Circles: radius, diameter, circumference and Pi

WEBVTT
Kind: captions
Language: en

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The circle is arguably the most
fundamental shape in our

00:00:04.880 --> 00:00:08.490
universe, whether you look at
the shapes of orbits of

00:00:08.490 --> 00:00:11.140
planets, whether you look at
wheels, whether you look at

00:00:11.140 --> 00:00:12.840
things on kind of a
molecular level.

00:00:12.840 --> 00:00:15.860
The circle just keeps
showing up over and

00:00:15.860 --> 00:00:17.350
over and over again.

00:00:17.350 --> 00:00:21.110
So it's probably worthwhile for
us to understand some of the

00:00:21.110 --> 00:00:23.330
properties of the circle.

00:00:23.330 --> 00:00:26.200
So the first thing when people
kind of discovered the circle,

00:00:26.200 --> 00:00:28.960
and you just have a look at the
moon to see a circle, but the

00:00:28.960 --> 00:00:31.570
first time they said well, what
are the properties

00:00:31.570 --> 00:00:32.910
of any circle?

00:00:32.910 --> 00:00:36.150
So the first one they might
want to say is well, a circle

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is all of the points that are
equal distant from the

00:00:38.690 --> 00:00:40.440
center of the circle.

00:00:40.440 --> 00:00:43.710
All of these points along the
edge are equal distant from

00:00:43.710 --> 00:00:45.210
that center right there.

00:00:45.210 --> 00:00:47.620
So one of the first things
someone might want to ask is

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what is that distance, that
equal distance that everything

00:00:50.280 --> 00:00:51.770
is from the center?

00:00:51.770 --> 00:00:52.950
Right there.

00:00:52.950 --> 00:00:58.110
We call that the
radius of the circle.

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It's just the distance from
the center out to the edge.

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If that radius is 3
centimeters, then this radius

00:01:02.820 --> 00:01:04.490
is going to be 3 centimeters.

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And this radius is going
to be 3 centimeters.

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It's never going to change.

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By definition, a circle is all
of the points that are equal

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distant from the center point.

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And that distance
is the radius.

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Now the next most interesting
thing about that, people might

00:01:19.880 --> 00:01:22.040
say well, how fat
is the circle?

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How wide is it along
its widest point?

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Or if you just want to cut it
along its widest point, what

00:01:28.710 --> 00:01:30.390
is that distance right there?

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And it doesn't have to be just
right there, I could have just

00:01:32.340 --> 00:01:35.490
as easily cut it along its
widest point right there.

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I just wouldn't be cutting it
like some place like that

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because that wouldn't be
along its widest point.

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There's multiple places
where I could cut it

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along its widest point.

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Well, we just saw the radius
and we see that widest point

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goes through the center
and just keeps going.

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So it's essentially two radii.

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You got one radius there
and then you have another

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radius over there.

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We call this distance along
the widest point of the

00:02:01.380 --> 00:02:03.030
circle, the diameter.

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So that is the diameter
of the circle.

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It has a very easy
relationship with the radius.

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The diameter is equal to
two times the radius.

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Now, the next most interesting
thing that you might be

00:02:21.790 --> 00:02:24.560
wondering about a circle is how
far is it around the circle?

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So if you were to get your tape
measure out and you were to

00:02:27.340 --> 00:02:35.910
measure around the circle like
that, what's that distance?

00:02:35.910 --> 00:02:44.710
We call that word the
circumference of the circle.

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Now, we know how the diameter
and the radius relates, but how

00:02:47.440 --> 00:02:49.790
does the circumference relate
to, say, the diameter.

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And if you're not really used
to the diameter, it's very

00:02:51.550 --> 00:02:54.290
easy to figure out how it
relates to the radius.

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Well, many thousands of years
ago, people took their tape

00:02:57.130 --> 00:02:58.890
measures out and they keep
measuring circumferences

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and radiuses.

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And let's say when their tape
measures weren't so good,

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let's say they measured the
circumference of the circle

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and they would get well, it
looks like it's about 3.

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And then they measure the
radius of the circle right here

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or the diameter of that circle,
and they'd say oh, the diameter

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looks like it's about 1.

00:03:16.290 --> 00:03:17.740
So they would say -- let
me write this down.

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So we're worried about
the ratio -- let me

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write it like this.

00:03:22.660 --> 00:03:33.955
The ratio of the circumference
to the diameter.

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So let's say that somebody had
some circle over here -- let's

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say they had this circle, and
the first time with not that

00:03:43.170 --> 00:03:45.880
good of a tape measure, they
measured around the circle

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and they said hey, it's
roughly equal to 3 meters

00:03:49.340 --> 00:03:50.490
when I go around it.

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And when I measure the
diameter of the circle,

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it's roughly equal to 1.

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OK, that's interesting.

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Maybe the ratio of
the circumference of

00:03:57.520 --> 00:03:58.500
the diameter's 3.

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So maybe the circumference
is always three

00:04:00.820 --> 00:04:02.020
times the diameter.

00:04:02.020 --> 00:04:03.610
Well that was just for this
circle, but let's say they

00:04:03.610 --> 00:04:05.720
measured some other
circle here.

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It's like this -- I
drew it smaller.

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Let's say that on this circle
they measured around it and

00:04:11.200 --> 00:04:14.960
they found out that the
circumference is 6 centimeters,

00:04:14.960 --> 00:04:18.210
roughly -- we have a bad
tape measure right then.

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Then they find out
that the diameter is

00:04:21.710 --> 00:04:23.520
roughly 2 centimeters.

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And once again, the ratio of
the circumference of the

00:04:25.490 --> 00:04:30.230
diameter was roughly 3.

00:04:30.230 --> 00:04:32.140
OK, this is a neat
property of circles.

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Maybe the ratio of the
circumference to the diameters

00:04:35.430 --> 00:04:38.080
always fixed for any circle.

00:04:38.080 --> 00:04:40.260
So they said let me
study this further.

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So they got better
tape measures.

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When they got better tape
measures, they measured hey,

00:04:45.090 --> 00:04:47.630
my diameter's definitely 1.

00:04:47.630 --> 00:04:49.430
They say my diameter's
definitely 1, but when I

00:04:49.430 --> 00:04:51.810
measure my circumference
a little bit, I realize

00:04:51.810 --> 00:04:53.040
it's closer to 3.1.

00:04:56.000 --> 00:04:57.290
And the same thing
with this over here.

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They notice that this
ratio is closer to 3.1.

00:04:59.370 --> 00:05:01.830
Then they kept measuring it
better and better and better,

00:05:01.830 --> 00:05:05.200
and then they realized that
they were getting this number,

00:05:05.200 --> 00:05:07.300
they just kept measuring it
better and better and they were

00:05:07.300 --> 00:05:10.850
getting this number 3.14159.

00:05:10.850 --> 00:05:12.550
And they just kept adding
digits and it would

00:05:12.550 --> 00:05:13.620
never repeat.

00:05:13.620 --> 00:05:16.640
It was a strange fascinating
metaphysical number

00:05:16.640 --> 00:05:18.300
that kept showing up.

00:05:18.300 --> 00:05:20.940
So since this number was so
fundamental to our universe,

00:05:20.940 --> 00:05:23.500
because the circle is so
fundamental to our universe,

00:05:23.500 --> 00:05:26.680
and it just showed up
for every circle.

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The ratio of the circumference
of the diameter was this

00:05:28.865 --> 00:05:32.390
kind of magical number,
they gave it a name.

00:05:32.390 --> 00:05:37.580
They called it pi, or you could
just give it the Latin or the

00:05:37.580 --> 00:05:41.880
Greek letter pi --
just like that.

00:05:41.880 --> 00:05:45.090
That represents this number
which is arguably the most

00:05:45.090 --> 00:05:46.790
fascinating number
in our universe.

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It first shows up as the ratio
of the circumference to the

00:05:50.430 --> 00:05:54.070
diameter, but you're going to
learn as you go through your

00:05:54.070 --> 00:05:57.160
mathematical journey, that
it shows up everywhere.

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It's one of these fundamental
things about the universe that

00:05:59.500 --> 00:06:03.060
just makes you think that
there's some order to it.

00:06:03.060 --> 00:06:07.750
But anyway, how can we
use this in I guess

00:06:07.750 --> 00:06:09.330
our basic mathematics?

00:06:09.330 --> 00:06:12.490
So we know, or I'm telling you,
that the ratio of the

00:06:12.490 --> 00:06:19.420
circumference to the diameter
-- when I say the ratio,

00:06:19.420 --> 00:06:21.390
literally I'm just saying if
you divide the circumference by

00:06:21.390 --> 00:06:28.400
the diameter, you're
going to get pi.

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Pi is just this number.

00:06:29.500 --> 00:06:33.570
I could write 3.14159 and just
keep going on and on and on,

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but that would be a waste of
space and it would just be hard

00:06:35.950 --> 00:06:38.570
to deal with, so people just
write this Greek

00:06:38.570 --> 00:06:40.330
letter pi there.

00:06:40.330 --> 00:06:41.850
So, how can we relate this?

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We can multiply both sides of
this by the diameter and we

00:06:44.920 --> 00:06:48.640
could say that the
circumference is equal to pi

00:06:48.640 --> 00:06:50.820
times the diameter.

00:06:50.820 --> 00:06:55.570
Or since the diameter is equal
to 2 times the radius, we could

00:06:55.570 --> 00:06:59.420
say that the circumference is
equal to pi times 2

00:06:59.420 --> 00:07:00.360
times the radius.

00:07:00.360 --> 00:07:03.450
Or the form that you're
most likely to see it,

00:07:03.450 --> 00:07:07.360
it's equal to 2 pi r.

00:07:07.360 --> 00:07:11.220
So let's see if we can apply
that to some problems.

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So let's say I have a circle
just like that, and I were to

00:07:17.240 --> 00:07:22.600
tell you it has a radius --
it's radius right there is 3.

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So, 3 -- let me write this down
-- so the radius is equal to 3.

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Maybe it's 3 meters --
put some units in there.

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What is the circumference
of the circle?

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The circumference is equal to
2 times 5 times the radius.

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So it's going to be equal to 2
times pi times the radius,

00:07:42.090 --> 00:07:47.280
times 3 meters, which is
equal to 6 meters times

00:07:47.280 --> 00:07:49.520
pi or 6 pi meters.

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6 pi meters.

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Now I could multiply this out.

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Remember pi is just a number.

00:07:55.900 --> 00:07:59.680
Pi is 3.14159 going
on and on and on.

00:07:59.680 --> 00:08:03.460
So if I multiply 6 times that,
maybe I'll get 18 point

00:08:03.460 --> 00:08:05.600
something something something.

00:08:05.600 --> 00:08:07.850
If you have your calculator you
might want to do it, but for

00:08:07.850 --> 00:08:10.490
simplicity people just tend to
leave our numbers

00:08:10.490 --> 00:08:12.120
in terms of pi.

00:08:12.120 --> 00:08:14.020
Now I don't know what this is
if you multiply 6 times

00:08:14.020 --> 00:08:18.510
3.14159, I don't know if you
get something close to 19 or

00:08:18.510 --> 00:08:20.910
18, maybe it's approximately
18 point something

00:08:20.910 --> 00:08:21.720
something something.

00:08:21.720 --> 00:08:23.450
I don't have my calculator
in front of me.

00:08:23.450 --> 00:08:25.300
But instead of writing
that number, you just

00:08:25.300 --> 00:08:27.060
write 6 pi there.

00:08:27.060 --> 00:08:29.770
Actually, I think it
wouldn't quite cross the

00:08:29.770 --> 00:08:31.430
threshold to 19 yet.

00:08:31.430 --> 00:08:33.770
Now, let's ask
another question.

00:08:33.770 --> 00:08:35.270
What is the diameter
of the circle?

00:08:38.580 --> 00:08:42.690
Well if this radius is 3, the
diameter is just twice that.

00:08:42.690 --> 00:08:45.730
So it's just going to be 3
times 2 or 3 plus 3, which

00:08:45.730 --> 00:08:47.170
is equal to 6 meters.

00:08:47.170 --> 00:08:50.750
So the circumference is 6 pi
meters, the diameter is 6

00:08:50.750 --> 00:08:53.620
meters, the radius is 3 meters.

00:08:53.620 --> 00:08:55.110
Now let's go the other way.

00:08:55.110 --> 00:08:57.310
Let's say I have
another circle.

00:08:57.310 --> 00:09:01.220
Let's say I have
another circle here.

00:09:01.220 --> 00:09:04.620
And I were to tell you that
its circumference is equal

00:09:04.620 --> 00:09:08.560
to 10 meters -- that's the
circumference of the circle.

00:09:08.560 --> 00:09:10.990
If you were to put a tape
measure to go around it and

00:09:10.990 --> 00:09:18.370
someone were to ask you what is
the diameter of the circle?

00:09:18.370 --> 00:09:22.810
Well, we know that the diameter
times pi, we know that pi times

00:09:22.810 --> 00:09:26.830
the diameter is equal to
the circumference; is

00:09:26.830 --> 00:09:28.700
equal to 10 meters.

00:09:28.700 --> 00:09:31.020
So to solve for this we would
just divide both sides

00:09:31.020 --> 00:09:32.520
of this equation by pi.

00:09:32.520 --> 00:09:35.860
The diameter would equal
10 meters over pi or

00:09:35.860 --> 00:09:38.710
10 over pi meters.

00:09:38.710 --> 00:09:40.020
And that is just a number.

00:09:40.020 --> 00:09:42.540
If you have your calculator,
you could actually divide 10

00:09:42.540 --> 00:09:46.030
divided by 3.14159, you're
going to get 3 point something

00:09:46.030 --> 00:09:47.500
something something meters.

00:09:47.500 --> 00:09:48.960
I can't do it in my head.

00:09:48.960 --> 00:09:50.070
But this is just a number.

00:09:50.070 --> 00:09:53.320
But for simplicity we often
just leave it that way.

00:09:53.320 --> 00:09:55.270
Now what is the radius?

00:09:55.270 --> 00:09:58.590
Well, the radius is equal
to 1/2 the diameter.

00:09:58.590 --> 00:10:02.870
So this whole distance right
here is 10 over pi meters.

00:10:02.870 --> 00:10:06.230
If we just 1/2 of that, if
we just want the radius, we

00:10:06.230 --> 00:10:07.580
just multiply it times 1/2.

00:10:07.580 --> 00:10:13.160
So you have 1/2 times 10 over
pi, which is equal to 1/2 times

00:10:13.160 --> 00:10:16.770
10, or you just divide the
numerator and the

00:10:16.770 --> 00:10:18.140
denominator by 2.

00:10:18.140 --> 00:10:21.130
You get 5 there, so
you get 5 over pi.

00:10:21.130 --> 00:10:23.890
So the radius over
here is 5 over pi.

00:10:23.890 --> 00:10:25.690
Nothing super fancy about this.

00:10:25.690 --> 00:10:29.760
I think the thing that confuses
people the most is to just

00:10:29.760 --> 00:10:31.820
realize that pi is a number.

00:10:31.820 --> 00:10:38.640
Pi is just 3.14159 and it just
keeps going on and on and on.

00:10:38.640 --> 00:10:41.950
There's actually thousands of
books written about pi, so

00:10:41.950 --> 00:10:45.100
it's not like -- I don't know
if there's thousands, I'm

00:10:45.100 --> 00:10:48.340
exaggerating, but you could
write books about this number.

00:10:48.340 --> 00:10:49.340
But it's just a number.

00:10:49.340 --> 00:10:52.480
It's a very special number, and
if you wanted to write it in a

00:10:52.480 --> 00:10:54.390
way that you're used to writing
numbers, you could literally

00:10:54.390 --> 00:10:55.680
just multiply this out.

00:10:55.680 --> 00:10:58.530
But most the time people just
realize they like leaving

00:10:58.530 --> 00:11:00.640
things in terms of pi.

00:11:00.640 --> 00:11:01.680
Anyway, I'll leave you there.

00:11:01.680 --> 00:11:05.090
In the next video we'll figure
out the area of a circle.


Area and Perimeter

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:03.000
I've got a square here.

00:00:03.000 --> 00:00:04.790
It's right like that.

00:00:04.790 --> 00:00:08.060
What makes it a square is
all of the sides are equal.

00:00:08.060 --> 00:00:10.380
I haven't gone in depth into
angles yet, but these are at

00:00:10.380 --> 00:00:12.520
right angles to each other.

00:00:12.520 --> 00:00:13.470
I'll just draw it like that.

00:00:13.470 --> 00:00:16.760
That means that if this bottom
side goes straight left and

00:00:16.760 --> 00:00:19.880
right, that this left side
will go straight up and down.

00:00:19.880 --> 00:00:22.210
That's all the right
angle really means.

00:00:22.210 --> 00:00:27.290
Let's say that the side down
here is equal to 8 meters.

00:00:27.290 --> 00:00:28.540
This side right here.

00:00:28.540 --> 00:00:30.100
And this is a square.

00:00:30.100 --> 00:00:35.980
And I were to ask you what
is the area of the square?

00:00:35.980 --> 00:00:39.040
Well, the area is essentially
how much space the square

00:00:39.040 --> 00:00:41.430
takes up, let's say, on
your screen right now.

00:00:41.430 --> 00:00:46.040
So it's essentially a way of
measuring how much space

00:00:46.040 --> 00:00:49.110
something takes up on kind of
a two-dimensional surface.

00:00:49.110 --> 00:00:52.170
A two-dimensional surface would
just be this computer screen or

00:00:52.170 --> 00:00:55.530
your piece of paper, if you're
also doing this problem.

00:00:55.530 --> 00:00:58.680
An analogy would be if you had
an 8 meter by 8 meter room, how

00:00:58.680 --> 00:01:01.570
much carpeting would you need
is kind of the size of the

00:01:01.570 --> 00:01:04.240
space you need to fill out in
two dimensions on some

00:01:04.240 --> 00:01:05.500
type of surface.

00:01:05.500 --> 00:01:09.750
So the area here is literally
how much is this size that

00:01:09.750 --> 00:01:11.980
you're filling up, and it's
very easy to figure

00:01:11.980 --> 00:01:12.605
out for a square.

00:01:12.605 --> 00:01:15.830
It's literally going to be your
base times your height -- and

00:01:15.830 --> 00:01:18.570
this is true for any rectangle
-- but since this is a square,

00:01:18.570 --> 00:01:20.650
your base and your height are
going to be the same number.

00:01:20.650 --> 00:01:22.340
It's going to be 8 meters.

00:01:22.340 --> 00:01:27.930
So your area is going to be 8
meters times 8 meters, which is

00:01:27.930 --> 00:01:32.020
equal to 8 times 8 is 64, and
then your meters times your

00:01:32.020 --> 00:01:34.580
meters -- you have to do the
same thing with your units --

00:01:34.580 --> 00:01:37.200
you get 64 meters squared.

00:01:37.200 --> 00:01:40.860
Or another way of saying,
this is 64 square meters.

00:01:40.860 --> 00:01:44.390
You might be asking where
are those 64 square meters?

00:01:44.390 --> 00:01:46.615
Well, you can actually
break it out here.

00:01:46.615 --> 00:01:48.470
So let me draw it a
little bit bigger than

00:01:48.470 --> 00:01:49.630
I originally drew it.

00:01:49.630 --> 00:01:51.890
I probably should have drawn
it this big to begin with.

00:01:51.890 --> 00:01:55.940
So let's say that's
my same square.

00:01:55.940 --> 00:01:58.100
I'm going to draw a little
bit, so let me divide

00:01:58.100 --> 00:02:00.240
it in the middle.

00:02:00.240 --> 00:02:03.770
Let me see, I have -- and
we divide them again.

00:02:03.770 --> 00:02:07.142
Then we divide each side
again just like that.

00:02:07.142 --> 00:02:08.410
I could probably do it neater.

00:02:08.410 --> 00:02:10.930
And let me do it one more time.

00:02:10.930 --> 00:02:16.840
Divide these just like that,
and then divide these

00:02:16.840 --> 00:02:19.010
just like that.

00:02:19.010 --> 00:02:20.940
There you go.

00:02:20.940 --> 00:02:21.480
OK.

00:02:21.480 --> 00:02:23.980
Now the reason why I did this
is to show you the dimensions

00:02:23.980 --> 00:02:27.030
along the base and the height.

00:02:27.030 --> 00:02:30.650
We said this is 8 meters,
and notice I have 1, 2,

00:02:30.650 --> 00:02:34.610
3, 4, 5, 6, 7, 8 meters.

00:02:34.610 --> 00:02:36.620
And the same thing
along this side.

00:02:36.620 --> 00:02:42.050
1, 2, 3, 4, 5, 6, 7, 8 meters.

00:02:42.050 --> 00:02:45.340
So when we're talking about
64 square meters, we're

00:02:45.340 --> 00:02:47.520
literally counting each
of the square meters.

00:02:47.520 --> 00:02:50.380
A square meter is a
two-dimensional measurement,

00:02:50.380 --> 00:02:51.780
that's 1 meter on each side.

00:02:51.780 --> 00:02:53.490
That's 1 meter, that's 1 meter.

00:02:53.490 --> 00:02:56.480
What I'm shading here in
yellow is 1 square meter.

00:02:56.480 --> 00:02:59.030
And you could imagine just
counting the square meters.

00:02:59.030 --> 00:03:05.070
In each row we're going to
have 1, 2, 3, 4, 5, 6,

00:03:05.070 --> 00:03:07.080
7, 8 square meters.

00:03:07.080 --> 00:03:08.610
And then we have 8 rows.

00:03:08.610 --> 00:03:11.200
So we're going to have 8
times 8 square meters

00:03:11.200 --> 00:03:12.760
or 64 meters square.

00:03:12.760 --> 00:03:14.840
Which is essentially if you sat
here and just counted each of

00:03:14.840 --> 00:03:19.050
these, you would count
64 square meters.

00:03:19.050 --> 00:03:21.540
Now, what happens if I
were to ask you the

00:03:21.540 --> 00:03:24.690
perimeter of my square?

00:03:28.000 --> 00:03:30.620
The perimeter is the distance
you need to go to go

00:03:30.620 --> 00:03:31.950
around the square.

00:03:31.950 --> 00:03:33.990
It's not measuring,
for example, how much

00:03:33.990 --> 00:03:35.070
carpeting you need.

00:03:35.070 --> 00:03:37.520
It's measuring, for example, if
you wanted to put a fence

00:03:37.520 --> 00:03:40.050
around your carpet -- I'm kind
of mixing the indoor and

00:03:40.050 --> 00:03:42.400
outdoor analogies -- it would
be how much fencing

00:03:42.400 --> 00:03:43.110
you would need.

00:03:43.110 --> 00:03:46.210
So it would be the
distance around.

00:03:46.210 --> 00:03:48.950
So it would be that distance
plus that distance plus that

00:03:48.950 --> 00:03:50.980
distance plus that distance.

00:03:50.980 --> 00:03:53.830
But we already know this
distance right here on the

00:03:53.830 --> 00:03:58.020
bottom, we already know
this distance is 8 meters.

00:03:58.020 --> 00:04:01.480
Then we know that the height
right here is 8 meters.

00:04:01.480 --> 00:04:02.180
It's a square.

00:04:02.180 --> 00:04:04.570
This distance up here is going
to be the same as this distance

00:04:04.570 --> 00:04:07.710
down here -- it's going
to be another 8 meters.

00:04:07.710 --> 00:04:09.450
Then when you go down the
left hand side it's going

00:04:09.450 --> 00:04:11.380
to be another 8 meters.

00:04:11.380 --> 00:04:15.670
We have four sides -- 1, 2, 3,
4 -- each of them are 8 meters.

00:04:15.670 --> 00:04:18.660
So you add 8 to itself 4 times,
that's the same thing as 8

00:04:18.660 --> 00:04:21.070
times 4, you get 36 meters.

00:04:21.070 --> 00:04:25.050
Now notice, when we measured
just the amount of fencing we

00:04:25.050 --> 00:04:28.530
needed, we ended up just with
meters, just with kind of a

00:04:28.530 --> 00:04:30.680
one-dimensional measurement.

00:04:30.680 --> 00:04:33.080
That's because we're not
measuring square meters here.

00:04:33.080 --> 00:04:35.310
We're not measuring how
much area we're taking up.

00:04:35.310 --> 00:04:38.560
We're measuring a distance
-- a distance to go around.

00:04:38.560 --> 00:04:40.920
We are taking turns, but you
can imagine straightening out

00:04:40.920 --> 00:04:44.570
this fence, and it would just
become one big fence like this,

00:04:44.570 --> 00:04:48.160
which would have the same
length of 36 meters.

00:04:48.160 --> 00:04:51.010
So that's why we just have
meters there for perimeter.

00:04:51.010 --> 00:04:53.640
But for area we got square
meters, because we're counting

00:04:53.640 --> 00:04:56.220
these two-dimensional
measurements.

00:04:56.220 --> 00:04:58.840
Now, let's make it a little
bit more interesting.

00:04:58.840 --> 00:05:02.070
What happens if instead
of a square I have a

00:05:02.070 --> 00:05:05.780
rectangle like this?

00:05:09.700 --> 00:05:15.280
Let's say that this side
over here is 7 centimeters.

00:05:15.280 --> 00:05:23.170
And let's say that the height
right here is 4 centimeters.

00:05:23.170 --> 00:05:25.845
So what is the area of this
rectangle going to be?

00:05:25.845 --> 00:05:28.280
It's going to be 7
times 4 centimeters.

00:05:28.280 --> 00:05:31.490
7 centimeters times
4 centimeters.

00:05:31.490 --> 00:05:36.390
Remember, we could draw 7 rows,
right, and each of them is

00:05:36.390 --> 00:05:39.540
going to have 4 square
centimeters -- each of those

00:05:39.540 --> 00:05:40.380
is a square centimeter.

00:05:40.380 --> 00:05:42.360
So if you were to count them
all out, you'd have 7 times

00:05:42.360 --> 00:05:44.170
4 square centimeters.

00:05:44.170 --> 00:05:45.140
It's 4 centimeters.

00:05:45.140 --> 00:05:50.390
So it's equal to 28 centimeters
square or squared centimeters.

00:05:50.390 --> 00:05:51.070
What's the perimeter?

00:05:55.260 --> 00:05:58.660
Well, it's going to be equal to
this distance down here, which

00:05:58.660 --> 00:06:03.670
is 7 centimeters, plus this
distance over here which is 4

00:06:03.670 --> 00:06:07.480
centimeters, plus the distance
on the top -- this is a

00:06:07.480 --> 00:06:09.170
rectangle, it's going to be
the same distance as

00:06:09.170 --> 00:06:10.440
this one over here.

00:06:10.440 --> 00:06:13.170
So plus another 7 centimeters.

00:06:13.170 --> 00:06:16.300
Then you're going to have this
distance on the left hand side.

00:06:16.300 --> 00:06:18.870
But this distance on the left
hand side is the same as this

00:06:18.870 --> 00:06:21.810
distance right here -- this
is also 4 centimeters.

00:06:21.810 --> 00:06:24.450
So plus another 4 centimeters.

00:06:24.450 --> 00:06:25.450
And what do you get?

00:06:25.450 --> 00:06:27.570
You get 7 plus 4 which is
11, and then you have

00:06:27.570 --> 00:06:29.020
another 7 plus 4.

00:06:29.020 --> 00:06:33.020
You have 11 plus 11, so
you have 22 centimeters.

00:06:33.020 --> 00:06:36.300
Once again, it's not
a square centimeter.

00:06:36.300 --> 00:06:42.300
Now let's divert -- let's go
away from our rectangle analogy

00:06:42.300 --> 00:06:43.760
or our rectangle examples.

00:06:43.760 --> 00:06:46.930
So let's see if we can do
the same with triangles.

00:06:46.930 --> 00:06:49.940
So let's say I have
a triangle here.

00:06:49.940 --> 00:06:52.100
I have a triangle like this.

00:06:54.990 --> 00:06:58.720
Let's say that this distance
right here -- actually

00:06:58.720 --> 00:06:59.760
let me draw it like this.

00:06:59.760 --> 00:07:02.210
I think this'll make it a
little bit easier for you

00:07:02.210 --> 00:07:04.550
to see how this relates
to a rectangle.

00:07:04.550 --> 00:07:05.810
Let me draw it like this.

00:07:09.360 --> 00:07:09.810
There you go.

00:07:09.810 --> 00:07:11.300
That's my triangle.

00:07:11.300 --> 00:07:14.510
And let's say that this
distance right here is 7

00:07:14.510 --> 00:07:17.210
centimeters right down there.

00:07:17.210 --> 00:07:21.090
And let's say that the
height of this triangle

00:07:21.090 --> 00:07:23.520
is 4 centimeters.

00:07:23.520 --> 00:07:26.160
And I were to ask you what is
the area of the triangle?

00:07:33.690 --> 00:07:36.590
Well, when we had a rectangle
like this, we just

00:07:36.590 --> 00:07:38.660
multiplied 7 times 4.

00:07:38.660 --> 00:07:39.600
But what would that give us?

00:07:39.600 --> 00:07:42.610
That would give us the area
of an entire rectangle.

00:07:42.610 --> 00:07:44.610
If we did 7 times 4, that
would give us the area of

00:07:44.610 --> 00:07:46.050
this entire rectangle.

00:07:46.050 --> 00:07:49.640
You could imagine extending
my triangle up like this.

00:07:49.640 --> 00:07:51.880
This is a right triangle --
this is going straight up and

00:07:51.880 --> 00:07:54.420
down, this is going straight
left and right on the

00:07:54.420 --> 00:07:55.910
bottom right here.

00:07:55.910 --> 00:07:58.910
It's a 90 degree angle, if
you've been exposed to the

00:07:58.910 --> 00:08:00.040
idea of angles already.

00:08:00.040 --> 00:08:03.460
So you could almost view it as
it's 1/2 of this rectangle.

00:08:03.460 --> 00:08:04.610
Not really almost, it is.

00:08:04.610 --> 00:08:07.580
Because if you just double this
guy, you could imagine if you

00:08:07.580 --> 00:08:12.190
flip this triangle over, you
get the same triangle but

00:08:12.190 --> 00:08:14.910
it's just upside down
and flipped over.

00:08:14.910 --> 00:08:17.650
So if you think about when you
multiply 7 times 4, you're

00:08:17.650 --> 00:08:25.140
getting the area of this entire
rectangle, which we

00:08:25.140 --> 00:08:26.800
just did up here.

00:08:26.800 --> 00:08:30.210
But we want to know the
area of the triangle.

00:08:30.210 --> 00:08:33.190
We want to know just
this area right here.

00:08:33.190 --> 00:08:36.290
You can see, hopefully, from
this drawing that the area of

00:08:36.290 --> 00:08:39.390
this triangle is exactly 1/2
of the area of the

00:08:39.390 --> 00:08:40.990
entire rectangle.

00:08:40.990 --> 00:08:47.040
So the area for a triangle is
equal to the base times the

00:08:47.040 --> 00:08:50.490
height -- now this so far,
base times height is the

00:08:50.490 --> 00:08:52.150
area of a rectangle.

00:08:52.150 --> 00:08:53.755
So in order to get the area of
the triangle, you're going

00:08:53.755 --> 00:08:55.910
to multiply that times 1/2.

00:08:55.910 --> 00:08:58.160
So 1/2 base times height.

00:08:58.160 --> 00:09:04.320
So in our example it's going
to be 1/2 times 7 centimeters

00:09:04.320 --> 00:09:07.020
times 4 centimeters.

00:09:07.020 --> 00:09:10.780
We know what 7 times 4 is.

00:09:10.780 --> 00:09:13.880
We already know it's
28 centimeters -- we

00:09:13.880 --> 00:09:15.710
did that up there.

00:09:15.710 --> 00:09:19.050
So this right here
is 28 centimeters.

00:09:19.050 --> 00:09:22.070
Then we want centimeters and we
want to multiply that by 1/2.

00:09:22.070 --> 00:09:26.720
So that's going to be 14
centimeters just like that.

00:09:26.720 --> 00:09:29.950
So the area of this triangle
is exactly 1/2 of the

00:09:29.950 --> 00:09:31.700
area of that rectangle.

00:09:31.700 --> 00:09:35.670
Now, the perimeter of this
triangle becomes a little bit

00:09:35.670 --> 00:09:43.380
more complicated because
figuring out this distance

00:09:43.380 --> 00:09:45.320
isn't the easiest
thing in the world.

00:09:45.320 --> 00:09:47.965
Well, it will be easy for
you once you get exposed to

00:09:47.965 --> 00:09:48.870
the Pythagorean Theorem.

00:09:48.870 --> 00:09:50.290
But I'm going to skip
that right now.

00:09:50.290 --> 00:09:54.010
I'm going to leave that for the
Pythagorean Theorem video.

00:09:54.010 --> 00:09:58.450
Let me just give you one
more area of a triangle.

00:09:58.450 --> 00:10:00.120
Let's say I have a triangle
that looks like this.

00:10:00.120 --> 00:10:03.190
This was a very special case
that I drew to make it look

00:10:03.190 --> 00:10:04.520
like half of a rectangle.

00:10:04.520 --> 00:10:07.220
Let's say we had a triangle
that looks like this.

00:10:07.220 --> 00:10:11.650
It's a little bit more
skewed looking like this.

00:10:11.650 --> 00:10:19.346
And let's say that this
distance down here is 3 meters

00:10:19.346 --> 00:10:21.950
-- that distance is 3 meters.

00:10:21.950 --> 00:10:25.230
Let's say we don't know what
that distance is and we don't

00:10:25.230 --> 00:10:26.570
know what that distance is.

00:10:26.570 --> 00:10:30.660
But we do know that if we were
to kind of drop a line straight

00:10:30.660 --> 00:10:32.670
down like this -- if you
imagine this was a building or

00:10:32.670 --> 00:10:34.760
some type of mountain and you
just drop something straight

00:10:34.760 --> 00:10:38.850
down onto the ground like that,
we know that this distance

00:10:38.850 --> 00:10:43.770
is equal to -- let's say
it's equal to 4 meters.

00:10:43.770 --> 00:10:46.140
So what is the area of this
triangle going to be?

00:10:50.420 --> 00:10:52.910
Well, we apply the
same formula.

00:10:52.910 --> 00:10:57.170
Area is equal to 1/2
base times height.

00:10:57.170 --> 00:11:00.490
So it's equal to 1/2 -- the
base is literally this base

00:11:00.490 --> 00:11:02.260
right here of this triangle.

00:11:02.260 --> 00:11:07.380
So 1/2 times 3 times the
height of the triangle.

00:11:07.380 --> 00:11:08.740
I guess a better way
to think of it is an

00:11:08.740 --> 00:11:10.570
altitude of the triangle.

00:11:10.570 --> 00:11:12.760
So this thing isn't even in
the triangle, but it is

00:11:12.760 --> 00:11:13.820
literally the height.

00:11:13.820 --> 00:11:15.850
If you imagine this was a
building, you say how high is

00:11:15.850 --> 00:11:18.360
the building, it would be
this height right there.

00:11:18.360 --> 00:11:20.395
So 1/2 times 3 times 4.

00:11:20.395 --> 00:11:22.880
You use that distance
right there.

00:11:22.880 --> 00:11:27.860
Which is equal to 3 times 4 is
12 times 1/2 is equal to 6.

00:11:27.860 --> 00:11:30.830
We're going to be dealing
with square meters.

00:11:30.830 --> 00:11:34.140
I really want to highlight the
idea, because if I gave you a

00:11:34.140 --> 00:11:40.000
triangle that looked like this,
where if this was 3 meters down

00:11:40.000 --> 00:11:44.250
here, and then if I were to
tell you that this side over

00:11:44.250 --> 00:11:50.930
here is 4 meters, this is not
something that you can just

00:11:50.930 --> 00:11:52.820
apply this formula
to and figure out.

00:11:52.820 --> 00:11:54.790
In fact, you'd have to know
some of the angles and whatnot

00:11:54.790 --> 00:11:56.840
to really be able to figure out
the area, or you'd have to

00:11:56.840 --> 00:11:58.350
know this other side here.

00:11:58.350 --> 00:12:02.480
So this is not easy.

00:12:02.480 --> 00:12:05.890
You have to know what the
altitude or the height

00:12:05.890 --> 00:12:06.720
of the triangle is.

00:12:06.720 --> 00:12:07.900
You need to know this distance.

00:12:07.900 --> 00:12:11.330
In this case, it was one of
the sides, but in this case

00:12:11.330 --> 00:12:12.290
it's not one of the sides.

00:12:12.290 --> 00:12:15.840
You'd have to figure out what
that side right there on the

00:12:15.840 --> 00:12:19.590
right hand side is in order
to apply this formula.


dim(v) + dim(orthogonal complement of v) = n

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:04.720
Let's say I've got some subspace
of Rn called V.

00:00:04.720 --> 00:00:10.370
So V is a subspace of Rn.

00:00:10.370 --> 00:00:12.920
And let's say that
I know its basis.

00:00:12.920 --> 00:00:14.732
Let's say the set.

00:00:14.732 --> 00:00:17.410
So I have a bunch of-- let me
make that bracket a little

00:00:17.410 --> 00:00:22.970
nicer-- so let's say the set of
the vectors v1, v2, all the

00:00:22.970 --> 00:00:29.180
way to vk, let's say that this
is equal to-- this is the

00:00:29.180 --> 00:00:32.630
basis for V.

00:00:32.630 --> 00:00:35.880
And just as a reminder, that
means that V's vectors both

00:00:35.880 --> 00:00:38.360
span V and they're linearly
independent.

00:00:38.360 --> 00:00:42.080
You can kind of see there's a
minimum set of vectors in Rn

00:00:42.080 --> 00:00:43.420
that span V.

00:00:43.420 --> 00:00:49.660
So, if I were to ask you what
the dimension of V is, that's

00:00:49.660 --> 00:00:51.490
just the number of vectors
you have in your

00:00:51.490 --> 00:00:52.900
basis for the subspace.

00:00:52.900 --> 00:00:56.180
So we have 1, 2 and we
count to k vectors.

00:00:56.180 --> 00:00:59.290
So it is equal to k.

00:00:59.290 --> 00:01:02.140
Now let's think about, if we
can somehow figure out what

00:01:02.140 --> 00:01:05.000
the dimension of the orthogonal

00:01:05.000 --> 00:01:07.070
complement of V can be.

00:01:07.070 --> 00:01:09.330
And to do that, let's
construct a matrix.

00:01:09.330 --> 00:01:12.980
Let's construct a matrix
whose column vectors

00:01:12.980 --> 00:01:14.690
are these basis vectors.

00:01:14.690 --> 00:01:17.115
So let's construct a matrix
A, and let's say

00:01:17.115 --> 00:01:18.790
it looks like this.

00:01:18.790 --> 00:01:21.090
First column is v1.

00:01:21.090 --> 00:01:23.370
This first basis vector
right there.

00:01:23.370 --> 00:01:28.390
v2 is the second one, and then
you go all the way to vk.

00:01:32.280 --> 00:01:34.360
Just to make sure we remember
the dimensions, we have k of

00:01:34.360 --> 00:01:36.790
these vectors, so we're going
to have k columns.

00:01:36.790 --> 00:01:38.580
And then how many rows
are we going to have?

00:01:38.580 --> 00:01:41.710
Well, as a member of Rn, so
these are all going to have n

00:01:41.710 --> 00:01:43.560
entries in each of these
vectors, there's going to be

00:01:43.560 --> 00:01:46.600
an n-- we're going to have
n rows and k columns.

00:01:46.600 --> 00:01:49.740
It's an n by k matrix.

00:01:49.740 --> 00:01:54.780
Now, what's another way of
expressing the subspace V?

00:01:54.780 --> 00:01:59.570
Well, the basis for V is-- or
V is spanned by these basis

00:01:59.570 --> 00:02:01.770
vectors, which is the
columns of these.

00:02:01.770 --> 00:02:05.360
So if I talk about the span--
so let me write out this-- V

00:02:05.360 --> 00:02:11.030
is equal to the span of
these guys, v1, v2,

00:02:11.030 --> 00:02:12.970
all the way to vk.

00:02:12.970 --> 00:02:15.380
And that's just the same thing
as the column space of A.

00:02:15.380 --> 00:02:15.620
Right?

00:02:15.620 --> 00:02:17.520
These are the column vectors,
and the span of them, that's

00:02:17.520 --> 00:02:23.190
equal to the column
space of A.

00:02:23.190 --> 00:02:26.030
Now, I said a little while ago,
we want to somehow relate

00:02:26.030 --> 00:02:29.220
to the orthogonal
complement of V.

00:02:29.220 --> 00:02:31.590
Well, what's the orthogonal
complement of the

00:02:31.590 --> 00:02:33.690
column space of A?

00:02:33.690 --> 00:02:37.230
The orthogonal complement of
the column space of A, I

00:02:37.230 --> 00:02:40.130
showed you-- I think it was two
or three videos ago-- that

00:02:40.130 --> 00:02:45.420
the column space of A's
orthogonal complement is equal

00:02:45.420 --> 00:02:49.690
to-- you could either view
it as the null space of A

00:02:49.690 --> 00:02:52.650
transpose, or another way
you call it is the left

00:02:52.650 --> 00:02:54.070
null space of A.

00:02:54.070 --> 00:02:57.870
This is equivalent to the
orthogonal complement of the

00:02:57.870 --> 00:03:01.760
column space of A, which is
also going to be equal to,

00:03:01.760 --> 00:03:04.700
which is also since this piece
right here is the same thing

00:03:04.700 --> 00:03:07.560
as V, you take it's orthogonal
complement, that's the same

00:03:07.560 --> 00:03:11.240
thing as V's orthogonal
complement.

00:03:11.240 --> 00:03:14.340
So if we want to figure out
there orthogonal complement

00:03:14.340 --> 00:03:19.590
of-- if we want to figure out
the dimension-- if we want to

00:03:19.590 --> 00:03:22.370
figure out the dimensional of
the orthogonal complement of

00:03:22.370 --> 00:03:27.230
V, we just need to figure out
the dimension of the left null

00:03:27.230 --> 00:03:31.210
space of A, or the null space
of A's transpose.

00:03:31.210 --> 00:03:32.555
Let me write that down.

00:03:32.555 --> 00:03:38.030
So the dimension-- get you
tongue-tied sometimes-- the

00:03:38.030 --> 00:03:43.280
dimension of the orthogonal
complement of V is going to be

00:03:43.280 --> 00:03:48.600
equal do the dimension
of A transpose.

00:03:48.600 --> 00:03:51.160
Or another way to think of it
is-- sorry, not just the

00:03:51.160 --> 00:03:54.350
dimension of A transpose, the
dimension of the null space of

00:03:54.350 --> 00:03:55.600
A transpose.

00:03:58.120 --> 00:04:00.620
And if you have a good memory,
I don't use the word a lot,

00:04:00.620 --> 00:04:03.160
this thing is the nullity--
this is the

00:04:03.160 --> 00:04:04.700
nullity of A transpose.

00:04:10.380 --> 00:04:12.640
The dimension of your null
space is nullity, the

00:04:12.640 --> 00:04:16.079
dimension of your column
space is your rank.

00:04:16.079 --> 00:04:17.610
Now let's see what
we can do here.

00:04:17.610 --> 00:04:21.079
So let's just take A transpose,
so you can just

00:04:21.079 --> 00:04:25.230
imagine A transpose
for a second.

00:04:25.230 --> 00:04:26.140
I can just even draw it out.

00:04:26.140 --> 00:04:33.890
It's going to be a k by n matrix
that looks like this.

00:04:33.890 --> 00:04:35.760
These columns are going
to turn into rows.

00:04:35.760 --> 00:04:40.240
This is going to be v1
transpose, v2 transpose, all

00:04:40.240 --> 00:04:44.270
the way down to vk transpose
vectors.

00:04:44.270 --> 00:04:47.640
These are all now row vectors.

00:04:47.640 --> 00:04:48.820
So we know one thing.

00:04:48.820 --> 00:04:52.120
We know one relationship between
the rank and nullity

00:04:52.120 --> 00:04:53.470
of any matrix.

00:04:53.470 --> 00:04:59.720
We know that they're equal to
the number of columns we have.

00:04:59.720 --> 00:05:10.910
We know that the rank of A
transpose plus the nullity of

00:05:10.910 --> 00:05:13.520
A transpose is equal
to the number of

00:05:13.520 --> 00:05:15.280
columns of A transpose.

00:05:15.280 --> 00:05:16.460
We have n columns.

00:05:16.460 --> 00:05:17.635
Each of these have n entries.

00:05:17.635 --> 00:05:19.120
It is equal to n.

00:05:19.120 --> 00:05:20.800
We saw this a while ago.

00:05:20.800 --> 00:05:23.970
And if you want just a bit of a
reminder of where that comes

00:05:23.970 --> 00:05:28.060
from, when you take a-- if I
wrote A transpose as a bunch

00:05:28.060 --> 00:05:31.480
of column vectors, which I can,
or maybe let me take some

00:05:31.480 --> 00:05:34.060
other vector B, because I want
to just remind you where this,

00:05:34.060 --> 00:05:35.480
why this made sense.

00:05:35.480 --> 00:05:39.290
If I take some vector B here,
and it has got a bunch of

00:05:39.290 --> 00:05:44.840
column vectors, b1, b2, all the
way to bn, and I put it

00:05:44.840 --> 00:05:51.090
into reduced row echelon form,
you're going to have some

00:05:51.090 --> 00:05:53.730
pivot columns and some
non-pivot columns.

00:05:53.730 --> 00:05:55.100
So let's say this is
a pivot column.

00:05:55.100 --> 00:05:57.490
You know, I got a 1 and a bunch
of 0's, let's say that

00:05:57.490 --> 00:06:00.510
this is one of them, and then
let's say I got one other one

00:06:00.510 --> 00:06:03.400
that's out, and it would be a 0
there, it's a 1 down there,

00:06:03.400 --> 00:06:06.610
and everything else is
a non-pivot column.

00:06:06.610 --> 00:06:10.580
I showed you in the last video
that your basis for your

00:06:10.580 --> 00:06:14.350
column space is the number of
pivot columns you have. So

00:06:14.350 --> 00:06:17.300
these guys are pivot columns.

00:06:17.300 --> 00:06:20.400
The corresponding column vectors
form a basis for your

00:06:20.400 --> 00:06:21.310
column space.

00:06:21.310 --> 00:06:23.290
I showed you that in
the last video.

00:06:23.290 --> 00:06:25.550
And so, if you want to know the
dimension of your column

00:06:25.550 --> 00:06:27.190
space, you just have to
count these things.

00:06:27.190 --> 00:06:28.440
You just count these things.

00:06:28.440 --> 00:06:31.490
This was equal to the number of,
well, for this B's case,

00:06:31.490 --> 00:06:34.080
the rank of B is just equal to
the number of pivot columns I

00:06:34.080 --> 00:06:37.230
have. Now the nullity is the
dimension of your null space.

00:06:37.230 --> 00:06:39.830
We've done multiple problems
where we found the null space

00:06:39.830 --> 00:06:40.850
of matrices.

00:06:40.850 --> 00:06:44.840
And every time, the dimension,
it's a bit obvious, and I

00:06:44.840 --> 00:06:46.760
actually showed you this proof,
it's related to the

00:06:46.760 --> 00:06:50.990
number of free columns you have,
or non-pivot columns.

00:06:50.990 --> 00:06:55.040
So, if you have no pivot
columns, then you are -- if

00:06:55.040 --> 00:06:57.370
all of your columns are pivot
columns, and none of them have

00:06:57.370 --> 00:06:59.170
free variables or are
associated with free

00:06:59.170 --> 00:07:01.740
variables, then you're null
space is going to be trivial.

00:07:01.740 --> 00:07:03.540
It's just going to have
the 0 vector.

00:07:03.540 --> 00:07:05.240
But the more free variables
you have, the more

00:07:05.240 --> 00:07:06.800
dimensionality your
null space has.

00:07:06.800 --> 00:07:09.810
So the free columns correspond
to the null space, and they

00:07:09.810 --> 00:07:12.100
form actually a basis
for your null space.

00:07:12.100 --> 00:07:15.420
And because of that, the basis
for your null space vectors,

00:07:15.420 --> 00:07:18.520
plus the basis for your column
space, is equal to the total

00:07:18.520 --> 00:07:21.040
number of columns you have. I
showed that to you in the

00:07:21.040 --> 00:07:23.410
past, but it's always good
to remind ourselves

00:07:23.410 --> 00:07:24.760
where things come from.

00:07:24.760 --> 00:07:26.100
But this was just
a bit of a side.

00:07:26.100 --> 00:07:28.260
I did this with a separate
vector B.

00:07:28.260 --> 00:07:31.220
Just to remind ourselves
where this thing

00:07:31.220 --> 00:07:32.820
right here came from.

00:07:32.820 --> 00:07:37.750
Now, in the last video, I showed
you that the rank of A

00:07:37.750 --> 00:07:41.040
transpose is the same thing
is the rank of A.

00:07:41.040 --> 00:07:43.540
This is equal to, this part
right here, is the same thing

00:07:43.540 --> 00:07:46.790
as the rank of A.

00:07:46.790 --> 00:07:47.990
I showed you that in
the last video.

00:07:47.990 --> 00:07:50.510
When you transpose a matrix, it
doesn't change its rank, or

00:07:50.510 --> 00:07:53.090
it doesn't change the dimension
of its column space.

00:07:53.090 --> 00:07:56.350
So we can rewrite this
statement, right here, as the

00:07:56.350 --> 00:08:07.130
rank of A plus the nullity of A
transpose is equal to n, and

00:08:07.130 --> 00:08:10.840
the rank of A is the same thing
as the dimension of the

00:08:10.840 --> 00:08:13.470
column space of A.

00:08:13.470 --> 00:08:16.290
And then the nullity of A
transpose is the same thing as

00:08:16.290 --> 00:08:21.580
the dimension of the null space
of A transpose-- that's

00:08:21.580 --> 00:08:24.100
just the definition of nullity--
they're going to be

00:08:24.100 --> 00:08:25.310
equal to n.

00:08:25.310 --> 00:08:28.700
Now what's the dimension--
what's the column space of A?

00:08:28.700 --> 00:08:31.620
The column space of A, that's
what's spanned by these

00:08:31.620 --> 00:08:34.429
vectors right here, which
were the basis for V.

00:08:34.429 --> 00:08:37.464
So this is the same thing
as the dimension of V.

00:08:37.464 --> 00:08:40.179
The column space of A is the
same thing as the dimension of

00:08:40.179 --> 00:08:43.270
my subspace V that I started
this video with.

00:08:43.270 --> 00:08:46.460
And what is the null space
of A transpose?

00:08:46.460 --> 00:08:51.880
The null space of A transpose,
we saw already, that's the

00:08:51.880 --> 00:08:54.160
orthogonal complement of V.

00:08:54.160 --> 00:08:58.670
So I could write this as plus
the dimension of the

00:08:58.670 --> 00:09:03.050
orthogonal complement
of V is equal to n.

00:09:03.050 --> 00:09:04.770
And that's the result
we wanted.

00:09:04.770 --> 00:09:17.670
If V is a subspace of Rn, that n
is the same thing as that n,

00:09:17.670 --> 00:09:21.890
then the dimension of V plus
the dimension of the

00:09:21.890 --> 00:09:26.490
orthogonal complement of V is
going to be equal to n.


rank(a) = rank(transpose of a)

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:04.360
A couple of videos ago, I made
the statement that the rank of

00:00:04.360 --> 00:00:08.280
a matrix A is equal to the
rank of its transpose.

00:00:08.280 --> 00:00:09.910
And I made a bit of a
hand wavy argument.

00:00:09.910 --> 00:00:12.310
It was at the end of the
video, and I was tired.

00:00:12.310 --> 00:00:13.710
It was actually the
end of the day.

00:00:13.710 --> 00:00:16.810
And I thought it'd be worthwhile
to maybe flush this

00:00:16.810 --> 00:00:17.300
out a little bit.

00:00:17.300 --> 00:00:18.620
Because it's an important
take away.

00:00:18.620 --> 00:00:21.550
It'll help us understand
everything we've learned a

00:00:21.550 --> 00:00:23.200
little bit better.

00:00:23.200 --> 00:00:25.560
So, let's understand-- I'm
actually going to start with

00:00:25.560 --> 00:00:26.830
the rank of A transpose.

00:00:31.630 --> 00:00:37.080
The rank of A transpose is equal
to the dimension of the

00:00:37.080 --> 00:00:40.020
column space of A transpose.

00:00:40.020 --> 00:00:42.690
That's the definition
of the rank.

00:00:42.690 --> 00:00:46.810
The dimension of the column
space of A transpose is the

00:00:46.810 --> 00:00:53.950
number of basis vectors for the

00:00:53.950 --> 00:00:55.330
column space of A transpose.

00:00:55.330 --> 00:00:56.810
That's what dimension is.

00:00:56.810 --> 00:00:59.900
For any subspace, you figure out
how many basis vectors you

00:00:59.900 --> 00:01:01.910
need in that subspace,
and you count them,

00:01:01.910 --> 00:01:02.830
and that's your dimension.

00:01:02.830 --> 00:01:07.180
So, it's the number of basis
vectors for the column space

00:01:07.180 --> 00:01:10.150
of A transpose, which is, of
course, the same thing.

00:01:10.150 --> 00:01:12.530
This thing we've seen multiple
times, is the same thing as

00:01:12.530 --> 00:01:13.780
the row space of A.

00:01:17.690 --> 00:01:17.950
Right?

00:01:17.950 --> 00:01:20.220
The columns of A transpose
are the same thing

00:01:20.220 --> 00:01:21.785
as the rows of A.

00:01:21.785 --> 00:01:24.310
It's because You switch the
rows and the columns.

00:01:24.310 --> 00:01:27.480
Now, how can we figure out the
number of basis vectors we

00:01:27.480 --> 00:01:30.390
need for the column space
of A transpose, or the

00:01:30.390 --> 00:01:32.040
row space of A?

00:01:32.040 --> 00:01:34.160
Let's just think about what
the column space of A

00:01:34.160 --> 00:01:36.330
transpose is telling us.

00:01:36.330 --> 00:01:38.290
So, it's equivalent to--
so let's say, let

00:01:38.290 --> 00:01:39.690
me draw A like this.

00:01:43.400 --> 00:01:44.420
That's a matrix A.

00:01:44.420 --> 00:01:47.160
Let's say it's an
m by n matrix.

00:01:47.160 --> 00:01:49.210
Let me just write it as a
bunch of row vectors.

00:01:49.210 --> 00:01:51.040
I could also write it as a bunch
of column vectors, but

00:01:51.040 --> 00:01:53.150
right now let's stick
to the row vectors.

00:01:53.150 --> 00:01:55.420
So we have row one.

00:01:55.420 --> 00:01:57.420
The transpose of
column vectors.

00:01:57.420 --> 00:02:02.460
That's row one, and we're going
to have row two, and

00:02:02.460 --> 00:02:05.710
we're going to go all the
way down to row m.

00:02:05.710 --> 00:02:06.010
Right?

00:02:06.010 --> 00:02:06.970
It's an m by n matrix.

00:02:06.970 --> 00:02:10.289
Each of these vectors are
members of rn, because they're

00:02:10.289 --> 00:02:11.690
going to have n entries in them

00:02:11.690 --> 00:02:13.800
because we have n columns.

00:02:13.800 --> 00:02:15.730
So, that's what A is
going to look like.

00:02:15.730 --> 00:02:17.050
A is going look like that.

00:02:17.050 --> 00:02:20.660
And then A transpose, all
of these rows are

00:02:20.660 --> 00:02:22.480
going to become columns.

00:02:22.480 --> 00:02:27.670
A transpose is going to look
like this. r1, r2,

00:02:27.670 --> 00:02:30.890
all the way to rm.

00:02:30.890 --> 00:02:33.880
And this is of course going
to be an n by m matrix.

00:02:33.880 --> 00:02:35.485
You swap these out.

00:02:35.485 --> 00:02:38.670
So, all these rows are
going to be columns.

00:02:38.670 --> 00:02:39.650
Right?

00:02:39.650 --> 00:02:41.590
And, obviously the column
space-- or maybe not so

00:02:41.590 --> 00:02:47.350
obviously-- the column space of
A transpose is equal to the

00:02:47.350 --> 00:02:56.270
span of r1, r2, all
the way to rm.

00:02:56.270 --> 00:02:56.900
Right?

00:02:56.900 --> 00:02:58.390
It's equal to the span
of these things.

00:02:58.390 --> 00:03:00.780
Or you could equivocally call
it, it's equal to the span of

00:03:00.780 --> 00:03:01.470
the rows of A.

00:03:01.470 --> 00:03:03.740
That's why it's also called
the row space.

00:03:03.740 --> 00:03:12.560
This is equal to the span
of the rows of A.

00:03:12.560 --> 00:03:14.530
These two things
are equivalent.

00:03:14.530 --> 00:03:16.100
Now, these are the span.

00:03:16.100 --> 00:03:18.260
That means this is some subspace
that's all of the

00:03:18.260 --> 00:03:20.700
linear combinations of these
columns, or all of the linear

00:03:20.700 --> 00:03:22.090
combinations of these rows.

00:03:22.090 --> 00:03:26.030
If we want the basis for it, we
want to find a minimum set

00:03:26.030 --> 00:03:29.280
of linearly independent vectors
that we could use to

00:03:29.280 --> 00:03:30.880
construct any of
these columns.

00:03:30.880 --> 00:03:34.830
Or that we could use construct
any of these rows, right here.

00:03:34.830 --> 00:03:37.270
Now, what happens when
we put A into

00:03:37.270 --> 00:03:40.070
reduced row echelon form?

00:03:40.070 --> 00:03:46.290
We do a bunch of row operations
to put it into

00:03:46.290 --> 00:03:49.020
reduced row echelon form.

00:03:49.020 --> 00:03:49.140
Right?

00:03:49.140 --> 00:03:52.150
Do a bunch of row operations
and you eventually get

00:03:52.150 --> 00:03:53.050
something like this.

00:03:53.050 --> 00:03:57.410
You'll get the reduced row
echelon form of A.

00:03:57.410 --> 00:03:59.610
The reduced row echelon form
of A is going to look

00:03:59.610 --> 00:04:00.840
something like this.

00:04:00.840 --> 00:04:04.180
You're going to have some pivot
rows, some rows that

00:04:04.180 --> 00:04:05.650
have pivot entries.

00:04:05.650 --> 00:04:08.010
Let's say that's one of them.

00:04:08.010 --> 00:04:08.980
Let's say that's one of them.

00:04:08.980 --> 00:04:11.390
This will all have 0's
all the way down.

00:04:11.390 --> 00:04:12.770
This one will have 0's.

00:04:12.770 --> 00:04:14.760
Your pivot entry has to
be the only non-zero

00:04:14.760 --> 00:04:16.180
entry in it's column.

00:04:16.180 --> 00:04:18.220
And everything to the left
of it all has to be 0.

00:04:18.220 --> 00:04:19.790
Let's say that this one isn't.

00:04:19.790 --> 00:04:21.360
These are some non-zero
values.

00:04:21.360 --> 00:04:22.600
These are 0.

00:04:22.600 --> 00:04:24.690
We have another pivot
entry over here.

00:04:24.690 --> 00:04:25.340
Everything else is 0.

00:04:25.340 --> 00:04:29.350
Let's say everything else
are non-pivot entries.

00:04:29.350 --> 00:04:32.510
So you come here and you have
a certain number of pivot

00:04:32.510 --> 00:04:35.350
rows, or a certain number
of pivot entries, right?

00:04:35.350 --> 00:04:37.630
And you got there by performing
linear row

00:04:37.630 --> 00:04:38.880
operations on these guys.

00:04:38.880 --> 00:04:41.670
So those linear row operations--
you know, I take

00:04:41.670 --> 00:04:44.655
3 times row two, and I add it
to row one, and that's going

00:04:44.655 --> 00:04:45.790
to become my new row two.

00:04:45.790 --> 00:04:47.840
And you keep doing that and
you get these things here.

00:04:47.840 --> 00:04:49.170
So, these things
here are linear

00:04:49.170 --> 00:04:50.890
combinations of those guys.

00:04:50.890 --> 00:04:52.830
Or another way to do it, you
can reverse those row

00:04:52.830 --> 00:04:53.380
operations.

00:04:53.380 --> 00:04:56.170
I could start with these
guys right here.

00:04:56.170 --> 00:04:58.990
And I could just as easily
perform the reverse row

00:04:58.990 --> 00:05:00.420
operations.

00:05:00.420 --> 00:05:02.470
Any linear operation, you can
perform the reverse of it.

00:05:02.470 --> 00:05:04.040
We've seen that multiple
times.

00:05:04.040 --> 00:05:09.690
You could perform row operations
with these guys to

00:05:09.690 --> 00:05:11.420
get all of these guys.

00:05:11.420 --> 00:05:15.070
Or another way to view it is,
these vectors here, these row

00:05:15.070 --> 00:05:20.400
vectors right here, they span
all of these-- or all of these

00:05:20.400 --> 00:05:23.170
row vectors can be represented
of linear combinations of your

00:05:23.170 --> 00:05:24.190
pivot rows right here.

00:05:24.190 --> 00:05:29.280
Obviously, your non-pivot rows
are going to be all 0's.

00:05:29.280 --> 00:05:31.380
And those are useless.

00:05:31.380 --> 00:05:33.670
But, your pivot rows, if you
take linear combinations of

00:05:33.670 --> 00:05:37.870
them, you can clearly do reverse
row echelon form and

00:05:37.870 --> 00:05:39.190
get back to your matrix.

00:05:39.190 --> 00:05:41.280
So, all of these guys can
be represented as linear

00:05:41.280 --> 00:05:42.730
combinations of them.

00:05:42.730 --> 00:05:47.140
And all of these pivot entries
are by definition-- well,

00:05:47.140 --> 00:05:48.590
almost by definition--
they are linearly

00:05:48.590 --> 00:05:49.900
independent, right?

00:05:49.900 --> 00:05:50.970
Because I've got a 1 here.

00:05:50.970 --> 00:05:53.320
No one else has a 1 there.

00:05:53.320 --> 00:05:55.880
So this guy can definitely not
be represented as a linear

00:05:55.880 --> 00:05:57.990
combination of the other guy.

00:05:57.990 --> 00:06:00.710
So why am I going through
this whole exercise?

00:06:00.710 --> 00:06:02.300
Well, we started off
saying we wanted a

00:06:02.300 --> 00:06:05.470
basis for the row space.

00:06:05.470 --> 00:06:09.600
We wanted some minimum set of
linearly independent vectors

00:06:09.600 --> 00:06:12.610
that spans everything that
these guys can span.

00:06:12.610 --> 00:06:14.920
Well, if all of these guys can
be represented as linear

00:06:14.920 --> 00:06:17.660
combinations of these row
vectors in reduced row echelon

00:06:17.660 --> 00:06:23.090
form-- or these pivot rows in
reduced row echelon form-- and

00:06:23.090 --> 00:06:25.910
these guys are all linearly
independent, then they are a

00:06:25.910 --> 00:06:27.980
reasonable basis.

00:06:27.980 --> 00:06:30.810
So these pivot rows right here,
that's one of them, this

00:06:30.810 --> 00:06:33.750
is the second one, this is the
third one, maybe they're the

00:06:33.750 --> 00:06:34.380
only three.

00:06:34.380 --> 00:06:36.050
This is just my particular
example.

00:06:36.050 --> 00:06:38.715
That would be a suitable basis
for the row space.

00:06:38.715 --> 00:06:40.520
So let me write this down.

00:06:40.520 --> 00:06:57.480
The pivot rows in reduced row
echelon form of A are a basis

00:06:57.480 --> 00:07:03.470
for the row space of A.

00:07:03.470 --> 00:07:07.180
And the row space of A is the
same thing, or the column

00:07:07.180 --> 00:07:08.230
space of A transpose.

00:07:08.230 --> 00:07:10.370
The row space of A is the
same thing as the

00:07:10.370 --> 00:07:11.490
column space of A transpose.

00:07:11.490 --> 00:07:13.150
We've see that multiple times.

00:07:13.150 --> 00:07:16.870
Now, if we want to know the
dimension of your column

00:07:16.870 --> 00:07:20.770
space, we just count the number
of pivot rows you have.

00:07:20.770 --> 00:07:22.530
So you just count the number
of pivot rows.

00:07:22.530 --> 00:07:25.740
So the dimension of your row
space, which is the same thing

00:07:25.740 --> 00:07:28.360
as the column space of A
transpose, is going to be the

00:07:28.360 --> 00:07:32.420
number of pivot rows you have
in reduced row echelon form.

00:07:32.420 --> 00:07:35.010
Or, even simpler, the number
of pivot entries you have

00:07:35.010 --> 00:07:37.430
because every pivot entry
has a pivot row.

00:07:37.430 --> 00:07:46.760
So we can write that the rank of
A transpose is equal to the

00:07:46.760 --> 00:07:57.180
number of pivot entries in
reduced row echelon form of A.

00:07:57.180 --> 00:07:57.490
Right?

00:07:57.490 --> 00:07:59.950
Because every pivot entry
corresponds to a pivot row.

00:07:59.950 --> 00:08:03.840
Those pivot rows are a suitable
basis for the entire

00:08:03.840 --> 00:08:06.260
row space, because every row
could be made with a linear

00:08:06.260 --> 00:08:07.910
combination of these guys.

00:08:07.910 --> 00:08:10.270
And since all these can be, then
anything that these guys

00:08:10.270 --> 00:08:12.970
can construct, these
guys can construct.

00:08:12.970 --> 00:08:13.930
Fair enough.

00:08:13.930 --> 00:08:16.350
Now, what is the rank of A?

00:08:16.350 --> 00:08:18.160
This is the rank of A transpose
that we've been

00:08:18.160 --> 00:08:20.440
dealing with so far.

00:08:20.440 --> 00:08:30.350
The rank of A is equal to
the dimension of the

00:08:30.350 --> 00:08:32.620
column space of A.

00:08:32.620 --> 00:08:41.669
Or, you could say it's the
number of vectors in the basis

00:08:41.669 --> 00:08:44.450
for the column space of A.

00:08:44.450 --> 00:08:50.910
So if we take that same matrix
A that we used above, and we

00:08:50.910 --> 00:08:55.860
instead we write it as a bunch
of column vectors, so c1, c2,

00:08:55.860 --> 00:08:57.720
all the way to cn.

00:08:57.720 --> 00:09:00.440
We have n columns right there.

00:09:00.440 --> 00:09:02.490
The column space is essentially
the subspace

00:09:02.490 --> 00:09:05.150
that's spanned by all of these
characters right here, right?

00:09:05.150 --> 00:09:06.790
Spanned by each of these
column vectors.

00:09:06.790 --> 00:09:13.810
So the column space of A is
equal to the span of c1, c2,

00:09:13.810 --> 00:09:15.810
all the way to cn.

00:09:15.810 --> 00:09:17.410
That's the definition of it.

00:09:17.410 --> 00:09:19.280
But we want to know the number
of basis vectors.

00:09:19.280 --> 00:09:23.020
And we've seen before-- we've
done this multiple times--

00:09:23.020 --> 00:09:25.170
what suitable basis
vectors could be.

00:09:25.170 --> 00:09:28.800
If you put this into reduced row
echelon form, and you have

00:09:28.800 --> 00:09:33.480
some pivot entries and their
corresponding pivot columns,

00:09:33.480 --> 00:09:35.820
so some pivot entries with
their corresponding pivot

00:09:35.820 --> 00:09:37.380
columns just like that.

00:09:37.380 --> 00:09:41.540
Maybe that's like that, and then
maybe this one isn't one,

00:09:41.540 --> 00:09:42.620
and then this one is.

00:09:42.620 --> 00:09:44.210
So you have a certain number
of pivot columns.

00:09:47.040 --> 00:09:49.450
Let me do it with another
color right here.

00:09:49.450 --> 00:09:53.190
When you put A into reduced row
echelon form, we learned

00:09:53.190 --> 00:09:56.660
that the basis vectors, or the
basis columns that form a

00:09:56.660 --> 00:09:59.090
basis for your column space,
are the columns that

00:09:59.090 --> 00:10:02.000
correspond to the
pivot columns.

00:10:02.000 --> 00:10:04.750
So the first column here is a
pivot column, so this guy

00:10:04.750 --> 00:10:05.780
could be a basis vector.

00:10:05.780 --> 00:10:08.010
The second column is, so this
guy could be a pivot vector.

00:10:08.010 --> 00:10:10.720
Or maybe the fourth one right
here, so this guy could be a

00:10:10.720 --> 00:10:11.880
pivot vector.

00:10:11.880 --> 00:10:15.690
So, in general, you just say
hey, if you want to count the

00:10:15.690 --> 00:10:17.290
number basis vectors-- because
we don't even have to know

00:10:17.290 --> 00:10:18.400
what they are to figure
out the rank.

00:10:18.400 --> 00:10:20.230
We just have to know the
number they are.

00:10:20.230 --> 00:10:22.960
Well you say, well for every
pivot column here, we have a

00:10:22.960 --> 00:10:24.530
basis vector over there.

00:10:24.530 --> 00:10:26.990
So we could just count the
number pivot columns.

00:10:26.990 --> 00:10:29.510
But the number of pivot columns
is equivalent to just

00:10:29.510 --> 00:10:31.510
the number of pivot entries we
have. Because every pivot

00:10:31.510 --> 00:10:33.200
entry gets its own column.

00:10:33.200 --> 00:10:42.220
So we could say that the rank of
A is equal to the number of

00:10:42.220 --> 00:10:49.870
pivot entries in the reduced
row echelon form of A.

00:10:49.870 --> 00:10:53.000
And, as you can see very
clearly, that's the exact same

00:10:53.000 --> 00:10:55.940
thing that we deduced was
equivalent to the rank of A

00:10:55.940 --> 00:10:57.480
transpose-- the dimension
of the

00:10:57.480 --> 00:10:59.720
columns space of A transpose.

00:10:59.720 --> 00:11:02.240
Or the dimension of the
row space of A.

00:11:02.240 --> 00:11:04.450
So we can now write
our conclusion.

00:11:04.450 --> 00:11:11.100
The rank of A is definitely the
same thing as the rank of

00:11:11.100 --> 00:11:12.350
A transpose.


Orthogonal complements

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.090
Say I've got a subspace V.

00:00:03.090 --> 00:00:13.710
So V is some subspace,
maybe of Rn.

00:00:13.710 --> 00:00:18.180
I'm going to define the
orthogonal complement of V,

00:00:18.180 --> 00:00:23.040
let me write that
down, orthogonal

00:00:23.040 --> 00:00:29.370
complement of V is the set.

00:00:29.370 --> 00:00:31.905
And, this is shorthand notation
right here, would be

00:00:31.905 --> 00:00:34.780
the orthogonal complement
of V.

00:00:34.780 --> 00:00:37.360
So we write this little
orthogonal notation as a

00:00:37.360 --> 00:00:38.440
superscript on V.

00:00:38.440 --> 00:00:42.200
And you can pronounce this
as 'V perp', not for

00:00:42.200 --> 00:00:44.770
'perpetrator' but for
'perpendicular.'

00:00:44.770 --> 00:00:55.200
So V perp is equal to the set of
all x's, all the vectors x

00:00:55.200 --> 00:01:06.440
that are a member of our Rn,
such that x dot V is equal to

00:01:06.440 --> 00:01:17.230
0 for every vector V that is
a member of our subspace.

00:01:17.230 --> 00:01:20.100
So we're essentially saying,
look, you have some subspace,

00:01:20.100 --> 00:01:21.760
it's got a bunch of
vectors in it.

00:01:21.760 --> 00:01:26.240
Now if I can find some other
set of vectors where every

00:01:26.240 --> 00:01:29.670
member of that set is orthogonal
to every member of

00:01:29.670 --> 00:01:34.000
the subspace in question, then
the set of those vectors is

00:01:34.000 --> 00:01:39.480
called the orthogonal
complement of V.

00:01:39.480 --> 00:01:42.520
And you write it this way,
V perp, right there.

00:01:42.520 --> 00:01:45.670
So the first thing that we just
tend to do when we are

00:01:45.670 --> 00:01:49.930
defining a space or defining
some set is to see, hey, is

00:01:49.930 --> 00:01:51.950
this a subspace?

00:01:51.950 --> 00:01:55.530
Is V perp, or the orthogonal
complement of

00:01:55.530 --> 00:01:57.040
V, is this a subspace?

00:02:00.730 --> 00:02:03.450
Well, you might remember from
many, many videos ago, that we

00:02:03.450 --> 00:02:07.040
had just a couple of conditions
for a subspace.

00:02:07.040 --> 00:02:13.600
That if-- let's say that a and b
are both a member of V perp,

00:02:13.600 --> 00:02:16.620
then we have to wonder
whether a plus b is

00:02:16.620 --> 00:02:18.750
a member of V perp.

00:02:18.750 --> 00:02:19.710
That's our first condition.

00:02:19.710 --> 00:02:23.470
It needs to be closed under
addition in order for this to

00:02:23.470 --> 00:02:24.490
be a subspace.

00:02:24.490 --> 00:02:27.390
And the next condition as well,
if a is a member of V

00:02:27.390 --> 00:02:36.570
perp, is some scalar multiple of
a also a member of V perp?

00:02:36.570 --> 00:02:38.920
And the last one, it has to
contain the zero vector.

00:02:38.920 --> 00:02:40.780
Which is a little bit redundant
with this, because

00:02:40.780 --> 00:02:45.810
if any scalar multiple of a is
a member of our orthogonal

00:02:45.810 --> 00:02:48.610
complement of V, you could
just multiply it by 0.

00:02:48.610 --> 00:02:53.950
So it would imply that the zero
vector is a member of V.

00:02:53.950 --> 00:02:56.740
So what does this imply?

00:02:56.740 --> 00:03:00.670
What is the fact that a and
b are members of V perp?

00:03:00.670 --> 00:03:07.170
That means that a dot V, where
this V is any member of our

00:03:07.170 --> 00:03:14.360
original subspace V, is equal
to 0 for any V that is a

00:03:14.360 --> 00:03:16.650
member of our subspace V.

00:03:16.650 --> 00:03:20.122
And it also means that b, since
b is also a member of V

00:03:20.122 --> 00:03:24.790
perp, that V dot any member of
our subspace is also going to

00:03:24.790 --> 00:03:32.130
be 0, or any b that
is a member of V.

00:03:32.130 --> 00:03:36.170
So what happens if we
take a plus b dot V?

00:03:36.170 --> 00:03:37.330
Let's do that.

00:03:37.330 --> 00:03:42.300
So if I do a plus b dot
V, what is this

00:03:42.300 --> 00:03:43.100
going to be equal to?

00:03:43.100 --> 00:03:49.860
This is going to be equal
to a dot V plus b dot V.

00:03:49.860 --> 00:03:53.290
And we just said, the fact that
both a and b are members

00:03:53.290 --> 00:03:56.210
of our orthogonal complement
means that both of these

00:03:56.210 --> 00:03:57.970
quantities are going
to be equal to 0.

00:03:57.970 --> 00:04:01.250
So this is going to be
equal to 0 plus 0

00:04:01.250 --> 00:04:02.910
which is equal to 0.

00:04:02.910 --> 00:04:06.860
So a plus b is definitely a
member of our orthogonal

00:04:06.860 --> 00:04:07.440
complement.

00:04:07.440 --> 00:04:10.620
So we got our check box right
there I'll do it in a

00:04:10.620 --> 00:04:12.430
different color than
the question mark.

00:04:12.430 --> 00:04:13.690
Check, for the first

00:04:13.690 --> 00:04:16.950
condition, for being a subspace.

00:04:16.950 --> 00:04:19.870
Now is ca a member of V perp?

00:04:19.870 --> 00:04:21.959
Well let's just take c.

00:04:21.959 --> 00:04:25.900
If we take ca and dot it with
any member of our original

00:04:25.900 --> 00:04:33.740
subspace this is the same thing
as c times a dot V.

00:04:33.740 --> 00:04:34.850
And what is this equal to?

00:04:34.850 --> 00:04:38.000
By definition a was a member of
our orthogonal complement,

00:04:38.000 --> 00:04:39.230
so this is going to
be equal to 0.

00:04:39.230 --> 00:04:43.750
So this is going to be c times
0, which is equal to 0.

00:04:43.750 --> 00:04:47.830
So this is also a member
of our orthogonal

00:04:47.830 --> 00:04:49.470
complement to V.

00:04:49.470 --> 00:04:52.730
And of course, I can multiply
c times 0 and I

00:04:52.730 --> 00:04:53.810
would get to 0.

00:04:53.810 --> 00:04:56.250
Or you could just say, look, 0
is orthogonal to everything.

00:04:56.250 --> 00:04:58.350
You take the zero vector, dot
it with anything, you're

00:04:58.350 --> 00:04:59.270
going to get 0.

00:04:59.270 --> 00:05:02.040
So the zero vector is always
going to be a member of any

00:05:02.040 --> 00:05:05.250
orthogonal complement, because
it obviously is always going

00:05:05.250 --> 00:05:07.570
to be true for this condition
right here.

00:05:07.570 --> 00:05:16.920
So we know that V perp, or the
orthogonal complement of V, is

00:05:16.920 --> 00:05:18.170
a subspace.

00:05:21.000 --> 00:05:24.040
Which is nice because now we
can apply to it all of the

00:05:24.040 --> 00:05:27.280
properties that we know
of subspaces.

00:05:27.280 --> 00:05:29.670
Now the next question, and I
touched on this in the last

00:05:29.670 --> 00:05:40.260
video, I said that if I have
some matrix A, and lets just

00:05:40.260 --> 00:05:42.820
say it's an m by n matrix.

00:05:42.820 --> 00:05:53.540
In the last video I said that
the row space of A is -- well,

00:05:53.540 --> 00:05:54.960
let me write this way.

00:05:54.960 --> 00:06:03.510
I wrote that the null space of
A is equal to the orthogonal

00:06:03.510 --> 00:06:25.570
complement of the
row space of A.

00:06:25.570 --> 00:06:28.290
And the way that we can write
the row space of A, this thing

00:06:28.290 --> 00:06:31.590
right here, the row space of
A, is the same thing as the

00:06:31.590 --> 00:06:35.510
column space of A transpose.

00:06:35.510 --> 00:06:39.180
So one way you can rewrite this
sentence right here, is

00:06:39.180 --> 00:06:44.710
that the null space of A is the
orthogonal complement of

00:06:44.710 --> 00:06:45.440
the row space.

00:06:45.440 --> 00:06:49.830
The row space is the column
space of the transpose matrix.

00:06:49.830 --> 00:06:52.810
And the claim, which I have
not proven to you, is that

00:06:52.810 --> 00:06:54.580
this is the orthogonal
complement of this.

00:06:54.580 --> 00:06:59.170
This is equal to that, the
little perpendicular

00:06:59.170 --> 00:07:00.450
superscript.

00:07:00.450 --> 00:07:03.770
That's the claim, and at least
in the particular example that

00:07:03.770 --> 00:07:06.160
I did in the last two videos
this was the case, where I

00:07:06.160 --> 00:07:08.580
actually showed you that
2 by 3 matrix.

00:07:08.580 --> 00:07:10.436
But let's see if this
applies generally.

00:07:13.780 --> 00:07:17.160
So let me write my matrix
A like this.

00:07:17.160 --> 00:07:19.270
So my matrix A, I can
write it as just a

00:07:19.270 --> 00:07:21.870
bunch of row vectors.

00:07:21.870 --> 00:07:26.260
But just to be consistent with
our notation, with vectors we

00:07:26.260 --> 00:07:31.240
tend to associate as column
vectors, so to represent the

00:07:31.240 --> 00:07:33.060
row vectors here I'm just
going to write them as

00:07:33.060 --> 00:07:34.530
transpose vectors.

00:07:34.530 --> 00:07:37.630
Because in our reality, vectors
will always be column

00:07:37.630 --> 00:07:40.520
vectors, and row vectors are
just transposes of those.

00:07:40.520 --> 00:07:46.720
r1 transpose, r2 transpose and
you go all the way down.

00:07:46.720 --> 00:07:47.830
We have m rows.

00:07:47.830 --> 00:07:52.300
So you're going to
get rm transpose.

00:07:52.300 --> 00:07:55.530
Don't let the transpose
part confuse you.

00:07:55.530 --> 00:07:58.690
I'm just saying that these
are row vectors.

00:07:58.690 --> 00:08:02.310
I'm writing transposes there
just to say that, look these

00:08:02.310 --> 00:08:04.710
are the transposes of
column vectors that

00:08:04.710 --> 00:08:06.230
represent these rows.

00:08:06.230 --> 00:08:09.530
But if it's helpful for you to
imagine them, just imagine

00:08:09.530 --> 00:08:13.050
this is the first row of the
matrix, this is the second row

00:08:13.050 --> 00:08:15.830
of that matrix, so
on and so forth.

00:08:15.830 --> 00:08:18.750
Now, what is the null
space of A?

00:08:18.750 --> 00:08:22.690
Well that's all of
the vectors here.

00:08:22.690 --> 00:08:24.350
Let me do it like this.

00:08:24.350 --> 00:08:31.660
The null space of A is all of
the vectors x that satisfy the

00:08:31.660 --> 00:08:36.230
equation that this is going to
be equal to the zero vector.

00:08:36.230 --> 00:08:38.610
Now to solve this equation,
what can we do?

00:08:38.610 --> 00:08:39.980
We've seen this multiple
times.

00:08:39.980 --> 00:08:44.340
This matrix-vector product is
essentially the same thing as

00:08:44.340 --> 00:08:50.710
saying-- let me write it like
this-- it's going to be equal

00:08:50.710 --> 00:08:52.970
to the zero vector in rm.

00:08:52.970 --> 00:08:57.980
You're going to have m 0's all
the way down to the m'th 0.

00:08:57.980 --> 00:09:00.640
So another way to write this
equation, you've seen it

00:09:00.640 --> 00:09:05.540
before, is when you take the
matrix-vector product, you

00:09:05.540 --> 00:09:07.550
essentially are taking
the dot product.

00:09:07.550 --> 00:09:12.100
So to get to this entry right
here, this entry right here is

00:09:12.100 --> 00:09:15.426
going to be this row dotted
with my vector x.

00:09:15.426 --> 00:09:21.320
So this is r1, we're calling
this row vector r1 transpose.

00:09:21.320 --> 00:09:23.640
This is the transpose of some
column vector that can

00:09:23.640 --> 00:09:25.040
represent that row.

00:09:25.040 --> 00:09:31.770
But that dot, dot my vector x,
this vector x is going to be

00:09:31.770 --> 00:09:33.730
equal to that 0.

00:09:33.730 --> 00:09:35.980
Now, if I take this guy-- let
me do it in a different

00:09:35.980 --> 00:09:40.190
color-- if I take this guy and
I dot him with vector x, it's

00:09:40.190 --> 00:09:42.190
going to be equal to that 0.

00:09:42.190 --> 00:09:49.060
So r2 transpose dot x is
going to be equal to

00:09:49.060 --> 00:09:50.440
that 0 right there.

00:09:53.960 --> 00:10:05.470
So, another way to write this
equation is that r1 transpose

00:10:05.470 --> 00:10:13.490
dot x is equal to 0, r2
transpose dot x is equal to 0,

00:10:13.490 --> 00:10:19.950
all the way down to rn transpose
dot x is equal to 0.

00:10:19.950 --> 00:10:28.070
And by definition the null space
of A is equal to all of

00:10:28.070 --> 00:10:33.710
the x's that are members of--
well in this case it's an m by

00:10:33.710 --> 00:10:36.250
n matrix, you're going to have
n columns-- so it's all the

00:10:36.250 --> 00:10:41.730
x's that are members of rn, such
that Ax is equal to 0.

00:10:41.730 --> 00:10:44.930
Or, you could alternately write
it this way: that if you

00:10:44.930 --> 00:10:47.720
were to dot each of the rows
with x, you're going to be

00:10:47.720 --> 00:10:49.830
equal to 0.

00:10:49.830 --> 00:10:53.150
So you could write it
this way, such that

00:10:53.150 --> 00:10:55.470
Ax is equal to 0.

00:10:55.470 --> 00:10:57.190
That's an easier way
to write it.

00:10:57.190 --> 00:10:59.220
So let's think about it.

00:10:59.220 --> 00:11:03.110
If someone is a member, if
by definition I give

00:11:03.110 --> 00:11:04.450
you some vector V.

00:11:08.950 --> 00:11:15.590
If I were to tell you that
V is a member of the

00:11:15.590 --> 00:11:18.280
null space of A.

00:11:18.280 --> 00:11:19.370
Let's call it V1.

00:11:19.370 --> 00:11:21.630
V1 is a member of
null space of A.

00:11:21.630 --> 00:11:24.800
That means it satisfies this
equation right here.

00:11:24.800 --> 00:11:27.410
That means A times
V is equal to 0.

00:11:27.410 --> 00:11:30.610
A times V is equal to 0 means
that when you dot each of

00:11:30.610 --> 00:11:34.070
these rows with V, you
get equal to 0.

00:11:34.070 --> 00:11:42.740
Or another way of saying that
is that V1 is orthogonal to

00:11:42.740 --> 00:11:47.060
all of these rows, to r1
transpose-- that's just the

00:11:47.060 --> 00:11:55.100
first row-- r2 transpose, all
the way to rm transpose.

00:11:55.100 --> 00:11:58.670
So this is orthogonal to all of
these guys, by definition,

00:11:58.670 --> 00:12:00.530
any member of the null space.

00:12:00.530 --> 00:12:04.880
Well, if you're orthogonal to
all of these members, all of

00:12:04.880 --> 00:12:08.110
these rows in your matrix,
you're also orthogonal to any

00:12:08.110 --> 00:12:09.630
linear combination of them.

00:12:25.340 --> 00:12:29.000
You can imagine, let's say that
we have some vector that

00:12:29.000 --> 00:12:31.830
is a linear combination of
these guys right here.

00:12:31.830 --> 00:12:49.480
So let's say vector w is equal
to some linear combination of

00:12:49.480 --> 00:12:51.280
these vectors right here.

00:12:51.280 --> 00:12:52.590
I wrote them as transposes,
just

00:12:52.590 --> 00:12:53.360
because they're row vectors.

00:12:53.360 --> 00:12:55.885
But I can just write them as
regular column vectors, just

00:12:55.885 --> 00:12:59.030
to show that w could be just
a regular column vector.

00:12:59.030 --> 00:13:07.210
So let's say w is equal to c1
times r1, plus c2 times r2,

00:13:07.210 --> 00:13:12.290
all the way to cm times rm.

00:13:12.290 --> 00:13:13.800
That's what w is equal to.

00:13:13.800 --> 00:13:16.383
So what happens when you take
V, which is a member of our

00:13:16.383 --> 00:13:19.400
null space, and you
dot it with w?

00:13:19.400 --> 00:13:24.750
So if you take V, and dot it
with w, it's going to be V

00:13:24.750 --> 00:13:28.060
dotted with each of these guys,
because our dot product

00:13:28.060 --> 00:13:29.370
has the distributive property.

00:13:29.370 --> 00:13:31.630
So if you dot V with each of
these guys, it's going to be

00:13:31.630 --> 00:13:35.750
equal to c1-- I'm just going
to take the scalar out-- c1

00:13:35.750 --> 00:13:47.100
times V dot r1, plus c2 times V
dot r2-- this is an r right

00:13:47.100 --> 00:13:51.130
here, not a V-- plus,
all the way to, plus

00:13:51.130 --> 00:13:56.030
cm times V dot rm.

00:13:56.030 --> 00:13:59.640
And we know, we already just
said, that V dot each of these

00:13:59.640 --> 00:14:01.435
r's are going to
be equal to 0.

00:14:01.435 --> 00:14:04.020
So all of these are going
to be equal to 0.

00:14:04.020 --> 00:14:06.700
So this whole expression is
going to be equal to 0.

00:14:06.700 --> 00:14:09.530
So if you have any vector that's
a linear combination of

00:14:09.530 --> 00:14:12.700
these row vectors, if you dot
it with any member of your

00:14:12.700 --> 00:14:15.590
null space, you're
going to get 0.

00:14:15.590 --> 00:14:19.840
So let me write this way, what
is any vector that's any

00:14:19.840 --> 00:14:21.630
linear combination
of these guys?

00:14:21.630 --> 00:14:24.150
Well, that's the span
of these guys.

00:14:24.150 --> 00:14:28.170
Or you could say that the row
space, that's the row space.

00:14:28.170 --> 00:14:33.360
So if w is a member of the row
space, which you can just

00:14:33.360 --> 00:14:39.920
represent as a column space of A
transpose, then we know that

00:14:39.920 --> 00:14:44.250
V is a member of
our null space.

00:14:44.250 --> 00:14:49.930
We know that V dot w is going
to be equal to 0, I just

00:14:49.930 --> 00:14:51.490
showed that to you
right there.

00:14:51.490 --> 00:14:56.390
So every member of our null
space is definitely orthogonal

00:14:56.390 --> 00:14:59.210
to every member of
our row space.

00:14:59.210 --> 00:15:01.240
So that's what we know so far.

00:15:01.240 --> 00:15:17.030
Every member of null space of
A is orthogonal to every

00:15:17.030 --> 00:15:27.970
member of the row space of A.

00:15:27.970 --> 00:15:29.390
Now, that only gets
us halfway.

00:15:29.390 --> 00:15:33.010
That still doesn't tell us that
this is equivalent to the

00:15:33.010 --> 00:15:35.530
orthogonal complement
of the null space.

00:15:35.530 --> 00:15:38.510
For example, there might be
members of our orthogonal

00:15:38.510 --> 00:15:43.820
complement of the row space that
aren't a member of our

00:15:43.820 --> 00:15:44.690
null space.

00:15:44.690 --> 00:15:51.710
So let's say that I have
some other vector u.

00:15:51.710 --> 00:16:01.160
Let's say that u is a member of
the orthogonal complement

00:16:01.160 --> 00:16:03.170
of our row space.

00:16:03.170 --> 00:16:05.110
I know the notation is a little
convoluted, maybe I

00:16:05.110 --> 00:16:06.080
should write an r there.

00:16:06.080 --> 00:16:09.470
But I want to really get set
into your mind that the row

00:16:09.470 --> 00:16:12.770
space is just the column
space of the transpose.

00:16:12.770 --> 00:16:16.090
Let's say that u is some member
of our orthogonal

00:16:16.090 --> 00:16:16.860
complement.

00:16:16.860 --> 00:16:18.990
What I want to do is show
you that u has to be

00:16:18.990 --> 00:16:19.750
in your null space.

00:16:19.750 --> 00:16:21.990
And when I show you that,
then we know.

00:16:21.990 --> 00:16:24.230
So far we just said that, OK
then, everything in the null

00:16:24.230 --> 00:16:27.810
space is orthogonal to the row
space, but we don't know that

00:16:27.810 --> 00:16:29.910
everything that's orthogonal
to the row space, which is

00:16:29.910 --> 00:16:32.230
represented by this set,
is also going to be

00:16:32.230 --> 00:16:32.970
in your null space.

00:16:32.970 --> 00:16:35.235
That's what we have to show, in
order for those two sets to

00:16:35.235 --> 00:16:37.230
be equivalent, in order
for the null space

00:16:37.230 --> 00:16:39.520
to be equal to this.

00:16:39.520 --> 00:16:49.990
So if we know this is true, then
this means that u dot w,

00:16:49.990 --> 00:16:58.670
where w is a member of our
row space, is going

00:16:58.670 --> 00:16:59.920
to be equal to 0.

00:17:02.950 --> 00:17:06.700
Let me write this down right
here, that is going to be

00:17:06.700 --> 00:17:08.030
equal to 0.

00:17:08.030 --> 00:17:09.329
And what does that mean?

00:17:09.329 --> 00:17:15.920
That means that u is
also orthogonal.

00:17:15.920 --> 00:17:24.440
So this implies that u dot--
well, r, j, any of the row

00:17:24.440 --> 00:17:32.330
vectors-- is also equal to 0,
where j is equal to 1, through

00:17:32.330 --> 00:17:33.890
all the way through m.

00:17:33.890 --> 00:17:35.090
How do I know that?

00:17:35.090 --> 00:17:38.090
Well, I'm saying that look, you
take u as a member of the

00:17:38.090 --> 00:17:41.850
orthogonal complement of the row
space, so that means u is

00:17:41.850 --> 00:17:44.170
orthogonal to any member
of your row space.

00:17:44.170 --> 00:17:46.960
So in particular the basis
vectors of your row space-- we

00:17:46.960 --> 00:17:50.810
don't know whether all of these
guys are basis vectors--

00:17:50.810 --> 00:17:53.380
these guys are definitely all
members of the row space.

00:17:53.380 --> 00:17:55.990
Some of them are actually the
basis for the row space.

00:17:55.990 --> 00:17:58.765
So that means if you take u dot
any of these guys, it's

00:17:58.765 --> 00:18:00.910
going to be equal to 0.

00:18:00.910 --> 00:18:06.240
So if u dot any of these guys is
equal to 0, that means that

00:18:06.240 --> 00:18:20.240
u dot r1 is 0, u dot r2 is equal
to 0, all the way to u

00:18:20.240 --> 00:18:23.460
dot rm is equal to 0.

00:18:23.460 --> 00:18:28.990
Well, if all of this is true,
that means that A times the

00:18:28.990 --> 00:18:31.280
vector u is equal to 0.

00:18:31.280 --> 00:18:32.860
That implies this, right?

00:18:32.860 --> 00:18:34.740
You stick u there, you take
all the dot products, it's

00:18:34.740 --> 00:18:36.410
going to satisfy
this equation.

00:18:36.410 --> 00:18:40.460
Which implies that u is a member
of our null space.

00:18:40.460 --> 00:18:42.750
So we've just shown you that
every member of your null

00:18:42.750 --> 00:18:46.020
space is definitely a member of
the orthogonal complement.

00:18:46.020 --> 00:18:47.760
And now we've said that every
member of our orthogonal

00:18:47.760 --> 00:18:49.900
complement is a member
of our null space.

00:18:52.590 --> 00:18:53.955
And actually I just noticed
that I made a

00:18:53.955 --> 00:18:54.785
slight error here.

00:18:54.785 --> 00:18:57.260
This dot product, I don't have
to write the transpose here,

00:18:57.260 --> 00:19:00.880
because we've defined our dot
product as the dot product of

00:19:00.880 --> 00:19:01.630
column vectors.

00:19:01.630 --> 00:19:03.680
So this is the transpose
of some column vectors.

00:19:03.680 --> 00:19:06.390
So you can un-transpose
it here and just

00:19:06.390 --> 00:19:07.250
take the dot product.

00:19:07.250 --> 00:19:08.540
Anyway, minor error there.

00:19:08.540 --> 00:19:12.130
But that diverts me from my main
takeaway, my punch line,

00:19:12.130 --> 00:19:13.230
the big picture.

00:19:13.230 --> 00:19:17.530
We now showed you, any member of
our null space is a member

00:19:17.530 --> 00:19:19.740
of the orthogonal complement.

00:19:19.740 --> 00:19:22.280
So we just showed you, this
first statement here is

00:19:22.280 --> 00:19:25.710
another way of saying, any
member of the null space-- or

00:19:25.710 --> 00:19:28.950
that the null space is a subset
of the orthogonal

00:19:28.950 --> 00:19:31.310
complement of the row space.

00:19:31.310 --> 00:19:33.570
So that's our row space, and
that's the orthogonal

00:19:33.570 --> 00:19:35.140
complement of our row space.

00:19:35.140 --> 00:19:42.360
And here we just showed that any
member of the orthogonal

00:19:42.360 --> 00:19:46.040
complement of our row space
is also a member

00:19:46.040 --> 00:19:47.900
of your null space.

00:19:47.900 --> 00:19:50.390
Well, if these two guys are
subsets of each other, they

00:19:50.390 --> 00:19:52.470
must be equal to each other.

00:19:52.470 --> 00:19:59.060
So we now know that the null
space of A is equal to the

00:19:59.060 --> 00:20:03.310
orthogonal complement of the row
space of A or the column

00:20:03.310 --> 00:20:05.270
space of A transpose.

00:20:05.270 --> 00:20:09.720
Now, I related the null space
with the row space.

00:20:09.720 --> 00:20:13.000
I could just as easily make a
bit of a substitution here.

00:20:13.000 --> 00:20:19.380
Let's say that A is
equal to some

00:20:19.380 --> 00:20:21.900
other matrix, B transpose.

00:20:21.900 --> 00:20:22.190
Right?

00:20:22.190 --> 00:20:24.190
It's going to be the transpose
of some matrix, you could

00:20:24.190 --> 00:20:25.620
transpose either way.

00:20:25.620 --> 00:20:27.010
So if I just make that
substitution

00:20:27.010 --> 00:20:28.220
here, what do we get?

00:20:28.220 --> 00:20:34.200
We get, the null space of B
transpose is equal to the

00:20:34.200 --> 00:20:39.380
column space of B transpose,
right?

00:20:39.380 --> 00:20:41.560
A transpose is B transpose
transposed.

00:20:41.560 --> 00:20:44.420
Let me get my parentheses
right.

00:20:44.420 --> 00:20:47.400
And then that thing's orthogonal
complement.

00:20:47.400 --> 00:20:49.510
So what is this equal to?

00:20:49.510 --> 00:20:52.280
The transpose of the transpose
is just equal to B.

00:20:52.280 --> 00:20:57.730
So I can write it as, the null
space of B transpose is equal

00:20:57.730 --> 00:21:02.190
to the orthogonal complement
of the column space of B.

00:21:06.150 --> 00:21:10.540
So just like this, we just show
that the left-- B and A

00:21:10.540 --> 00:21:12.470
are just arbitrary matrices.

00:21:12.470 --> 00:21:17.790
So this showed us that the null
space, sometimes it's

00:21:17.790 --> 00:21:24.150
nice to write in words,
is the orthogonal

00:21:24.150 --> 00:21:32.290
complement of row space.

00:21:32.290 --> 00:21:42.260
And this right here is showing
us, that the left null space

00:21:42.260 --> 00:21:45.090
which is just the same thing as
a null space of a transpose

00:21:45.090 --> 00:21:50.840
matrix, is equal to,
orthogonal-- I'll just

00:21:50.840 --> 00:21:58.610
shorthand it-- complement
of the column space.

00:21:58.610 --> 00:22:00.460
Which are two pretty
neat takeaways.

00:22:00.460 --> 00:22:03.310
We saw a particular example of
it a couple of videos ago, and

00:22:03.310 --> 00:22:05.460
now you see that it's true
for all matrices.