Introduction to ratios

WEBVTT
Kind: captions
Language: en

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Let's try to learn a thing
or two about ratios.

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So ratios are just expressions
that compare quantities.

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So that might just be a fancy
way-- let me just --of saying

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something that you may
or may not understand.

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So let me give you
actual examples.

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If I have 10 horses
and I have 5 dogs.

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And someone were to come to
me and say Sal, what is the

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ratio is of horses to dogs?

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So I want to know how many
horses do I have for

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some number of dogs.

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So I could say I have 10
horses for every 5 dogs.

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So I could say the ratio of
horses to dogs is 10:5.

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Or I could also write
that as a fraction.

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I could say the ratio of
horses to dogs is 10/5.

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Or I could just write it out.

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I could say it is 10 to 5.

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These all are saying
the same thing.

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And the thing I write first,
or the thing that I write on

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top, is the number of horses.

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So this is the number
of horses right there.

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That's the number of horses.

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And that's all the
number of horses.

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If I wanted to talk about the
number of dogs, this is the

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number of dogs, that's the
number of dogs, or that's

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the number of dogs.

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I'm just-- These are all
just expressions that are

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comparing two quantities.

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Now, I just said I have 10
horses for every 5 dogs.

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But what does that mean?

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That means I have 5
horses for every 1 dog.

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Sorry.

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Not 5 horses for every 1 dog.

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It means I have 2 horses
for every 1 dog, right?

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If for 5 dogs I have 10, that
means that for every 1 of these

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dogs, there are 2 horses.

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For every one of-- every 2 of
these horses, there's 1 dog.

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I just kind of reasoned
through that.

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So this is-- I have 2
horses for every 1 dog.

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But how do you get there?

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How do you get
from 10:5 to 2:1?

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Well you can think of what's
the biggest number that divides

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into both of these numbers?

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What's their greatest
common divisor?

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I have a whole video on that.

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The biggest number that divides
into both of these guys is 5.

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So you divide both of them by 5
and you can kind of get this

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ratio into a reduced form.

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And if I write it here, it
would be the same thing

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as 2/1 or 2 to 1.

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And so what's interesting about
ratios, it isn't literally, or

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doesn't always have to be
literally, the number of horses

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and the numbers of
dogs you have.

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What a ratio tells you
is how many horses do

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I have for every dog.

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Or how many dogs do I
have for every horse.

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Now, just to make things clear,
what if someone asked me what

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is the ratio of dogs to horses?

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So what's the difference
in these two statements?

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Here I said horses to dogs.

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Here I'm saying dogs to horses.

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So, since I've switched the
ratio-- What I'm looking for--

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I'm looking for the ratio
of dogs to horses, I

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switch the numbers.

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So dogs-- For every 5
dogs, I have 10 horses.

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Or if I divide both of
these by 5, for every 1

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dog, I have 5 horses.

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So the ratio of dogs to
horses is 5:10 or 1:5.

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Or you could write it this way.

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1 to-- I can write it-- Let
me write it down here.

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1/5.

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Or I could write 1 to 5.

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And the general convention--
This wouldn't be

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necessarily incorrect.

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That's not wrong.

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But the general convention is
to get your ratio or your

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fraction, if you want to call
it that, into the simplest

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form or into this reduced
form right there.

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Let's just do a couple
of other examples.

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Let's say I have 20 apples.

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Let's say I have 40 oranges.

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And let's say that I
have 60 strawberries.

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Now what is the ratio of apples
to oranges to strawberries?

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I could write it like this.

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I could write what is the ratio
of-- I'll write it like this

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--apples:oranges:strawberries?

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Well I can start off by
literally saying, well for 60

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strawberries-- for every 60
strawberries, I have 40

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oranges and I have 20 apples.

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And this would be legitimate.

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You could say the ratio
of apples to oranges to

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strawberries are 20:40-- Sorry.

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20:40:60.

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And that wouldn't be wrong.

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But we saw before, we could
put into reduced form.

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So we think of what's the
largest number that divides

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into all three of these?

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We can't just do it into two of
these now because now my ratio

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has three actual quantities.

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Well the largest number
that divides into all

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of these guys is 20.

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If we divide all of them by 20,
we can then say for every 1

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apple, I now have-- you divide
this guy by 20 --I have 2

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oranges, and I have
3 strawberries.

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So the ratio of
apples to oranges to

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strawberries is 1:2:3.

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And I got that, in every
case, by just dividing

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these guys all by 20.

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I divided by 20.

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I think you get
the general idea.

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If someone were to ask you
what's the ratio of-- Let me

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just write it down because it
never hurts to have a little

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bit more clarification.

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If someone wanted to know
the ratio of strawberries

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to oranges-- Let me get
into my orange color.

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Strawberries to
oranges to apples.

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I thought I was going
to do that in yellow.

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To apples.

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What is this ratio going to be?

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Well for every 3 strawberries,
I have 2 oranges

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and I have 1 apple.

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So then it would be 3:2:1.

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The general idea is whatever
order someone asks you for the

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different items, you put-- the
ratio is going to be in

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that same exact order.

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Now, in all of the examples so
far I gave you the number of

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quantity-- the quantity of
things we had and I-- we

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figured out the ratio.

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What if it went the other way?

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What if I told you a ratio?

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What if I said the ratio of
boys to girls in a classroom

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is-- Let's say the ratio
of boys to girls is 2/3.

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Which I could've also written
as 2:3 just like that.

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So for every 2 boys, I have
3 girls or for every 3

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girls, I have 2 boys.

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And let's say that there are
40 students in the classroom.

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And then someone were to ask
you how many girls are there?

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How many girls are
in the classroom?

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So this seems a little
bit more convoluted than

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what we did before.

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We know the total number of
students and we know the ratio.

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But how many girls
are in the room?

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So let's think
about it this way.

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The fact that the ratio
of boys to girls-- I'll

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write it like this.

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Boys-- Maybe I'll be
stereotypical with the colors.

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The ratio of boys to
girls is equal to 2:3.

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Hate to be so stereotypical,
but it doesn't hurt.

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2:3.

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The ratio of boys
to girls is 2:3.

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So this stands for every
3 girls, there's 2 boys.

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For every 2 boys,
there's 3 girls.

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But what does it also say?

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It also says for every 5
students, there are what?

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There are 2 boys and 3 girls.

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Now why is this
statement helpful?

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Well how many groups of
5 students do I have?

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I have 40 students in my
class right there, right?

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I have 40 students in my class.

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And for every 5 students,
there are 2 boys and 3 girls.

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So how many groups of
5 students do I have?

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So I have a total
of 40 students.

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Let me do it in
this purple color.

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I have 40 students and then
there are 5 students per group.

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And I figured out that group
just by looking at the ratio.

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For every 5 students, I
have 2 boys and 3 girls.

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How many groups of 5
students do I have?

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So that means that I have 8
groups-- 40 divided by 5 --I

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have 8 groups of 5 students.

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Now we're wondering how
many girls there are.

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So each group is going
to have 3 girls.

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So how many girls do I have?

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I have 8 groups, each
of them have 3 girls.

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So I have 8 groups times 3
girls per group is equal to

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24 girls in the classroom.

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And you could do the same
exercise with boys.

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How many boys are there?

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There's a couple of
ways you could do it.

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You could say for every
group, there are 2 boys.

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There's 8 groups.

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There's 16 boys.

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Or you could say
there's 40 students.

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24 of them are girls.

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40 minus 24 is 16.

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So either way you
get to 16 boys.

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And if you want to pick
up a fast way to do it.

00:10:44.760 --> 00:10:45.670
It would be identical.

00:10:45.670 --> 00:10:48.940
You'd say look, 2 plus 3 is 5.

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For every 5 students,
2 boys, 3 girls.

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How many groups are there?

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You say 40 divided by 5
is equal to 8 groups.

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Every group has 3 girls.

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So you do 8 times 3 is
equal to 24 girls.

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Let's do one that's a little
bit harder than that.

00:11:08.130 --> 00:11:16.800
Let's do one where I say that
the ratio of let's say--

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Well let's go back
to the farm example.

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The ratio of sheep--
I'll do sheep in white.

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The ratio of sheep to--
I don't know --chickens

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to-- I don't know.

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What's another farm animal?

00:11:38.960 --> 00:11:40.920
--to pigs.

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The ratio of sheep to chicken
to pigs-- Maybe I should just

00:11:45.510 --> 00:11:46.560
say chicken right there.

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The ratio of sheep to chicken
to pigs-- Or chickens.

00:11:49.830 --> 00:11:51.390
I should say chickens.

00:11:51.390 --> 00:12:02.130
Is-- Let's say the
ratio is 2:5:10.

00:12:02.130 --> 00:12:03.990
And notice, I can't
reduce this anymore.

00:12:03.990 --> 00:12:06.470
There's no number that
divides into all of these.

00:12:06.470 --> 00:12:10.110
So this is the ratio if
sheep to chickens to pigs.

00:12:10.110 --> 00:12:12.880
And let's say that I have
a total of 51 animals.

00:12:21.510 --> 00:12:24.230
And I want to know how
many chickens do I have.

00:12:32.450 --> 00:12:34.490
Well we do the same idea.

00:12:34.490 --> 00:12:38.400
For every 2 sheep, I have 5
chickens and I have 10 pigs.

00:12:38.400 --> 00:12:48.770
That tells me for every 17
animals-- So every group of

00:12:48.770 --> 00:12:51.350
17 animals, what do I have?

00:12:51.350 --> 00:12:52.730
And where did I get 17 from?

00:12:52.730 --> 00:12:55.590
I just added 2 plus 5 plus 10.

00:12:55.590 --> 00:12:59.370
For every 17 animals,
I'm going to have-- Let

00:12:59.370 --> 00:13:01.010
me pick a new color.

00:13:01.010 --> 00:13:12.330
I'm going to have 2 sheep,
5 chickens, and 10 pigs.

00:13:12.330 --> 00:13:16.290
Now, how many groups of
17 animals do I have?

00:13:16.290 --> 00:13:18.600
I have a total of 51 animals.

00:13:18.600 --> 00:13:22.820
So if there's 17 animals per
group, 51 animals divided

00:13:22.820 --> 00:13:24.840
by 17 animals per group.

00:13:24.840 --> 00:13:30.470
I have 3 groups of 17 animals.

00:13:35.210 --> 00:13:37.140
Now I want to know
how many chickens.

00:13:37.140 --> 00:13:40.960
Every group has 5 chickens.

00:13:40.960 --> 00:13:42.020
We already know that.

00:13:42.020 --> 00:13:43.630
And I have 3 groups.

00:13:43.630 --> 00:13:44.560
So I have 3 groups.

00:13:44.560 --> 00:13:45.970
Every group has 5 chickens.

00:13:45.970 --> 00:13:49.110
So I'm going to have 3
times 5 chickens, which

00:13:49.110 --> 00:13:52.085
is equal to 15 chickens.

00:13:54.970 --> 00:13:55.800
Not too bad.

00:13:55.800 --> 00:13:58.660
All I did is add these up and
say for every 17 animals,

00:13:58.660 --> 00:14:00.120
I've got 5 chickens.

00:14:00.120 --> 00:14:02.270
I've got 3 groups of 17.

00:14:02.270 --> 00:14:05.260
So for each of those
groups, I have 5 chickens.

00:14:05.260 --> 00:14:07.540
3 times 5 is 15 chicken.

00:14:07.540 --> 00:14:10.380
You could use the same process
to figure out the number sheep

00:14:10.380 --> 00:14:12.630
or pigs you might have.


Transpose of a vector

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:04.070
Say I have a vector v that's
a member of Rn.

00:00:04.070 --> 00:00:06.260
So it's got n components
in it.

00:00:06.260 --> 00:00:10.850
So v1, v2, all the
way down to vn.

00:00:10.850 --> 00:00:13.990
I've touched on the idea before,
but now that we've

00:00:13.990 --> 00:00:17.090
seen what a transpose is, and
we've taken transposes of

00:00:17.090 --> 00:00:19.610
matrices, there's no reason
why we can't take the

00:00:19.610 --> 00:00:23.900
transpose of a vector, or a
column vector in this case.

00:00:23.900 --> 00:00:27.800
So what would v transpose
look like?

00:00:27.800 --> 00:00:32.659
Well if you think of this as a n
by 1 matrix, which it is, it

00:00:32.659 --> 00:00:35.200
has n rows and one column.

00:00:35.200 --> 00:00:35.990
Then what are we going to get?

00:00:35.990 --> 00:00:39.160
We're going to have a 1 by n
matrix when you take the

00:00:39.160 --> 00:00:40.060
transpose of it.

00:00:40.060 --> 00:00:43.125
And this one column is going
to turn into the one row.

00:00:43.125 --> 00:00:47.380
So you're going to have
it be equal to v1, v2,

00:00:47.380 --> 00:00:51.270
all the way to vn.

00:00:51.270 --> 00:00:53.830
And you might remember, we've
already touched on this in a

00:00:53.830 --> 00:00:56.630
lot of matrices before.

00:00:56.630 --> 00:00:58.980
Let's say that's
some matrix A.

00:00:58.980 --> 00:01:01.960
We called the row vectors of
those matrix, we called them

00:01:01.960 --> 00:01:07.500
the transpose of some column
vectors, a1 transpose, a2

00:01:07.500 --> 00:01:11.480
transpose, all the way
down to an transpose.

00:01:11.480 --> 00:01:14.820
In fact, not so many videos ago
I had those row vectors,

00:01:14.820 --> 00:01:17.510
and I could have just called
them the transpose of column

00:01:17.510 --> 00:01:18.660
vectors, just like that.

00:01:18.660 --> 00:01:21.430
And that would have been, in
some ways, a better way to do

00:01:21.430 --> 00:01:24.160
it because we've defined all
these operations around column

00:01:24.160 --> 00:01:27.760
vectors, so you could always
refer to the transpose of the

00:01:27.760 --> 00:01:30.140
transpose and then do some
operations on them.

00:01:30.140 --> 00:01:32.480
But anyway, I don't want
to get too diverted.

00:01:32.480 --> 00:01:35.030
But let's think a little bit
of what happens when you

00:01:35.030 --> 00:01:37.730
operate this vector, or you take
some operation of this

00:01:37.730 --> 00:01:39.230
vector with some
other vectors.

00:01:39.230 --> 00:01:42.940
So let's say I have another
vector here that's w, and it's

00:01:42.940 --> 00:01:45.080
also a member of Rn.

00:01:45.080 --> 00:01:50.760
So you have w1, w2, all
the way down to wn.

00:01:50.760 --> 00:01:53.340
There's a couple of things that
we're already, I think,

00:01:53.340 --> 00:01:55.140
reasonably familiar with.

00:01:55.140 --> 00:01:57.740
You could take the dot
product of v and w.

00:01:57.740 --> 00:02:00.710
v dot w is equal to what?

00:02:00.710 --> 00:02:10.759
It is equal to v1 times w1, plus
v2, w2, and you just keep

00:02:10.759 --> 00:02:13.760
going all the way to vn, wn.

00:02:13.760 --> 00:02:16.420
This is the definition
of the dot product

00:02:16.420 --> 00:02:18.540
of two-column vectors.

00:02:18.540 --> 00:02:23.470
Now, how can we relate that to
maybe the transpose of v?

00:02:23.470 --> 00:02:27.050
Well, we could take the
transpose of v-- let me write

00:02:27.050 --> 00:02:30.410
it this way-- what is-- if I did
a matrix multiplication,

00:02:30.410 --> 00:02:36.160
so I did v1, v2, all the way
to vn-- so this is v

00:02:36.160 --> 00:02:39.490
transpose, that's v transpose--
and I take the

00:02:39.490 --> 00:02:40.750
product of that with w.

00:02:44.010 --> 00:02:49.830
So I have w1, w2, all
the way down to wn.

00:02:49.830 --> 00:02:52.740
Now, if I view these as just
matrices-- this is w right

00:02:52.740 --> 00:02:56.220
here-- if I viewed these just
as matrices, is this

00:02:56.220 --> 00:02:59.730
matrix-matrix product
well-defined?

00:02:59.730 --> 00:03:04.640
Over here I have a
n by 1 matrix.

00:03:04.640 --> 00:03:07.250
Here I have a 1-- sorry.

00:03:07.250 --> 00:03:12.230
Here, the first one I have is
a 1 by n matrix-- I have one

00:03:12.230 --> 00:03:14.140
row and n columns.

00:03:14.140 --> 00:03:18.350
And here I have an
n by 1 matrix.

00:03:18.350 --> 00:03:20.930
I have n rows and
only one column.

00:03:20.930 --> 00:03:22.190
So this is well-defined.

00:03:22.190 --> 00:03:24.490
I have the same number
of columns here

00:03:24.490 --> 00:03:26.740
as I have rows here.

00:03:26.740 --> 00:03:29.050
This is going to result
in a 1 by 1 matrix.

00:03:29.050 --> 00:03:30.520
And what's it going
to look like?

00:03:30.520 --> 00:03:37.270
It's going to equal to v1 times
w1-- let me write it

00:03:37.270 --> 00:03:43.530
like this-- v1, w1 plus
v2, w2-- it's only

00:03:43.530 --> 00:03:44.250
going to have one entry.

00:03:44.250 --> 00:03:47.670
We could write it is as just
a 1 by 1 matrix like that.

00:03:47.670 --> 00:03:50.280
Let me just do it-- 1 by
1 matrix like that.

00:03:50.280 --> 00:03:56.270
v1, w1 plus v2, w2-- let me
just, I could write v2 there--

00:03:56.270 --> 00:04:00.260
plus all the way to vn, wn.

00:04:00.260 --> 00:04:00.820
That's what it'll be.

00:04:00.820 --> 00:04:03.370
It'll just be a 1 by 1 matrix
that looks like that.

00:04:03.370 --> 00:04:06.360
But you might notice that
these two things are

00:04:06.360 --> 00:04:07.420
equivalent.

00:04:07.420 --> 00:04:15.950
So we can make the statement
that v dot w, which is the

00:04:15.950 --> 00:04:21.980
same thing as w dot v, these
things are equivalent to-- v

00:04:21.980 --> 00:04:25.450
dot w is the equivalent of-- let
me just write it once over

00:04:25.450 --> 00:04:31.860
here-- v dot w is the same thing
as the transpose of v, v

00:04:31.860 --> 00:04:39.040
transpose times w as just
a matrix-matrix product.

00:04:39.040 --> 00:04:41.470
So if you view v as a matrix,
take its transpose and then

00:04:41.470 --> 00:04:44.150
just take that matrix and take
the product of that with w,

00:04:44.150 --> 00:04:46.440
it's the same thing
as v dot w.

00:04:46.440 --> 00:04:48.140
So that's an interesting
take-away.

00:04:48.140 --> 00:04:50.620
I guess you could argue somewhat
obvious, and we've

00:04:50.620 --> 00:04:54.420
already been referring this--
when I defined matrix-matrix

00:04:54.420 --> 00:04:56.850
products, I kind of said you're
taking the dot product

00:04:56.850 --> 00:04:59.960
of each row with each column,
and you can see that it really

00:04:59.960 --> 00:05:02.180
is, it's really the dot product
of the transpose of

00:05:02.180 --> 00:05:05.640
that row with each column, but
you got the general idea.

00:05:05.640 --> 00:05:08.390
But let's see if we can build
on this a little bit.

00:05:08.390 --> 00:05:14.380
Let's say I have some matrix
A-- let me save our little

00:05:14.380 --> 00:05:17.560
outcome that I have there--
let's say I have some-- let me

00:05:17.560 --> 00:05:20.820
get a good color here-- let's
say I have some matrix A and

00:05:20.820 --> 00:05:23.653
it's an m by n matrix.

00:05:26.170 --> 00:05:31.980
Now if I were to multiply that
times a vector x, so I'm going

00:05:31.980 --> 00:05:36.220
to multiply it by some vector
x-- and let's say that x is a

00:05:36.220 --> 00:05:43.575
member-- let me write it this
way-- x is a member of Rn.

00:05:43.575 --> 00:05:44.950
So it has n elements.

00:05:44.950 --> 00:05:49.670
Or another way you could view it
is, it's an n by 1 matrix.

00:05:49.670 --> 00:05:52.260
Now when I take the product of
these, what am I going to get?

00:05:52.260 --> 00:05:56.850
Or another way to say it is,
what is the vector Ax?

00:05:56.850 --> 00:05:58.520
When I take this product, I'm
just going to get another

00:05:58.520 --> 00:06:00.090
vector, and what's
it going to be?

00:06:00.090 --> 00:06:03.640
It's going to be an
m by 1 vector.

00:06:06.190 --> 00:06:11.400
So we could say that Ax
is a member of Rm.

00:06:11.400 --> 00:06:13.160
It's going to have m
elements, right?

00:06:13.160 --> 00:06:16.790
If this was equal to, if you
said that Ax equal to, I don't

00:06:16.790 --> 00:06:20.780
know, let's say it's equal to
z, z would have m elements.

00:06:20.780 --> 00:06:26.370
You would have z1, z2, all
the way down to zm.

00:06:26.370 --> 00:06:30.420
And I know that because you have
m rows in A, and you have

00:06:30.420 --> 00:06:33.870
only one-- well you could
say this is m by n,

00:06:33.870 --> 00:06:36.140
this is n by 1.

00:06:36.140 --> 00:06:39.680
The resulting product will be
m by 1, or it'll be a vector

00:06:39.680 --> 00:06:43.440
that is a member of Rm-- it'll
have exactly m elements.

00:06:43.440 --> 00:06:47.890
Now, if that's a vector of Rm,
then the idea of dotting this

00:06:47.890 --> 00:06:50.800
with another member of
Rm is well-defined.

00:06:50.800 --> 00:06:52.870
So let's say that I have
another member of Rm.

00:06:52.870 --> 00:06:55.370
Let's say I have a vector y.

00:06:55.370 --> 00:07:00.120
Let's say y is also
a member of Rm.

00:07:00.120 --> 00:07:03.800
This has-- the vector Ax, the
vector that you get when you

00:07:03.800 --> 00:07:07.130
take this product, has m
elements, this has m elements.

00:07:07.130 --> 00:07:10.480
So the idea of taking their dot
product is well-defined.

00:07:10.480 --> 00:07:11.270
Let me write that.

00:07:11.270 --> 00:07:18.870
So you could take Ax, that's a
vector, and now we are dotting

00:07:18.870 --> 00:07:22.290
it with this vector right here
and we'll get a number.

00:07:22.290 --> 00:07:25.420
We just take each of their
terms, multiply the

00:07:25.420 --> 00:07:27.210
corresponding terms, add
them all up, and you

00:07:27.210 --> 00:07:28.510
get their dot product.

00:07:28.510 --> 00:07:32.460
But what is this equal to?

00:07:32.460 --> 00:07:35.980
We can just use this little,
I guess you could call it a

00:07:35.980 --> 00:07:39.080
result, that we got earlier
on in this video.

00:07:39.080 --> 00:07:42.380
Using this result, the dot
product of two matrices-- or

00:07:42.380 --> 00:07:46.900
sorry, the dot product of two
vectors is equal to the

00:07:46.900 --> 00:07:50.650
transpose of the first vector
as a kind of a matrix.

00:07:50.650 --> 00:07:55.160
So you can view this
as Ax transpose.

00:07:55.160 --> 00:08:02.600
This is a m by 1,
this is m by 1.

00:08:02.600 --> 00:08:06.370
Now this is now a 1 by m
matrix, and now we can

00:08:06.370 --> 00:08:09.165
multiply 1 by m matrix
times y.

00:08:11.880 --> 00:08:13.290
Just like that.

00:08:13.290 --> 00:08:17.390
Now what is this
thing equal to?

00:08:17.390 --> 00:08:21.040
We saw a while ago, I think it
was two or three videos ago,

00:08:21.040 --> 00:08:25.100
we saw that if we take the
product of two matrices and

00:08:25.100 --> 00:08:28.400
take its transpose, that's equal
to the reverse product

00:08:28.400 --> 00:08:30.140
of the transposes.

00:08:30.140 --> 00:08:32.870
You switch the order and then
take the transposes.

00:08:32.870 --> 00:08:36.280
So this is going to be equal
to-- this purple part-- is

00:08:36.280 --> 00:08:43.090
going to be equal to
x transpose times A

00:08:43.090 --> 00:08:47.470
transpose times y.

00:08:47.470 --> 00:08:50.070
And this is just matrix
products.

00:08:50.070 --> 00:08:51.230
These are matrix products.

00:08:51.230 --> 00:08:52.990
These aren't necessarily
vector operations.

00:08:52.990 --> 00:08:55.265
We're treating all of these
vectors as matrices.

00:08:55.265 --> 00:08:57.890
And of course, we're treating
the matrix as a matrix.

00:08:57.890 --> 00:08:59.640
So what is this equal to?

00:08:59.640 --> 00:09:03.480
Well we know that matrix
products are associative.

00:09:03.480 --> 00:09:05.540
You could put a parentheses--
right now we have a

00:09:05.540 --> 00:09:09.370
parentheses around there from
there, but we could just take

00:09:09.370 --> 00:09:10.390
another association.

00:09:10.390 --> 00:09:18.060
We could say that that is equal
to x transpose times

00:09:18.060 --> 00:09:20.070
these two matrices
times each other.

00:09:20.070 --> 00:09:21.815
This is a vector, but
you can represent it

00:09:21.815 --> 00:09:23.980
as an m by 1 matrix.

00:09:23.980 --> 00:09:28.485
Times A transpose y.

00:09:31.180 --> 00:09:32.580
Just like that.

00:09:32.580 --> 00:09:35.440
Now let's think about what
A transpose y is.

00:09:35.440 --> 00:09:36.840
Let's think about it.

00:09:36.840 --> 00:09:41.930
A transpose-- we have
here a is m by n.

00:09:41.930 --> 00:09:43.870
What is A transpose?

00:09:43.870 --> 00:09:47.880
A transpose is going to
be n by m, right?

00:09:47.880 --> 00:09:49.020
It's going to be an m by n.

00:09:49.020 --> 00:09:51.760
So this is an m by n.

00:09:51.760 --> 00:09:55.230
And then what is this vector
y going to be?

00:09:55.230 --> 00:09:58.890
This is an m by 1.

00:09:58.890 --> 00:10:00.960
So when you take this product,
you're going to

00:10:00.960 --> 00:10:04.390
get an n by 1 matrix.

00:10:04.390 --> 00:10:09.040
Or you could imagine this as a
vector that is a member of Rn.

00:10:09.040 --> 00:10:12.280
So this is a member of Rn.

00:10:12.280 --> 00:10:15.050
The entire product is going to
result with a vector that's a

00:10:15.050 --> 00:10:16.740
member of Rn.

00:10:16.740 --> 00:10:18.810
And of course, it's well-defined
because this is a

00:10:18.810 --> 00:10:22.900
1 by n vector right there.

00:10:22.900 --> 00:10:24.540
Now we can go back
to our identity.

00:10:24.540 --> 00:10:28.100
We have the transpose of some
vector times some other

00:10:28.100 --> 00:10:31.010
vector-- they have the same,
well I guess you could say

00:10:31.010 --> 00:10:33.730
this has as many horizontal
entries as this guy has

00:10:33.730 --> 00:10:35.390
vertical entries,
just like that.

00:10:35.390 --> 00:10:36.330
So what is this equal to?

00:10:36.330 --> 00:10:37.830
We just use that identity.

00:10:37.830 --> 00:10:42.090
This is equal to, the just
regular x in this case,

00:10:42.090 --> 00:10:44.740
instead of x transpose
we'll just have x.

00:10:44.740 --> 00:10:52.350
So this is equal to x dot--
remember, we just un-transpose

00:10:52.350 --> 00:10:57.075
it, I guess you can view it that
way-- dot A transpose y.

00:11:01.090 --> 00:11:02.860
Which is a pretty
neat outcome.

00:11:02.860 --> 00:11:04.470
We got this being
equal to that.

00:11:04.470 --> 00:11:07.350
We can kind of change the
associativity, although we

00:11:07.350 --> 00:11:11.880
have to essentially change the
order a bit and take the

00:11:11.880 --> 00:11:13.630
transpose of our matrix.

00:11:13.630 --> 00:11:15.630
So let me re-write that
just so that you can

00:11:15.630 --> 00:11:16.670
remember the outcome.

00:11:16.670 --> 00:11:20.100
So the two big outcomes of this
video are-- I'll rewrite

00:11:20.100 --> 00:11:27.290
this one up here-- v dot w is
equal to the matrix product of

00:11:27.290 --> 00:11:30.050
v transpose times w.

00:11:30.050 --> 00:11:34.540
And if I have some matrix--
you assume all of these

00:11:34.540 --> 00:11:36.920
matrix-vector products are
well-defined and all the dot

00:11:36.920 --> 00:11:38.340
products are well-defined.

00:11:38.340 --> 00:11:48.000
If I have Ax dot y, some other
vector y, this is equivalent

00:11:48.000 --> 00:11:53.680
to x dot-- you're essentially
putting the A with the other

00:11:53.680 --> 00:11:57.650
vector-- A transpose times y.

00:11:57.650 --> 00:12:00.620
And this just might be a useful
outcome, or a useful

00:12:00.620 --> 00:12:04.260
result, that we could build
upon later in the linear

00:12:04.260 --> 00:12:06.190
algebra playlist.


Transposes of sums and inverses

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:03.890
Let's see if we can prove to
ourselves some more reasonably

00:00:03.890 --> 00:00:06.415
interesting transpose
properties.

00:00:06.415 --> 00:00:10.660
So let's define some matrix C,
that's equal to the sum of two

00:00:10.660 --> 00:00:14.280
other matrices, A and B.

00:00:14.280 --> 00:00:19.040
And so any entry in C, I can
denote with a lowercase cij.

00:00:19.040 --> 00:00:22.830
So if I want the ith row in jth
column it would be cij,

00:00:22.830 --> 00:00:25.880
and so each of its entries are
going to be the sum of the

00:00:25.880 --> 00:00:28.300
corresponding columns that
are matrices A and B.

00:00:28.300 --> 00:00:33.140
So our ij entry in C is going to
be equal to the ij entry in

00:00:33.140 --> 00:00:36.450
A, plus the ij entry in B.

00:00:36.450 --> 00:00:38.470
That's our definition
of matrix addition.

00:00:38.470 --> 00:00:40.710
You just get the corresponding
entry in the same row and

00:00:40.710 --> 00:00:44.130
column, add them up, and you get
your entry in the same row

00:00:44.130 --> 00:00:46.400
and column, and your new
matrix is the sum

00:00:46.400 --> 00:00:48.140
of the other two.

00:00:48.140 --> 00:00:52.610
Now, let's think a little bit
about the transposes of these

00:00:52.610 --> 00:00:53.790
guys right here.

00:00:53.790 --> 00:00:59.130
So, if A looks like this.

00:00:59.130 --> 00:01:00.310
I won't draw all
of the entries.

00:01:00.310 --> 00:01:01.540
It takes forever.

00:01:01.540 --> 00:01:06.100
But each of its entries are
ij, just like that.

00:01:06.100 --> 00:01:08.210
Let's say that A transpose
looks like this.

00:01:08.210 --> 00:01:12.230
Each of its entries, we would
call it, that's if you've got

00:01:12.230 --> 00:01:15.790
that same entry, we're going
to call it a-prime ij.

00:01:15.790 --> 00:01:18.220
And these things aren't probably
going to be the same.

00:01:18.220 --> 00:01:20.350
There's some chance they are,
but they're probably not going

00:01:20.350 --> 00:01:21.650
to be the same.

00:01:21.650 --> 00:01:23.810
But that its ijth entry.

00:01:23.810 --> 00:01:25.930
In the ith row, jth column.

00:01:25.930 --> 00:01:27.490
In A transpose.

00:01:27.490 --> 00:01:29.900
Now, the fact that this is the
transpose of that means that

00:01:29.900 --> 00:01:32.800
everything that's in some row
and column here is going to be

00:01:32.800 --> 00:01:35.160
in that column and row over
here, that the rows and

00:01:35.160 --> 00:01:36.580
columns get switched.

00:01:36.580 --> 00:01:43.330
So we know that we could write
that a-prime ij, we're going

00:01:43.330 --> 00:01:49.560
to have the same entry
that was in aji.

00:01:49.560 --> 00:01:53.220
Maybe aji is over here.
aji is over here.

00:01:53.220 --> 00:01:56.230
So, this thing over here, which
is in the same position

00:01:56.230 --> 00:01:58.940
as this one, is going to be
equal to this guy over here if

00:01:58.940 --> 00:02:00.820
you switched the rows
and columns.

00:02:00.820 --> 00:02:02.060
I think you can accept that.

00:02:02.060 --> 00:02:04.630
And you can make the same
argument for B.

00:02:04.630 --> 00:02:05.820
Let me actually draw it out.

00:02:05.820 --> 00:02:07.070
So if I make B transpose.

00:02:09.949 --> 00:02:12.960
The entry in the ith row and
jth column, I'll call it

00:02:12.960 --> 00:02:15.380
b-prime ij.

00:02:15.380 --> 00:02:16.090
Just like that.

00:02:16.090 --> 00:02:18.450
Just like I did for A.

00:02:18.450 --> 00:02:24.750
So we could say that b-prime ij
is equal to, you take the

00:02:24.750 --> 00:02:28.300
matrix B, what's going to be
the entry that's in the jth

00:02:28.300 --> 00:02:29.920
row and ith column.

00:02:29.920 --> 00:02:31.290
These are, you could
almost say, the

00:02:31.290 --> 00:02:33.060
definition of the transpose.

00:02:33.060 --> 00:02:35.910
If I'm in the third row and
second column now, it's going

00:02:35.910 --> 00:02:39.970
to be what was in the second
row and third column.

00:02:39.970 --> 00:02:40.790
Fair enough.

00:02:40.790 --> 00:02:44.980
So we already have what
cij is equal to.

00:02:44.980 --> 00:02:48.610
What's the transpose of cij
going to be equal to?

00:02:48.610 --> 00:02:50.850
Let me write that down.

00:02:50.850 --> 00:02:55.370
So C transpose, let me
write it over here.

00:02:55.370 --> 00:02:58.620
Write C transpose is equal to.

00:02:58.620 --> 00:02:59.780
I'll use the same notation.

00:02:59.780 --> 00:03:01.100
The prime means that
we're taking

00:03:01.100 --> 00:03:03.380
entries in the transpose.

00:03:03.380 --> 00:03:06.970
So C transpose is just going to
be a bunch of entries, ij.

00:03:06.970 --> 00:03:10.105
And I'll put a little prime
there showing that that's

00:03:10.105 --> 00:03:13.440
entries in the matrix
of the transpose,

00:03:13.440 --> 00:03:15.620
and not in C itself.

00:03:15.620 --> 00:03:23.440
And we know that c-prime
ij is equal to cji.

00:03:23.440 --> 00:03:24.430
Nothing new at all.

00:03:24.430 --> 00:03:26.480
We've just expressed kind
of the definition of the

00:03:26.480 --> 00:03:28.970
transpose for these
three matrices.

00:03:28.970 --> 00:03:32.230
Now what is cji equal to?

00:03:32.230 --> 00:03:34.220
So let's focus on this
a little bit.

00:03:34.220 --> 00:03:36.100
What is cji equal to?

00:03:36.100 --> 00:03:40.480
We know that cij is equal to a
sub ij plus b sub ij, so if

00:03:40.480 --> 00:03:43.320
you swap them around, this is
going to be equal to, you just

00:03:43.320 --> 00:03:45.050
swap the j's and the i's.

00:03:45.050 --> 00:03:48.990
a sub ji plus b sub ji.

00:03:48.990 --> 00:03:51.280
I just used this information
here-- you could almost view

00:03:51.280 --> 00:03:53.420
it as this assumption or
this definition-- to go

00:03:53.420 --> 00:03:54.370
from this to this.

00:03:54.370 --> 00:03:57.480
If I had an x and a y here, I'd
have an x and a y here,

00:03:57.480 --> 00:03:58.860
and a x and a y here.

00:03:58.860 --> 00:04:02.640
I have a j and an i here, so I
have a j and an i there, and a

00:04:02.640 --> 00:04:04.880
j and an i right there.

00:04:04.880 --> 00:04:06.740
Now what are these?

00:04:06.740 --> 00:04:08.620
What are these equal to?

00:04:08.620 --> 00:04:09.870
This is equal to.

00:04:13.310 --> 00:04:18.829
This guy right here is equal
to-- we do it in the green--

00:04:18.829 --> 00:04:26.560
the same entry for the
transpose of a at ij.

00:04:26.560 --> 00:04:28.800
And this is equal to the
same entry for the

00:04:28.800 --> 00:04:31.330
transpose of b at ij.

00:04:36.090 --> 00:04:39.020
Now, what is this telling us?

00:04:39.020 --> 00:04:42.030
It's telling us that the
transpose of C, which is the

00:04:42.030 --> 00:04:46.660
same thing is A plus B, so it's
saying that A plus B, A

00:04:46.660 --> 00:04:51.050
plus B transpose is the same
thing as C transpose.

00:04:51.050 --> 00:04:52.320
Let me write that.

00:04:52.320 --> 00:04:55.350
C transpose is the same thing
as A plus B transpose.

00:04:55.350 --> 00:04:59.030
So these are the entries in A
plus B transpose right here.

00:04:59.030 --> 00:05:00.410
And what is this over here?

00:05:00.410 --> 00:05:01.670
What are these?

00:05:01.670 --> 00:05:03.740
These are the entries
right there.

00:05:03.740 --> 00:05:06.260
We do the equal sign
over here.

00:05:06.260 --> 00:05:07.320
What are these?

00:05:07.320 --> 00:05:13.370
These are the entries in A
transpose plus B transpose.

00:05:13.370 --> 00:05:14.490
Right?

00:05:14.490 --> 00:05:15.600
These are the entries
in A transpose.

00:05:15.600 --> 00:05:17.620
These are the entries
in B transpose.

00:05:17.620 --> 00:05:20.150
If you take the sum of the two,
you're just adding up the

00:05:20.150 --> 00:05:21.000
corresponding entries.

00:05:21.000 --> 00:05:23.870
So that's straightforward to
show that if you take the sum

00:05:23.870 --> 00:05:26.610
of two matrices and then
transpose it, it's equivalent

00:05:26.610 --> 00:05:30.540
to transposing them first, and
then taking their sum.

00:05:30.540 --> 00:05:33.920
Which is a reasonably
neat outcome.

00:05:33.920 --> 00:05:36.510
Let's do one more and I think
we'll finish up all of our

00:05:36.510 --> 00:05:39.190
major transpose properties.

00:05:39.190 --> 00:05:42.360
Let's say that A inverse-- this
is going to be a slightly

00:05:42.360 --> 00:05:43.380
different take on things.

00:05:43.380 --> 00:05:45.130
We're still going to
take the transpose.

00:05:45.130 --> 00:05:51.760
So if we know that A inverse
is the inverse of A, that

00:05:51.760 --> 00:05:58.410
means that A times A inverse
is equal to the identity

00:05:58.410 --> 00:06:01.360
matrix, assuming that these
are n-by-n matrices.

00:06:01.360 --> 00:06:04.430
So it's the n-dimensional
identity matrix.

00:06:04.430 --> 00:06:10.730
And that A inverse times A is
also going to be equal to the

00:06:10.730 --> 00:06:12.250
identity matrix.

00:06:12.250 --> 00:06:14.790
Now, let's take the
transpose of both

00:06:14.790 --> 00:06:16.990
sides of this equation.

00:06:16.990 --> 00:06:18.910
I'll do them both
simultaneously.

00:06:18.910 --> 00:06:21.540
So if you take the transpose of
both sides of the equation,

00:06:21.540 --> 00:06:26.880
you get A times A inverse
transpose is equal to the

00:06:26.880 --> 00:06:28.950
identity matrix transpose.

00:06:28.950 --> 00:06:31.560
And what's the transpose
of the identity matrix?

00:06:31.560 --> 00:06:33.180
Let's draw it out.

00:06:33.180 --> 00:06:34.490
The identity matrix
looks like this.

00:06:34.490 --> 00:06:37.800
You have just ones all the
way down the diagonal and

00:06:37.800 --> 00:06:39.050
everything else is 0.

00:06:41.870 --> 00:06:45.480
Right, and you could view this
as i 1, 1 i 2, 2 all the way

00:06:45.480 --> 00:06:46.710
down to i n, n.

00:06:46.710 --> 00:06:48.285
Everything else is 0.

00:06:48.285 --> 00:06:50.040
So when you take the transpose,
you're just

00:06:50.040 --> 00:06:52.430
swapping out the
zeroes, right?

00:06:52.430 --> 00:06:53.800
These guys don't change.

00:06:53.800 --> 00:06:57.090
The diagonal does not change
when you take the transpose.

00:06:57.090 --> 00:06:59.570
So the transpose of the identity
matrix is equal to

00:06:59.570 --> 00:07:00.820
the identity matrix.

00:07:02.890 --> 00:07:04.490
And so we can apply that
same thing here.

00:07:04.490 --> 00:07:07.750
Let's take the transpose
for this statement.

00:07:07.750 --> 00:07:12.870
So we know that A inverse times
A transpose is equal to

00:07:12.870 --> 00:07:15.090
the identity matrix transpose,
which is equal to

00:07:15.090 --> 00:07:16.960
the identity matrix.

00:07:16.960 --> 00:07:18.610
And then we know what happens
when you take the

00:07:18.610 --> 00:07:19.960
transpose of a product.

00:07:19.960 --> 00:07:21.190
It's equal to the
product of the

00:07:21.190 --> 00:07:23.720
transposes in reverse order.

00:07:23.720 --> 00:07:28.050
So this thing right here we
can rewrite as A inverse

00:07:28.050 --> 00:07:32.450
transpose times A transpose,
which is going to be equal to

00:07:32.450 --> 00:07:33.540
the identity matrix.

00:07:33.540 --> 00:07:35.550
You could do the same
thing over here.

00:07:35.550 --> 00:07:39.930
This thing is going to be equal
to A transpose times A

00:07:39.930 --> 00:07:44.310
inverse transpose, which is also
going to be equal to the

00:07:44.310 --> 00:07:45.880
identity matrix.

00:07:45.880 --> 00:07:48.830
Now, this is an interesting
statement.

00:07:48.830 --> 00:07:55.340
The fact that, if I have this
guy right here, times the

00:07:55.340 --> 00:07:57.510
transpose of A is equal to the
identity matrix, and the

00:07:57.510 --> 00:08:00.885
transpose of A times that same
guy is equal to identity

00:08:00.885 --> 00:08:11.650
matrix, implies that A inverse
transpose is the

00:08:11.650 --> 00:08:16.150
inverse of A transpose.

00:08:16.150 --> 00:08:19.800
Or another way of writing that
is if I take A transpose, and

00:08:19.800 --> 00:08:24.510
if I take its inverse, that is
going to be equal to this guy.

00:08:24.510 --> 00:08:27.925
It's going to be equal to
A inverse transpose.

00:08:27.925 --> 00:08:31.840
So, another neat outcome dealing
with transposes.

00:08:31.840 --> 00:08:35.760
If you take the inverse of the
transpose, it's the same thing

00:08:35.760 --> 00:08:39.159
as the transpose
of the inverse.


Determinant of transpose

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.910
Let's see if taking the
transpose of a matrix does

00:00:03.910 --> 00:00:05.510
anything to its determinant.

00:00:05.510 --> 00:00:09.365
So a good place to start is just
with the 2-by-2 scenario.

00:00:09.365 --> 00:00:12.380
So let's do the 2-by-2
scenario.

00:00:12.380 --> 00:00:14.570
So if I start with
some matrix here.

00:00:14.570 --> 00:00:16.219
Let me just take its
determinant.

00:00:16.219 --> 00:00:20.280
So a, b, c, d.

00:00:20.280 --> 00:00:22.610
And then let me take
its determinant.

00:00:22.610 --> 00:00:27.930
So this is going to be
equal to ad minus bc.

00:00:27.930 --> 00:00:30.370
Let me take the transpose of
this and then take its

00:00:30.370 --> 00:00:30.980
determinant.

00:00:30.980 --> 00:00:36.810
So that would be the determinant
of ac, the columns

00:00:36.810 --> 00:00:40.870
turn into the rows, and
then bd, the rows

00:00:40.870 --> 00:00:42.150
turn into the columns.

00:00:42.150 --> 00:00:43.570
What is this going
to be equal to?

00:00:43.570 --> 00:00:48.620
This is going to be equal
to ad minus bc again.

00:00:48.620 --> 00:00:50.820
The only thing that happened is
these two guys got swapped

00:00:50.820 --> 00:00:52.610
and they multiply times
each other anyway.

00:00:52.610 --> 00:00:54.700
So these two things
are equivalent.

00:00:54.700 --> 00:01:01.090
So at least for the 2-by-2 case,
the determinant of some

00:01:01.090 --> 00:01:07.440
matrix is equal to the
determinant of the transpose

00:01:07.440 --> 00:01:08.100
of that matrix.

00:01:08.100 --> 00:01:10.380
Now, that's just the
2-by-2 case.

00:01:10.380 --> 00:01:14.210
Now, I'm going to make an
inductive argument, or I could

00:01:14.210 --> 00:01:21.170
just say an argument by
induction, to show that this

00:01:21.170 --> 00:01:27.090
works for all n by
n, for all cases.

00:01:27.090 --> 00:01:30.760
And the way you construct an
argument by induction is you

00:01:30.760 --> 00:01:34.990
assume that it's true
for the n-by-n case.

00:01:41.560 --> 00:01:48.373
So let's assume that for
n by n, and let's

00:01:48.373 --> 00:01:49.623
say I have some matrix.

00:01:52.590 --> 00:01:55.920
Let me call it matrix B, and
let's say it's an n-by-n

00:01:55.920 --> 00:02:01.270
matrix, we assume that the
determinant of any matrix B

00:02:01.270 --> 00:02:06.740
that's n by n is equal to the
determinant of B's transpose.

00:02:06.740 --> 00:02:09.289
That's where we started off with
our inductive argument.

00:02:09.289 --> 00:02:20.580
And then we see if given this,
if given that, is it true of n

00:02:20.580 --> 00:02:26.610
plus 1 by n plus 1 matrix?

00:02:26.610 --> 00:02:28.910
Because if we can, if we can
say, look, given that it's

00:02:28.910 --> 00:02:32.470
true for the n-by-n case, it's
going to be true for the n

00:02:32.470 --> 00:02:36.240
plus 1 by n plus 1 case, then
we're done because we know

00:02:36.240 --> 00:02:39.220
it's true for the base case,
the 2-by-2 case, which you

00:02:39.220 --> 00:02:41.130
could say, well, that's
your first n by n.

00:02:41.130 --> 00:02:43.460
So if it's true for the 2 by 2
case, then it'll be true for

00:02:43.460 --> 00:02:46.170
the 3-by-3 case, because that's
just one increment.

00:02:46.170 --> 00:02:48.530
But then if it's true for the
3-by-3 case, then it'll be

00:02:48.530 --> 00:02:50.170
true for the 4-by-4 case.

00:02:50.170 --> 00:02:52.210
And if it's true for the 4-by-4
case, it'll be true for

00:02:52.210 --> 00:02:55.190
the 5-by-5 case, and you just
keep going up like that.

00:02:55.190 --> 00:02:57.730
So when you do a proof by
induction, you prove a base

00:02:57.730 --> 00:03:03.050
case, and then you prove that if
it's true for n, or in this

00:03:03.050 --> 00:03:07.010
case an n-by-n determinant, if
you can prove that given it's

00:03:07.010 --> 00:03:08.690
true for an n-by-n determinant,
it's going to be

00:03:08.690 --> 00:03:12.560
true for an n plus 1 by n plus
1 determinant or an n plus 1

00:03:12.560 --> 00:03:15.850
by n plus 1 matrix, then you
have completed your proof.

00:03:15.850 --> 00:03:18.145
So let's see if this
is the case.

00:03:18.145 --> 00:03:24.742
So let me construct an n plus
1 by n plus 1 matrix.

00:03:24.742 --> 00:03:30.140
So let's say I have my matrix
A, my favorite letter to use

00:03:30.140 --> 00:03:33.330
for matrices, I think the
entirely linear algebra's

00:03:33.330 --> 00:03:35.180
favorite letter to
use for matrices.

00:03:35.180 --> 00:03:40.290
And let's say it's an n plus
1 by n plus 1 matrix.

00:03:40.290 --> 00:03:44.430
And just to simplify my notation
let's just say that m

00:03:44.430 --> 00:03:45.740
is equal to n plus 1.

00:03:45.740 --> 00:03:48.160
So we could call it
an m-by-m matrix.

00:03:48.160 --> 00:03:49.840
And what is it going
to look like?

00:03:49.840 --> 00:03:53.070
Let's draw its entries
right here.

00:03:53.070 --> 00:03:56.270
I'm going to draw more than the
normal amount of entries.

00:03:56.270 --> 00:04:03.970
a11, this is its first row,
a12, all the way to a1m.

00:04:03.970 --> 00:04:08.000
We have m columns, which is the
same thing as m plus 1.

00:04:08.000 --> 00:04:10.000
That's not m times 1 columns.

00:04:10.000 --> 00:04:11.830
This is m plus 1 as well.

00:04:11.830 --> 00:04:14.800
And then we do our second
row right here.

00:04:14.800 --> 00:04:22.700
a21, a22, a23, all
the way to a2m.

00:04:22.700 --> 00:04:31.120
Then you have your third row
right here: a31, a32, a33, all

00:04:31.120 --> 00:04:34.540
the way to a3m.

00:04:34.540 --> 00:04:38.050
And then you go all
the way down here.

00:04:38.050 --> 00:04:41.015
At the end you have your mth
row, which you could also say

00:04:41.015 --> 00:04:44.150
is your m plus 1 row.

00:04:44.150 --> 00:04:47.600
So it's your mth row, first
column, and then you have a

00:04:47.600 --> 00:04:56.820
sub m2, and then a sub m3,
all the way to a sub mm.

00:04:56.820 --> 00:04:58.280
Fair enough.

00:04:58.280 --> 00:05:02.340
Now, let me draw the
transpose of A.

00:05:02.340 --> 00:05:09.120
So a transpose is also going to
be an n plus 1 by n plus 1

00:05:09.120 --> 00:05:14.270
matrix, which you could also
write as an m-by-m matrix.

00:05:14.270 --> 00:05:16.960
I'm just going to have to take
the transpose of this.

00:05:16.960 --> 00:05:20.670
So the transpose of that, this
row becomes a column, so it

00:05:20.670 --> 00:05:25.690
becomes a11, and this entry
right here is a12.

00:05:25.690 --> 00:05:27.440
It's this entry right there.

00:05:27.440 --> 00:05:35.450
Then you can go all the
way down to a1m.

00:05:35.450 --> 00:05:42.480
Then this pink row becomes a
pink column here, a21-- I

00:05:42.480 --> 00:05:45.600
wanted to do it in pink.

00:05:45.600 --> 00:05:54.190
You have a21, a22, you have an
a23, and it goes all the way

00:05:54.190 --> 00:05:56.680
down to a2m.

00:05:56.680 --> 00:05:59.065
You have your green row right
here, it's your third one, so

00:05:59.065 --> 00:06:07.390
it's a31, a32, a33, all
the way down to a3m.

00:06:07.390 --> 00:06:10.820
And then we can just skip a
bunch of rows in this case,

00:06:10.820 --> 00:06:12.270
but it's columns in this case.

00:06:12.270 --> 00:06:18.910
So you just draw some dots
and you have am1, am2.

00:06:18.910 --> 00:06:21.290
I'm just going down this guy,
but this row is now going to

00:06:21.290 --> 00:06:22.875
become-- which was
the last row.

00:06:22.875 --> 00:06:24.970
It's now going to become
the last column.

00:06:24.970 --> 00:06:29.990
am3, all the way down to amm.

00:06:29.990 --> 00:06:33.320
And I have my transpose.

00:06:33.320 --> 00:06:36.100
Now, let's take the
determinant of A.

00:06:39.730 --> 00:06:41.270
Let me do it in purple.

00:06:41.270 --> 00:06:47.260
So the determinant of my matrix
A, we could just go

00:06:47.260 --> 00:06:49.120
down this first row up here.

00:06:49.120 --> 00:06:56.450
It's going to be equal to a11
times the determinant of its

00:06:56.450 --> 00:07:00.410
submatrix, so it's the
determinant of this submatrix

00:07:00.410 --> 00:07:01.660
right here.

00:07:05.110 --> 00:07:10.180
We could call that the
submatrix a sub 11.

00:07:10.180 --> 00:07:12.740
We've seen this notation
before.

00:07:12.740 --> 00:07:19.170
So it's the determinant of a sub
11, and then it is minus

00:07:19.170 --> 00:07:24.560
a12 times the determinant of
its submatrix, so you cross

00:07:24.560 --> 00:07:26.710
out that row and that column.

00:07:26.710 --> 00:07:30.230
So that is going to be a sub 12,
and you're going to go all

00:07:30.230 --> 00:07:33.140
the way to-- and I don't what
the sign on that is, so we

00:07:33.140 --> 00:07:37.650
could call it negative 1 to the
1 plus m-- that'll give us

00:07:37.650 --> 00:07:40.260
the right sign for the
checkerboard pattern-- times

00:07:40.260 --> 00:07:43.220
the determinant of the submatrix
for this guy.

00:07:43.220 --> 00:07:47.960
So we call it a1m, where you
cross out that guy's row, that

00:07:47.960 --> 00:07:49.540
guy's column, and you're
just left with all of

00:07:49.540 --> 00:07:51.250
this stuff over here.

00:07:51.250 --> 00:07:52.310
Fair enough.

00:07:52.310 --> 00:07:54.790
Now, let's look at the
determinant of a transpose.

00:07:58.820 --> 00:08:01.200
We learned earlier, you don't
have to go down the first row

00:08:01.200 --> 00:08:02.400
or you don't even
go down a row.

00:08:02.400 --> 00:08:03.840
You could go down a column.

00:08:03.840 --> 00:08:04.970
Let me be clear.

00:08:04.970 --> 00:08:09.080
So for our determinant of A, we
went down this row, and our

00:08:09.080 --> 00:08:11.140
submatrices, this was
my first submatrix.

00:08:11.140 --> 00:08:12.820
My second submatrix, you know
what it looks like.

00:08:12.820 --> 00:08:15.460
You would cross out the second
column and that row, and

00:08:15.460 --> 00:08:18.040
whatever's left over would be
the second submatrix and so on

00:08:18.040 --> 00:08:19.010
and so forth.

00:08:19.010 --> 00:08:22.420
But for our determinant of A
transpose, let's just go down

00:08:22.420 --> 00:08:27.520
this first column and get the
submatrices like that.

00:08:27.520 --> 00:08:29.790
So this is going to be equal
to-- let's get our first term

00:08:29.790 --> 00:08:31.070
right here.

00:08:31.070 --> 00:08:36.320
a sub 11 times the determinant
of its submatrix.

00:08:36.320 --> 00:08:38.039
So what's the determinant
of its submatrix?

00:08:38.039 --> 00:08:41.740
It's going to be-- its
submatrix, you cross out its

00:08:41.740 --> 00:08:46.000
row and its column, and you're
going to be left with this

00:08:46.000 --> 00:08:48.000
thing right here.

00:08:48.000 --> 00:08:51.120
Now, an interesting question
is how does this thing that

00:08:51.120 --> 00:08:54.760
I've just squared off, this
guy's submatrix, relate to

00:08:54.760 --> 00:08:56.450
this guy's submatrix?

00:08:56.450 --> 00:08:59.610
Well, if you look at it
carefully, this row from a22

00:08:59.610 --> 00:09:04.320
to a2m has now become a column
from a22 to a2m.

00:09:04.320 --> 00:09:07.820
This row, which is the next one,
from a32 to a3m has now

00:09:07.820 --> 00:09:10.540
become a column from
a32 to a3m.

00:09:10.540 --> 00:09:13.120
If you keep going down this last
row, it has now become

00:09:13.120 --> 00:09:14.570
this column.

00:09:14.570 --> 00:09:18.690
So this guy's submatrix or the
thing we're going to have to

00:09:18.690 --> 00:09:21.610
take the determinant of right
here is equal to the transpose

00:09:21.610 --> 00:09:22.710
of this guy.

00:09:22.710 --> 00:09:34.190
So this is equal to a
sub 11 transpose.

00:09:34.190 --> 00:09:40.350
And if you go through it, we go
minus this guy, minus a12

00:09:40.350 --> 00:09:42.590
times the determinant
of his submatrix.

00:09:42.590 --> 00:09:48.110
And if we cross out this guy's
row and that guy's column,

00:09:48.110 --> 00:09:49.400
what is that going
to look like?

00:09:49.400 --> 00:09:51.840
His submatrix is going
to look like this.

00:09:51.840 --> 00:09:55.030
It's going to have that there
and it's going to have that

00:09:55.030 --> 00:09:56.120
right there.

00:09:56.120 --> 00:09:58.060
How does that compare to a12?

00:09:58.060 --> 00:10:01.670
So a12 is if you crossed out
this and this, you're left

00:10:01.670 --> 00:10:03.260
with all of this right here.

00:10:06.040 --> 00:10:08.920
So once again, you see that this
row is the same thing as

00:10:08.920 --> 00:10:12.610
this column, that this row is
the same thing as this column,

00:10:12.610 --> 00:10:14.880
that that row is the same
thing as that column.

00:10:14.880 --> 00:10:17.980
So once again, the submatrix
we have to take the

00:10:17.980 --> 00:10:20.070
determinant of is equal
to the transpose of

00:10:20.070 --> 00:10:21.580
this thing over here.

00:10:21.580 --> 00:10:25.500
So it's equal to
a12 transpose.

00:10:25.500 --> 00:10:29.270
This thing-- I could draw it
shaded in-- is equal to the

00:10:29.270 --> 00:10:34.290
transpose of this thing, is
equal to the transpose of that

00:10:34.290 --> 00:10:35.000
thing right there.

00:10:35.000 --> 00:10:38.330
So, in general, each of these
submatrices when we go down

00:10:38.330 --> 00:10:40.860
this row is equal to the
transpose of each of these.

00:10:40.860 --> 00:10:43.950
So you keep going, so then
you're going to go all the way

00:10:43.950 --> 00:10:46.440
to plus minus 1.

00:10:46.440 --> 00:10:50.570
We're going to go all the way
down to the minus 1, 1 plus m

00:10:50.570 --> 00:10:55.150
times the determinant of--
it's going to be this guy

00:10:55.150 --> 00:10:55.530
transposed.

00:10:55.530 --> 00:10:56.680
You can even do it.

00:10:56.680 --> 00:11:00.560
If you go all the way to-- if
you cross that guy out and

00:11:00.560 --> 00:11:04.190
that guy out, you're left with
everything else on this

00:11:04.190 --> 00:11:07.800
matrix, and that's equal to the
transpose of if you cross

00:11:07.800 --> 00:11:10.030
this guy out and that guy out.

00:11:10.030 --> 00:11:12.770
This row turns into that column,
that row turns into

00:11:12.770 --> 00:11:13.660
that column.

00:11:13.660 --> 00:11:15.030
I think you see the point.

00:11:15.030 --> 00:11:16.290
I don't want to beat
a dead horse.

00:11:16.290 --> 00:11:21.230
So that's going to be equal
to a sub 1m transpose.

00:11:21.230 --> 00:11:25.440
Now, remember, going into this
inductive proof, or proof by

00:11:25.440 --> 00:11:29.120
induction, I assume that for--
remember, this is an n plus 1

00:11:29.120 --> 00:11:30.630
by n plus 1 matrix.

00:11:30.630 --> 00:11:34.200
But going into it, I assumed
that for an n-by-n matrix, the

00:11:34.200 --> 00:11:36.330
determinant of B
is equal to the

00:11:36.330 --> 00:11:39.180
determinant of B transpose.

00:11:39.180 --> 00:11:43.430
Well, these guys right
here, these are

00:11:43.430 --> 00:11:45.490
n-by-n matrices, right?

00:11:45.490 --> 00:11:50.330
This guy right here is an
n plus 1 by n plus 1.

00:11:50.330 --> 00:11:52.230
Same thing for this
guy right here.

00:11:52.230 --> 00:11:54.560
But these guys right
here are n by n.

00:11:57.190 --> 00:12:02.060
So if we assume for the n-by-n
case that the determinant of a

00:12:02.060 --> 00:12:05.390
matrix is equal to the
determinant of a transpose--

00:12:05.390 --> 00:12:07.170
this is the determinant of
the matrix, this is the

00:12:07.170 --> 00:12:09.360
determinant of its transpose--
these two

00:12:09.360 --> 00:12:11.300
things have to be equal.

00:12:11.300 --> 00:12:15.500
So we can then say that the
determinant of A transpose is

00:12:15.500 --> 00:12:21.110
equal to this term A sub 11
times this, but this is equal

00:12:21.110 --> 00:12:23.040
to this for the n-by-n case.

00:12:23.040 --> 00:12:26.540
Remember, we're doing the n
plus 1 by n plus 1 case.

00:12:26.540 --> 00:12:29.450
But these submatrices
are one dimension

00:12:29.450 --> 00:12:30.960
smaller in each direction.

00:12:30.960 --> 00:12:32.700
It has one less row and
one less column.

00:12:32.700 --> 00:12:33.955
So these two things are equal.

00:12:33.955 --> 00:12:36.430
So instead of writing this, I
can just write this, so times

00:12:36.430 --> 00:12:39.820
the determinant of a sub 11.

00:12:39.820 --> 00:12:40.720
Then you keep going.

00:12:40.720 --> 00:12:43.920
Minus a sub 12 times
the determinant.

00:12:43.920 --> 00:12:45.730
Instead of writing this, I
could write that because

00:12:45.730 --> 00:12:46.840
they're equal.

00:12:46.840 --> 00:12:52.240
Determinant a sub 12, all the
way to plus minus 1 to the 1

00:12:52.240 --> 00:12:55.570
plus m times the determinant
of this.

00:12:55.570 --> 00:12:56.750
These two things are equal.

00:12:56.750 --> 00:12:58.730
That is equal to that.

00:12:58.730 --> 00:13:00.620
That was our assumption
in this inductive

00:13:00.620 --> 00:13:04.230
proof, a sub 1m.

00:13:04.230 --> 00:13:06.790
And then you see that, of
course, this line, this blue

00:13:06.790 --> 00:13:09.450
line here, is equivalent to
this blue line there.

00:13:09.450 --> 00:13:16.100
So we get that the determinant
of A, which is an n plus 1 by

00:13:16.100 --> 00:13:22.240
n plus 1, so this is the n
plus 1 by n plus 1 case.

00:13:22.240 --> 00:13:24.260
We get the determinant
of A is equal to the

00:13:24.260 --> 00:13:27.380
determinant of A transpose.

00:13:27.380 --> 00:13:36.310
And we got this assuming that it
is true-- let me write it--

00:13:36.310 --> 00:13:43.950
assuming that it's true
for n-by-n case.

00:13:43.950 --> 00:13:44.610
And then we're done.

00:13:44.610 --> 00:13:47.230
We've now proved that this is
true in general, because we've

00:13:47.230 --> 00:13:48.350
proven the base case.

00:13:48.350 --> 00:13:52.300
We've proven it for the 2-by-2
case, and then we showed that

00:13:52.300 --> 00:13:54.230
if it's true for the n
case, it's true for

00:13:54.230 --> 00:13:55.940
the n plus 1 case.

00:13:55.940 --> 00:13:57.980
So if it's true for the 2 case,
it's going to be true

00:13:57.980 --> 00:13:59.570
for the 3-by-3 case.

00:13:59.570 --> 00:14:02.110
If it's true for the 3-by-3
case, it's true for the 4-by-4

00:14:02.110 --> 00:14:03.810
case, so on and so forth.

00:14:03.810 --> 00:14:05.460
But the takeaway
is pretty neat.

00:14:05.460 --> 00:14:07.020
You can take the transpose, the

00:14:07.020 --> 00:14:09.160
determinant doesn't change.


Transpose of a matrix product

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.270
I've got a handful
of matrices here.

00:00:02.270 --> 00:00:05.890
I have the matrix A
that's an m by n matrix.

00:00:05.890 --> 00:00:09.895
You can see it has n
columns and m rows.

00:00:09.895 --> 00:00:11.770
And actually let me
throw in one entry there.

00:00:11.770 --> 00:00:12.930
It might be useful.

00:00:12.930 --> 00:00:15.600
This is the jth column.

00:00:15.600 --> 00:00:19.040
So the row is going to
look like this-- amj.

00:00:19.040 --> 00:00:20.840
That's that entry right there.

00:00:20.840 --> 00:00:23.100
And then I have matrix
B defined similarly,

00:00:23.100 --> 00:00:29.980
but instead of being an m by n
matrix, B is an n by m matrix.

00:00:29.980 --> 00:00:31.720
And so this entry right here.

00:00:31.720 --> 00:00:34.500
Let me just-- I realize
this might be useful.

00:00:34.500 --> 00:00:36.920
This is going to be my nth row.

00:00:36.920 --> 00:00:38.970
And it's going to
be my jth column.

00:00:38.970 --> 00:00:41.370
And then I also wrote
out their transposes.

00:00:41.370 --> 00:00:47.170
So if you look at the transpose
of B, B was an n by m matrix.

00:00:47.170 --> 00:00:51.010
Now the transpose is going
to be an m by n matrix.

00:00:51.010 --> 00:00:53.760
And each of its rows
become its columns.

00:00:53.760 --> 00:00:57.470
And the same thing I did for A.
It's transpose is right there,

00:00:57.470 --> 00:00:59.300
A was m by n.

00:00:59.300 --> 00:01:02.130
The transpose is n by m.

00:01:02.130 --> 00:01:05.200
And each of these rows
because each of these columns.

00:01:05.200 --> 00:01:06.450
Now fair enough.

00:01:06.450 --> 00:01:10.130
Let's define two new
matrices right now.

00:01:10.130 --> 00:01:17.380
Let's define the matrix
C. Let me do it over here.

00:01:17.380 --> 00:01:21.010
I think the real estate will
be valuable in this video.

00:01:21.010 --> 00:01:22.610
Let's define my
matrix C as being

00:01:22.610 --> 00:01:26.270
equal to the product of A and B.

00:01:26.270 --> 00:01:29.210
So what are going to
be the dimensions of C?

00:01:29.210 --> 00:01:33.490
Well, an m by n matrix times
an n by m matrix, these two

00:01:33.490 --> 00:01:37.640
have to be equal even for the
matrix product to be defined.

00:01:37.640 --> 00:01:40.265
And that's going to result
with an m by m matrix.

00:01:43.430 --> 00:01:44.820
Now let's define another matrix.

00:01:44.820 --> 00:01:52.000
Let's call it D. And
it's equal to B transpose

00:01:52.000 --> 00:01:54.976
times A transpose.

00:01:54.976 --> 00:01:57.350
And the dimensions are going
to be the same, because this

00:01:57.350 --> 00:02:00.020
is an m by n times an n by m.

00:02:00.020 --> 00:02:01.540
So these are the same.

00:02:01.540 --> 00:02:04.530
So which is a requirement for
this product to be defined.

00:02:04.530 --> 00:02:07.853
And so the dimensions of
B are going to be m by m.

00:02:10.910 --> 00:02:14.770
So let's explore a little bit
what the different entries of C

00:02:14.770 --> 00:02:16.085
are going to look like.

00:02:16.085 --> 00:02:19.250
So let me write my
matrix C right here.

00:02:19.250 --> 00:02:25.800
So it's just going to have a
bunch of entries-- c11, c12,

00:02:25.800 --> 00:02:29.189
all the way to c1m.

00:02:29.189 --> 00:02:31.105
You can imagine because
it's an m by n matrix,

00:02:31.105 --> 00:02:33.670
you're going to
have cmm over here.

00:02:33.670 --> 00:02:35.846
You know how this drill goes.

00:02:35.846 --> 00:02:37.220
But what I'm
curious about is how

00:02:37.220 --> 00:02:41.120
do we figure out what
the general cij is?

00:02:41.120 --> 00:02:44.390
How do we figure out what
a particular entry is?

00:02:44.390 --> 00:02:47.590
We know that C is the
product of A and B.

00:02:47.590 --> 00:02:50.640
So to get to a
particular entry in C--

00:02:50.640 --> 00:02:56.870
and we've seen this
before-- so cij--

00:02:56.870 --> 00:02:58.680
It's going to be--
you can view it

00:02:58.680 --> 00:03:05.204
as the dot product
of the ith row in A

00:03:05.204 --> 00:03:13.810
with the jth column
in B, just like that.

00:03:13.810 --> 00:03:15.710
And what's that
going to be equal to?

00:03:15.710 --> 00:03:29.280
It's going to be equal to ai1
times b1j plus ai2 times b2j.

00:03:31.702 --> 00:03:33.160
And you're just
going to keep going

00:03:33.160 --> 00:03:34.660
until you get to
the last term here,

00:03:34.660 --> 00:03:38.880
ain times the last
term here, bnj.

00:03:41.810 --> 00:03:42.950
Fair enough.

00:03:42.950 --> 00:03:46.330
Now what about our matrix D?

00:03:46.330 --> 00:03:49.100
What are its entries
going to look like?

00:03:49.100 --> 00:03:52.950
So D, similarly, it's
going to look like-- you're

00:03:52.950 --> 00:04:01.520
going to have d11, d12,
all the way to d1m.

00:04:01.520 --> 00:04:02.650
You're going to have dmm.

00:04:02.650 --> 00:04:04.410
I could keep putting
entries here.

00:04:04.410 --> 00:04:07.370
But I'm curious about just
that sum in general entry here.

00:04:07.370 --> 00:04:11.778
Let's say I want
to find d sub ji.

00:04:15.160 --> 00:04:16.300
That's what I want to find.

00:04:16.300 --> 00:04:20.215
So I want to find a general way
for any particular entry of d.

00:04:20.215 --> 00:04:21.605
The jth row and
ith column, which

00:04:21.605 --> 00:04:23.480
is a little bit different
than the convention

00:04:23.480 --> 00:04:25.210
we normally use
for these letters.

00:04:25.210 --> 00:04:26.500
But it's fine.

00:04:26.500 --> 00:04:28.560
The first one is D's row.

00:04:28.560 --> 00:04:32.290
The second one is D's--
is this entry's column.

00:04:32.290 --> 00:04:34.500
So how do we figure that out?

00:04:34.500 --> 00:04:38.900
So d sub j i.

00:04:38.900 --> 00:04:42.130
It's going to be equal to-- D is
the product of these two guys.

00:04:42.130 --> 00:04:46.550
So to get the jth row and
ith column entry here,

00:04:46.550 --> 00:04:49.370
we essentially take the
dot product of the jth row

00:04:49.370 --> 00:05:01.310
here, which is that right there,
with the ith column of A, which

00:05:01.310 --> 00:05:03.030
is that right there.

00:05:03.030 --> 00:05:04.950
So I'm going to take
the dot product of that.

00:05:04.950 --> 00:05:08.360
And you might already see
something interesting here.

00:05:08.360 --> 00:05:12.430
This thing right here is
equivalent to that thing

00:05:12.430 --> 00:05:13.592
right there.

00:05:13.592 --> 00:05:15.800
And this thing right here
is equivalent to that thing

00:05:15.800 --> 00:05:17.633
right there, because
we took the transposes.

00:05:17.633 --> 00:05:20.980
But let's actually
just write it out.

00:05:20.980 --> 00:05:24.705
So what is this dot product
going to be equal to?

00:05:24.705 --> 00:05:26.669
Well, it's going to be bij.

00:05:26.669 --> 00:05:27.710
Let me write it this way.

00:05:27.710 --> 00:05:31.150
It's going to bij times ai1.

00:05:31.150 --> 00:05:35.230
Or we could write
it as ai1 times b1j.

00:05:37.870 --> 00:05:40.980
And it's going to be
plus b2j times ai2, which

00:05:40.980 --> 00:05:43.120
is the same thing
as ai2 times b2j.

00:05:47.360 --> 00:05:51.050
And you're going to keep going
until you get b and j times

00:05:51.050 --> 00:05:51.950
ain.

00:05:51.950 --> 00:05:55.932
Or you could write
that as ain times bnj.

00:05:58.560 --> 00:05:59.920
Now notice something.

00:05:59.920 --> 00:06:01.450
These two things are equivalent.

00:06:01.450 --> 00:06:03.260
They're completely
equivalent statement.

00:06:03.260 --> 00:06:07.950
The d sub ji is
equivalent to c sub ij.

00:06:07.950 --> 00:06:08.880
Let me write that.

00:06:11.410 --> 00:06:18.630
Or I could write c sub ij
is equivalent to d sub ji.

00:06:18.630 --> 00:06:23.410
Or another way you could say
it is, all the entries that's

00:06:23.410 --> 00:06:31.210
at row i, column j in C is
now in row j, column i in D.

00:06:31.210 --> 00:06:32.840
And this is true
for all the entries.

00:06:37.390 --> 00:06:39.560
I stayed as general as possible.

00:06:39.560 --> 00:06:40.660
So what does this mean?

00:06:40.660 --> 00:06:43.360
This is the definition
of a transpose.

00:06:43.360 --> 00:06:48.870
So we now get that C
transpose is equal to D.

00:06:48.870 --> 00:06:53.267
Or you could say that C
is equal to D transpose.

00:06:53.267 --> 00:06:55.350
Now this is pretty
interesting, because how did we

00:06:55.350 --> 00:06:56.820
define these two?

00:06:56.820 --> 00:07:04.500
We said that our matrix C is
equal to the matrix product

00:07:04.500 --> 00:07:09.560
A and B. And we said that D is
equal to our matrix product B

00:07:09.560 --> 00:07:12.400
transpose times A transpose.

00:07:12.400 --> 00:07:15.750
I did those definitions
right there.

00:07:15.750 --> 00:07:17.560
Here are the definitions.

00:07:17.560 --> 00:07:23.620
And now we just found out that D
is equal to the transpose of C.

00:07:23.620 --> 00:07:26.230
So we could write that
C transpose, which

00:07:26.230 --> 00:07:32.292
is the same thing as A times
B transpose, is equal to D.

00:07:32.292 --> 00:07:37.720
So it is equal to D, which is
just B transpose A transpose.

00:07:37.720 --> 00:07:39.200
And this is a pretty
neat takeaway.

00:07:43.670 --> 00:07:46.710
If I take the product
of two matrices,

00:07:46.710 --> 00:07:50.490
and then transpose it, it's
equivalent to switching

00:07:50.490 --> 00:07:53.120
the order, or transposing
them, and then taking

00:07:53.120 --> 00:07:55.790
the product of the
reverse order-- B

00:07:55.790 --> 00:08:00.250
transpose, A transpose--
which is a pretty, pretty neat

00:08:00.250 --> 00:08:01.040
take away.

00:08:01.040 --> 00:08:02.790
And you could
actually extend this

00:08:02.790 --> 00:08:05.720
to an arbitrary
number of matrices

00:08:05.720 --> 00:08:07.891
that you're taking
the product of.

00:08:07.891 --> 00:08:09.640
I'm not proving it
here, but it's actually

00:08:09.640 --> 00:08:12.440
a very simple extension
from this right now.

00:08:12.440 --> 00:08:14.780
If you take the
matrices, let's say

00:08:14.780 --> 00:08:19.012
A-- let me do different
letters-- X, Y,

00:08:19.012 --> 00:08:23.460
Z, if you take their product
and then transpose it,

00:08:23.460 --> 00:08:29.780
it's equal to Z transpose,
Y transpose, X transpose.

00:08:29.780 --> 00:08:31.670
I haven't proven
this general case,

00:08:31.670 --> 00:08:33.549
and you could keep doing
it with four or five

00:08:33.549 --> 00:08:38.020
or n matrices multiplied by each
other, but it generally works.

00:08:38.020 --> 00:08:39.770
And you could
essentially prove it

00:08:39.770 --> 00:08:41.640
using what we proved
in this video right

00:08:41.640 --> 00:08:43.940
here, that you take the
product of two matrices,

00:08:43.940 --> 00:08:46.770
take their transpose,
it's equal to the product

00:08:46.770 --> 00:08:49.900
of their transposes
in reverse order.


Transpose of a matrix

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:04.730
I've got a matrix A, and
it's an m by n matrix.

00:00:04.730 --> 00:00:07.250
It has m rows and n columns.

00:00:07.250 --> 00:00:10.220
So I can write it in fairly
general terms like this.

00:00:10.220 --> 00:00:14.560
The first row would be a11.

00:00:14.560 --> 00:00:16.350
First row, first column.

00:00:16.350 --> 00:00:19.120
a12, First row, second column.

00:00:19.120 --> 00:00:22.100
All the way to-- I
have n columns.

00:00:22.100 --> 00:00:26.450
So a1n, first row,
n-th column.

00:00:26.450 --> 00:00:29.640
And then the second row
would look like this.

00:00:29.640 --> 00:00:32.540
a second row, first column.

00:00:32.540 --> 00:00:34.530
A second row, second column.

00:00:34.530 --> 00:00:39.190
All the way to a second
row, n column.

00:00:39.190 --> 00:00:42.060
And we'll just keep doing that
all the way down until you get

00:00:42.060 --> 00:00:43.890
to the m-th row.

00:00:43.890 --> 00:00:45.500
The m row would look
like this.

00:00:45.500 --> 00:00:47.780
Each of these are the entries
in each of the rows or

00:00:47.780 --> 00:00:50.400
columns, depending on how
you want to look at it.

00:00:50.400 --> 00:00:54.160
So this is going to
be a sub m 1.

00:00:54.160 --> 00:00:56.740
mth row, first column.

00:00:56.740 --> 00:00:59.110
a sub m 2.

00:00:59.110 --> 00:01:03.950
And you go all the
way to a sub m n.

00:01:03.950 --> 00:01:07.590
This is our matrix right here.

00:01:07.590 --> 00:01:09.500
That is my matrix A.

00:01:09.500 --> 00:01:24.600
Now, I'm going to define the
transpose of this matrix as a

00:01:24.600 --> 00:01:27.740
with this superscript t.

00:01:27.740 --> 00:01:29.880
And this is going to be my
definition, it is essentially

00:01:29.880 --> 00:01:33.930
the matrix A with all the rows
and the columns swapped.

00:01:33.930 --> 00:01:40.220
So my matrix A transpose is
going to be a n by m matrix.

00:01:40.220 --> 00:01:42.560
Notice I said m rows
and n columns.

00:01:42.560 --> 00:01:46.440
Now this is going to have
n rows and m columns.

00:01:46.440 --> 00:01:48.940
So what is this guy going
to look like?

00:01:48.940 --> 00:01:50.140
What is he going to look like?

00:01:50.140 --> 00:01:52.670
Well I'm gonna swap my
rows and my columns.

00:01:52.670 --> 00:01:55.440
So my first row becomes
my first column.

00:01:55.440 --> 00:01:58.470
So I'm going to have a11.

00:01:58.470 --> 00:02:00.680
That entry's still going
to be in that position.

00:02:00.680 --> 00:02:03.510
But now this entry is not
going to be right here.

00:02:03.510 --> 00:02:05.250
a12.

00:02:05.250 --> 00:02:09.289
And my second row I have what
I used to have in my second

00:02:09.289 --> 00:02:10.560
row, first column.

00:02:10.560 --> 00:02:14.200
I'm now going to have what
I had in my second

00:02:14.200 --> 00:02:15.630
column, first row.

00:02:15.630 --> 00:02:20.410
I'm just going to go down
all the way to a1n.

00:02:20.410 --> 00:02:24.150
And that makes--
not a i n, a1n.

00:02:24.150 --> 00:02:28.120
And that makes sense because
I'll now have n columns.

00:02:28.120 --> 00:02:29.780
Sorry I now have n rows.

00:02:29.780 --> 00:02:31.130
I had n columns before.

00:02:31.130 --> 00:02:32.620
Now I have n rows.

00:02:32.620 --> 00:02:37.540
Now this row, when I transpose
it is going look like this.

00:02:37.540 --> 00:02:44.510
a21, a22, all the
way down to a2n.

00:02:44.510 --> 00:02:47.370
It might be a little confusing
for you right now to have this

00:02:47.370 --> 00:02:50.590
notation right there because
everything we've done so far.

00:02:50.590 --> 00:02:53.250
We've always said, hey this
first number's the row and the

00:02:53.250 --> 00:02:54.620
second number is the column.

00:02:54.620 --> 00:02:56.380
That's what we did up here.

00:02:56.380 --> 00:02:59.170
What I'm doing here, you can
ignore that reference to the

00:02:59.170 --> 00:02:59.930
rows and columns.

00:02:59.930 --> 00:03:02.490
You can just say whatever we
had here in my first row,

00:03:02.490 --> 00:03:04.730
second column, I
now have here.

00:03:04.730 --> 00:03:07.370
When you look at this transpose,
don't take these

00:03:07.370 --> 00:03:08.800
subscripts too literally.

00:03:08.800 --> 00:03:11.280
Or now you can kind of reverse
your interpretation.

00:03:11.280 --> 00:03:14.670
This is now the first
column, second row.

00:03:14.670 --> 00:03:17.625
This was the second
row, first column.

00:03:20.340 --> 00:03:21.900
I don't want you get too
confused with these

00:03:21.900 --> 00:03:22.820
subscripts.

00:03:22.820 --> 00:03:25.370
Just keep in mind, we're
taking all the rows and

00:03:25.370 --> 00:03:28.120
turning them into the columns
to get the transpose.

00:03:28.120 --> 00:03:31.920
And then you just
keep doing this.

00:03:31.920 --> 00:03:36.110
And then this m-th row will now
become the m-th column.

00:03:36.110 --> 00:03:41.780
am1, am2, all the way
down to a m n.

00:03:45.880 --> 00:03:48.030
So, this entry is
now that entry.

00:03:48.030 --> 00:03:50.890
If you know, this entry
is now that entry.

00:03:50.890 --> 00:03:52.890
That entry is now
at that entry.

00:03:52.890 --> 00:03:54.330
I think you get the idea.

00:03:54.330 --> 00:03:55.870
This is what a transpose is.

00:03:55.870 --> 00:03:58.350
And sometimes when you do in
the abstract, it can be a

00:03:58.350 --> 00:03:59.140
little confusing.

00:03:59.140 --> 00:04:01.420
And we'll especially appreciate
that once we do

00:04:01.420 --> 00:04:03.650
some of the proofs involving
the transpose.

00:04:03.650 --> 00:04:06.390
But actually taking the
transpose of an actual matrix,

00:04:06.390 --> 00:04:08.990
with actual numbers, shouldn't
be too difficult.

00:04:08.990 --> 00:04:11.290
So, let's start with
the 2 by 2 case.

00:04:11.290 --> 00:04:13.720
I'll try to color code
it as best as I can.

00:04:13.720 --> 00:04:15.380
So let's say I have
the matrix.

00:04:15.380 --> 00:04:16.790
Let's say I defined A.

00:04:16.790 --> 00:04:17.760
Let's do B now.

00:04:17.760 --> 00:04:19.250
I already defined A.

00:04:19.250 --> 00:04:21.110
Let's say B.

00:04:21.110 --> 00:04:33.510
B is equal to the matrix
1, 2, 3, 4.

00:04:33.510 --> 00:04:35.150
Those colors are pretty close.

00:04:35.150 --> 00:04:37.430
But what is B transpose
going to look like?

00:04:37.430 --> 00:04:40.800
B transpose is going to be equal
to-- You switch the rows

00:04:40.800 --> 00:04:41.370
and columns.

00:04:41.370 --> 00:04:43.510
So the first row will now
become the first column.

00:04:43.510 --> 00:04:44.590
1, 2.

00:04:44.590 --> 00:04:46.200
And the second row will
now become the

00:04:46.200 --> 00:04:49.300
second column, 3, 4.

00:04:49.300 --> 00:04:50.770
Or you could view it
the other way.

00:04:50.770 --> 00:04:53.200
The first column now became
the first row.

00:04:53.200 --> 00:04:56.110
And the second column now
became the second row.

00:04:56.110 --> 00:04:57.970
Let's do an example.

00:04:57.970 --> 00:05:01.400
Instead of even doing a 2 by
3-- or a 3 by 3-- let me do

00:05:01.400 --> 00:05:04.910
one that might be a little
bit more challenging.

00:05:04.910 --> 00:05:07.340
I think this'll make
things clear.

00:05:07.340 --> 00:05:09.055
So let's say I have
the matrix C.

00:05:14.130 --> 00:05:15.890
Let me make it a pretty
big matrix.

00:05:15.890 --> 00:05:21.540
Let's say it is a 4 by
3 matrix right here.

00:05:21.540 --> 00:05:24.120
Let me just throw some
numbers in there.

00:05:24.120 --> 00:05:28.060
1, 0, minus 1.

00:05:28.060 --> 00:05:29.660
2, 7.

00:05:29.660 --> 00:05:31.250
Oh I want to do it in
different colors.

00:05:31.250 --> 00:05:32.310
Let me do that in a
different color.

00:05:32.310 --> 00:05:37.120
So then I get 2, 7, minus 5.

00:05:37.120 --> 00:05:42.780
Then I get 4 minus 3, 2.

00:05:42.780 --> 00:05:44.420
I have to do one
more row here.

00:05:44.420 --> 00:05:50.030
So let me just make that
minus 1, 3, and 0.

00:05:50.030 --> 00:05:52.280
That is my matrix C.

00:05:52.280 --> 00:05:55.300
So what is-- let me do that,
and I like to be

00:05:55.300 --> 00:05:56.210
aesthetically pleasing.

00:05:56.210 --> 00:05:58.510
So let me close the bracket
in the same color.

00:05:58.510 --> 00:06:01.310
So what is C transpose
going to be?

00:06:01.310 --> 00:06:03.295
So, C transpose.

00:06:03.295 --> 00:06:06.000
Let me do that in a
different color.

00:06:06.000 --> 00:06:11.620
C transpose is now going
to be a 3 by 4 matrix.

00:06:11.620 --> 00:06:13.980
And, essentially, it's going
to be the matrix C with all

00:06:13.980 --> 00:06:16.150
the rows swapped for the columns
or all the columns

00:06:16.150 --> 00:06:17.830
swapped for the rows.

00:06:17.830 --> 00:06:20.050
So, it's now going to
be a 3 by 4 matrix.

00:06:24.050 --> 00:06:26.110
And that first row there
is now going to

00:06:26.110 --> 00:06:27.360
become the first column.

00:06:27.360 --> 00:06:30.650
1, 0, minus 1.

00:06:30.650 --> 00:06:32.890
The second row here
is now going to

00:06:32.890 --> 00:06:34.180
become the second column.

00:06:34.180 --> 00:06:36.760
2, 7, minus 5.

00:06:36.760 --> 00:06:39.420
I didn't use the exact same
green, but you get the idea.

00:06:39.420 --> 00:06:41.830
This third row will become
the third column.

00:06:41.830 --> 00:06:45.420
4, minus 3, 2.

00:06:45.420 --> 00:06:47.620
And then, finally, the
fourth row will

00:06:47.620 --> 00:06:48.760
become the fourth column.

00:06:48.760 --> 00:06:52.470
Minus 1, 3, and 0.

00:06:52.470 --> 00:07:03.766
All we did is, this guy was in
the second row, third column.

00:07:06.340 --> 00:07:09.690
Now, that same guy
is in the what?

00:07:09.690 --> 00:07:19.450
He is in the second column
and the third row.

00:07:19.450 --> 00:07:21.920
All we did is switch the
rows and the columns.

00:07:21.920 --> 00:07:23.370
We could do it with another.

00:07:23.370 --> 00:07:23.920
Let's see.

00:07:23.920 --> 00:07:26.200
Let's do it with this
one right here.

00:07:26.200 --> 00:07:32.100
This guy right here is
in the third row.

00:07:32.100 --> 00:07:32.980
1, 2, 3.

00:07:32.980 --> 00:07:34.230
And the second column.

00:07:36.460 --> 00:07:43.930
And when you go down here this
guy is now in the third column

00:07:43.930 --> 00:07:48.600
and the second row.

00:07:48.600 --> 00:07:51.350
That's all a transpose is.

00:07:51.350 --> 00:07:53.670
And, just as a little
interesting thing, what

00:07:53.670 --> 00:07:56.860
happens if we take the transpose
of the transpose?

00:07:56.860 --> 00:07:59.660
So what happens if we
take C transpose and

00:07:59.660 --> 00:08:01.710
then transpose that?

00:08:01.710 --> 00:08:03.650
What is that going
to be equal to?

00:08:03.650 --> 00:08:06.070
Well to go from C to C
transpose, we switched all the

00:08:06.070 --> 00:08:07.150
rows and the columns.

00:08:07.150 --> 00:08:08.970
All the entries the
rows and columns.

00:08:08.970 --> 00:08:11.880
When you take the transpose
again, remember let's just

00:08:11.880 --> 00:08:12.580
focus on this guy.

00:08:12.580 --> 00:08:14.000
This was second row,
third column.

00:08:14.000 --> 00:08:15.690
You took the transpose,
it becomes second

00:08:15.690 --> 00:08:17.140
column and third row.

00:08:17.140 --> 00:08:18.940
If you were to take the
transpose again of that, he

00:08:18.940 --> 00:08:22.100
would then become the second
row and third column again.

00:08:22.100 --> 00:08:26.375
So C transpose, the transpose
of C transpose, is

00:08:26.375 --> 00:08:27.740
just equal to C.

00:08:27.740 --> 00:08:29.130
You're swapping all
the columns when

00:08:29.130 --> 00:08:30.120
you take the transpose.

00:08:30.120 --> 00:08:31.380
And when you take the
transpose again, you

00:08:31.380 --> 00:08:32.730
swap them all back.

00:08:32.730 --> 00:08:34.150
That's all that means.

00:08:34.150 --> 00:08:34.679
Anyway.

00:08:34.679 --> 00:08:36.630
Hope you found that useful.