you want to do it
algebraically, you

00:04:34.950 --> 00:04:36.250
could cross-multiply.

00:04:36.250 --> 00:04:38.280
So you get 35 plus g times 1.

00:04:38.280 --> 00:04:43.180
So you get 35 plus g is
equal to 2 times 20.

00:04:43.180 --> 00:04:45.270
Is equal to 40, which
is essentially what

00:04:45.270 --> 00:04:45.950
we did in our head.

00:04:45.950 --> 00:04:48.360
We said, hey, this thing
has to be equal to forty.

00:04:48.360 --> 00:04:50.590
So, if you subtract 35
from both sides, you

00:04:50.590 --> 00:04:52.370
get g is equal to 5.

00:04:52.370 --> 00:04:55.960
So if you add 5 girls to your
classroom, you're going

00:04:55.960 --> 00:04:58.540
to have 40 girls.

00:04:58.540 --> 00:05:01.100
So then you'd have 40 girls.

00:05:01.100 --> 00:05:04.110
And you have 20 boys.

00:05:04.110 --> 00:05:07.560
And your ratio is 20:40,
which is 2:4, which is

00:05:07.560 --> 00:05:09.480
the same thing is 1:2.

00:05:09.480 --> 00:05:11.720
And we've done our problem.