Fractional Reserve Banking Commentary 1

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:09.240
I've talked a lot about what
fractional reserve banking is,

00:00:09.240 --> 00:00:12.260
but I have kind of reserved
making any commentary on it,

00:00:12.260 --> 00:00:14.310
and that's what I'm actually
going to do in this video.

00:00:14.310 --> 00:00:17.760
So just as a bit of a review on
the whole banking playlist,

00:00:17.760 --> 00:00:19.610
I started off with a
very simple example

00:00:19.610 --> 00:00:21.390
of we're on an island.

00:00:21.390 --> 00:00:23.600
Let's say you had all of the
gold on an island and you give

00:00:23.600 --> 00:00:25.530
it to me for safekeeping.

00:00:25.530 --> 00:00:32.000
So now, I have this asset called
gold, and let's say

00:00:32.000 --> 00:00:35.740
it's 100 gold pieces
that you gave me.

00:00:35.740 --> 00:00:37.610
And I also have a liability.

00:00:37.610 --> 00:00:41.950
This is my primitive island
bank, and my liability is that

00:00:41.950 --> 00:00:45.580
you could demand this gold from
me whenever you want, so

00:00:45.580 --> 00:00:48.150
this is my liability.

00:00:48.150 --> 00:00:49.660
If I were to draw your
balance sheet, this

00:00:49.660 --> 00:00:50.510
would be your asset.

00:00:50.510 --> 00:00:55.435
So this would be an
IOU to depositor.

00:00:58.960 --> 00:01:01.730
So at any point in time, I've
told you, look, I have this

00:01:01.730 --> 00:01:04.290
nice safe, this vault there,
but you can come to me any

00:01:04.290 --> 00:01:07.180
time and take as much of
your gold as you want.

00:01:07.180 --> 00:01:10.640
And we, of course, know from the
fractional reserve banking

00:01:10.640 --> 00:01:14.150
playlist that there's a slight
discrepancy between what I

00:01:14.150 --> 00:01:17.790
told you, the depositor, and the
reality of the situation,

00:01:17.790 --> 00:01:20.440
because the first thing that I'm
going to do when you take

00:01:20.440 --> 00:01:23.490
this is, you know, I've taken
your deposits and I could

00:01:23.490 --> 00:01:26.220
charge you for keeping your
money with me, but I've come

00:01:26.220 --> 00:01:33.390
up with a slightly sneakier
scheme, where I say, hey, I've

00:01:33.390 --> 00:01:34.620
seen your behavior.

00:01:34.620 --> 00:01:37.990
You never withdraw more than 10
gold pieces out at a time.

00:01:37.990 --> 00:01:41.960
So what I do is I lend out all
but 10 of your gold pieces.

00:01:41.960 --> 00:01:43.210
So we end up with
the situation.

00:01:47.990 --> 00:01:55.810
I have an IOU to you of 100 gold
pieces like that, and I

00:01:55.810 --> 00:01:58.600
take the other 90 gold pieces
and I lend it out to random

00:01:58.600 --> 00:01:59.900
entrepreneurs on our island.

00:02:05.480 --> 00:02:10.930
So loan to A and that
is loan to B.

00:02:10.930 --> 00:02:13.910
And I think this is great,
because I obviously collect

00:02:13.910 --> 00:02:17.080
interest from these guys, and
assuming that these guys don't

00:02:17.080 --> 00:02:20.040
blow the money, I'll get all the
money back, so you'll get

00:02:20.040 --> 00:02:22.210
all your money back and I'll get
some interest, and I can

00:02:22.210 --> 00:02:23.880
say, hey, I'm keeping
your money for free.

00:02:23.880 --> 00:02:26.460
You get safekeeping, and I
collect the interest on it,

00:02:26.460 --> 00:02:29.450
and maybe that helps to fund
making the vault nice, and

00:02:29.450 --> 00:02:30.550
everyone's happy.

00:02:30.550 --> 00:02:33.720
But clearly here, there
is a discrepancy.

00:02:33.720 --> 00:02:35.650
What did I tell you?

00:02:35.650 --> 00:02:37.125
What did I tell you,
the depositor?

00:02:40.530 --> 00:02:43.990
I told you that you can
withdraw 100 gold

00:02:43.990 --> 00:02:45.240
pieces at any time.

00:02:54.030 --> 00:02:55.280
Now, what is the reality?

00:02:59.190 --> 00:03:07.830
The reality is you can withdraw
10 gold pieces at any

00:03:07.830 --> 00:03:12.250
time, and clearly, this is-- I
don't know if you want to call

00:03:12.250 --> 00:03:14.250
this a lie or not,
but this isn't

00:03:14.250 --> 00:03:16.460
completely true right here.

00:03:16.460 --> 00:03:19.660
And I know some of you might be
saying, hey, Sal, this is

00:03:19.660 --> 00:03:22.190
kind of a very primitive example
of gold pieces, but

00:03:22.190 --> 00:03:24.990
our current banking system
doesn't work quite like this.

00:03:24.990 --> 00:03:29.220
And just to make sure that you
understand that the analogy

00:03:29.220 --> 00:03:31.570
isn't completely off from our
current baking system, our

00:03:31.570 --> 00:03:35.660
current banking system is more
like this: You have 10 gold

00:03:35.660 --> 00:03:39.080
pieces that you give me.

00:03:39.080 --> 00:03:40.320
I'll draw it here.

00:03:40.320 --> 00:03:45.930
This is the modern
banking system.

00:03:45.930 --> 00:03:49.780
What you do is you give me a
deposit of 10 gold pieces like

00:03:49.780 --> 00:03:54.686
that, and so I say that,
yeah, sure, I

00:03:54.686 --> 00:03:56.960
owe you 10 gold pieces.

00:03:56.960 --> 00:03:58.610
So this is my IOU to you.

00:03:58.610 --> 00:04:02.120
Everything looks nice and clean
right now, but what I

00:04:02.120 --> 00:04:05.230
then do is then I issue checking
deposits to other

00:04:05.230 --> 00:04:09.030
people, and my limit on how many
checking deposits I can

00:04:09.030 --> 00:04:12.400
issue is based on the reserve
requirements and all of that.

00:04:12.400 --> 00:04:15.660
But let's say I have to keep
at least 10% reserves, just

00:04:15.660 --> 00:04:16.540
keep the math simple.

00:04:16.540 --> 00:04:24.470
So what I do in our modern
system is I make a loan to B

00:04:24.470 --> 00:04:30.190
like that and I gave B a
checking account that is

00:04:30.190 --> 00:04:31.910
equivalent to your
checking account.

00:04:31.910 --> 00:04:35.350
So this is an IOU to B.

00:04:35.350 --> 00:04:38.410
So this is essentially a promise
to B that, look, this

00:04:38.410 --> 00:04:40.290
is a checking account, and
any time you can come

00:04:40.290 --> 00:04:41.830
and get this much.

00:04:41.830 --> 00:04:47.680
Maybe this was 40, maybe
this is 50 gold pieces.

00:04:47.680 --> 00:04:50.760
This means that you can come
at any time and get 40 gold

00:04:50.760 --> 00:04:51.590
pieces from me.

00:04:51.590 --> 00:04:53.850
But we know the reality that
people actually don't actually

00:04:53.850 --> 00:04:54.940
get the gold pieces.

00:04:54.940 --> 00:04:57.820
In our modern system, we're not
even dealing with gold.

00:04:57.820 --> 00:05:01.270
This will be reserve currency,
but it's the same idea that

00:05:01.270 --> 00:05:05.480
people are getting money that
really isn't necessarily in

00:05:05.480 --> 00:05:05.980
the reserves.

00:05:05.980 --> 00:05:12.650
And, of course, I'm going to
make another loan to person A,

00:05:12.650 --> 00:05:14.130
and I'm just going to
create a checking

00:05:14.130 --> 00:05:15.890
account deposit for them.

00:05:15.890 --> 00:05:21.070
I'm just going to create an IOU
to A for 50, and the whole

00:05:21.070 --> 00:05:25.140
reason why this works is because
people use these

00:05:25.140 --> 00:05:28.980
checking account deposits
as a form of money.

00:05:28.980 --> 00:05:31.690
When I write you a check,
notice what's happening.

00:05:31.690 --> 00:05:34.420
I'm just transferring one of
these checking account

00:05:34.420 --> 00:05:41.080
deposits from me to you without
any gold or without

00:05:41.080 --> 00:05:43.720
any reserve currency actually
changing hands.

00:05:43.720 --> 00:05:44.530
So that's how it works.

00:05:44.530 --> 00:05:47.120
But if you look at these two
balance sheets, one is kind of

00:05:47.120 --> 00:05:48.790
the primitive example.

00:05:48.790 --> 00:05:51.570
On my island where I had 100
gold pieces and I lent

00:05:51.570 --> 00:05:53.460
out 90, I owe 100.

00:05:53.460 --> 00:05:56.660
I've told people, maybe it's
just one person, that hey, you

00:05:56.660 --> 00:05:59.560
can come to me at any time and
get your 100 gold pieces, but

00:05:59.560 --> 00:06:02.180
the reality is I've lent
all but 10 of it

00:06:02.180 --> 00:06:03.490
out to other people.

00:06:03.490 --> 00:06:05.460
Now what's the situation
in the modern?

00:06:05.460 --> 00:06:09.550
Well, I've told people that
there's 100 gold pieces that

00:06:09.550 --> 00:06:10.940
can come from me, right?

00:06:10.940 --> 00:06:14.190
10 to the original depositor,
40 to this guy that I gave a

00:06:14.190 --> 00:06:17.490
loan to, and another 50 to that
guy, but the reality is

00:06:17.490 --> 00:06:20.540
that I only have 10 gold pieces
in my vault and my

00:06:20.540 --> 00:06:22.340
other assets are just
loans to people.

00:06:22.340 --> 00:06:25.000
So if you look at these two
balance sheets, you see that

00:06:25.000 --> 00:06:26.980
they are equivalent.

00:06:26.980 --> 00:06:29.130
I always like to start with this
one because this one at

00:06:29.130 --> 00:06:30.320
least makes more sense.

00:06:30.320 --> 00:06:32.910
If you were to just walk up to a
five-year-old and say, look,

00:06:32.910 --> 00:06:36.230
this is what fractional reserve
banking is, this one

00:06:36.230 --> 00:06:38.830
starts to seem even a little
bit even shadier, because

00:06:38.830 --> 00:06:40.750
you're creating these loans and
you're creating these IOUs

00:06:40.750 --> 00:06:43.490
out of thin air, but your end
result is that they're equally

00:06:43.490 --> 00:06:46.320
shady because they both
have the same

00:06:46.320 --> 00:06:48.300
resulting balance sheet.

00:06:48.300 --> 00:06:51.850
So your next question is, hey
Sal, what's wrong with this?

00:06:51.850 --> 00:06:55.410
And I could ask my son, once
he learns to talk, what's

00:06:55.410 --> 00:06:57.640
wrong with this is that there's
a discrepancy between

00:06:57.640 --> 00:07:01.800
what I told the depositor-- so
in this case, I told the

00:07:01.800 --> 00:07:04.460
depositor you can withdraw 100
gold pieces at any time, when

00:07:04.460 --> 00:07:06.250
the reality is they
can withdraw 10.

00:07:06.250 --> 00:07:11.680
Here, I'm telling the depositor,
you can take your

00:07:11.680 --> 00:07:21.800
money whenever, where the
reality in this modern system

00:07:21.800 --> 00:07:24.150
is that you can demand
whenever-- and I would throw

00:07:24.150 --> 00:07:37.970
an asterisk there-- as long as
no more than 10% ask for the

00:07:37.970 --> 00:07:39.220
money at the same time.

00:07:47.270 --> 00:07:51.460
Now, obviously, I think this
is-- everyone already becomes

00:07:51.460 --> 00:07:53.270
a little uncomfortable when
you say this, but you say,

00:07:53.270 --> 00:07:56.310
look, this is what our modern
financial system is based on

00:07:56.310 --> 00:08:00.150
and what this allows is this
guy, this bank, however you

00:08:00.150 --> 00:08:02.680
want to view it, is allowed to
be an intermediary between the

00:08:02.680 --> 00:08:05.780
savers, the people who are
giving their deposits, and the

00:08:05.780 --> 00:08:09.360
people who need to invest the
capital in building new

00:08:09.360 --> 00:08:11.340
projects and whatever else.

00:08:11.340 --> 00:08:14.130
And the counterargument to that
for someone who's against

00:08:14.130 --> 00:08:16.160
fractional banking is there's
nothing to stop

00:08:16.160 --> 00:08:17.330
you from doing this.

00:08:17.330 --> 00:08:20.390
All you have to do is tell
this guy the truth.

00:08:20.390 --> 00:08:23.940
What you do is, you do this
exact same scenario.

00:08:27.580 --> 00:08:29.410
You don't tell him you
can withdraw 100 gold

00:08:29.410 --> 00:08:30.940
pieces at any time.

00:08:30.940 --> 00:08:39.330
You tell him you can withdraw 10
gold pieces at any time and

00:08:39.330 --> 00:08:47.030
that maybe another 40 gold
pieces will be available

00:08:47.030 --> 00:08:50.540
whenever this guy is supposed to
pay back his loan, maybe in

00:08:50.540 --> 00:08:58.210
a year, and that the other 50
gold pieces would be available

00:08:58.210 --> 00:08:59.970
whenever this guy pays
his loan back.

00:08:59.970 --> 00:09:02.460
Maybe it's in two years.

00:09:02.460 --> 00:09:04.960
So the question is why
doesn't this happen?

00:09:04.960 --> 00:09:11.830
Well, the reality is because if
you told the depositor that

00:09:11.830 --> 00:09:14.140
they can't have all of their
money on demand,

00:09:14.140 --> 00:09:15.760
he'd say, OK, fine.

00:09:15.760 --> 00:09:18.770
One, assuming that you're giving
these guys-- these are

00:09:18.770 --> 00:09:21.890
legitimate people to give your
money to-- I want some of the

00:09:21.890 --> 00:09:23.720
interest that you're taking
because you're just lending

00:09:23.720 --> 00:09:25.330
out my money.

00:09:25.330 --> 00:09:34.530
If you were to lock up your
money for longer, people would

00:09:34.530 --> 00:09:35.740
want interest for this.

00:09:35.740 --> 00:09:38.390
But when the bank is allowed
to kind of tell this

00:09:38.390 --> 00:09:41.220
half-truth, they don't have to
give as much interest on it,

00:09:41.220 --> 00:09:43.440
because when people say, oh, I
can get my money whenever I

00:09:43.440 --> 00:09:45.790
want, they say, well, then I
don't need that much interest

00:09:45.790 --> 00:09:47.160
on it, because it's on demand.

00:09:47.160 --> 00:09:48.730
But the reality is they
can't get all of their

00:09:48.730 --> 00:09:49.740
money at any time.

00:09:49.740 --> 00:09:53.680
They can only get some
fraction of their

00:09:53.680 --> 00:09:55.300
money at any time.

00:09:55.300 --> 00:09:58.110
So the next thought might be,
OK, you're being a little bit

00:09:58.110 --> 00:10:01.970
disingenuous when you tell the
depositor that it's on demand

00:10:01.970 --> 00:10:04.230
and because you're being
disingenuous, you're allowed

00:10:04.230 --> 00:10:07.220
to kind of not give them as
much interest as you would

00:10:07.220 --> 00:10:09.350
have to give them if you told
them that their money is

00:10:09.350 --> 00:10:12.980
locked up, but what's wrong
as long as you are a good

00:10:12.980 --> 00:10:17.030
investor and that you put
this money to good use?

00:10:17.030 --> 00:10:18.570
Doesn't the whole
world benefit?

00:10:18.570 --> 00:10:22.030
And that might be true most of
the time, assuming everything

00:10:22.030 --> 00:10:24.250
I just told you, but imagine
a situation where we have

00:10:24.250 --> 00:10:25.530
multiple banks in
our universe.

00:10:28.400 --> 00:10:29.800
Let me draw my multiple banks.

00:10:29.800 --> 00:10:39.390
So this is bank one, I have bank
two, and then bank three.

00:10:39.390 --> 00:10:41.360
And bank one and bank
two are very

00:10:41.360 --> 00:10:44.370
honest and good investors.

00:10:44.370 --> 00:10:47.470
And we all know it's very hard
to know who's a good or a bad

00:10:47.470 --> 00:10:50.170
investor, especially when times
are good, because then

00:10:50.170 --> 00:10:52.650
most loans tend to
come out good.

00:10:52.650 --> 00:10:55.190
So they're good investors.

00:10:55.190 --> 00:10:59.210
But bank three takes
on extra risk.

00:10:59.210 --> 00:11:01.870
Let me write it down.

00:11:01.870 --> 00:11:04.980
And what's their incentive
for taking extra risk?

00:11:04.980 --> 00:11:07.950
Let's say that bank three is
willing to lend to people that

00:11:07.950 --> 00:11:11.210
banks one and two weren't
willing to lend to.

00:11:11.210 --> 00:11:14.190
So bank three's balance sheet
here, these loans, are going

00:11:14.190 --> 00:11:15.800
to be riskier loans.

00:11:15.800 --> 00:11:17.830
Now why would they do that?

00:11:17.830 --> 00:11:19.950
Because when you give it to
riskier people, you're able to

00:11:19.950 --> 00:11:22.430
take more interest from them.

00:11:22.430 --> 00:11:29.820
So they take extra risk, which
leads to extra interest. And

00:11:29.820 --> 00:11:31.740
actually, if you were to look at
these banks, you would say

00:11:31.740 --> 00:11:34.050
that this guy's the most
profitable bank.

00:11:40.020 --> 00:11:43.180
Even worse, because he's getting
extra interest, he can

00:11:43.180 --> 00:11:45.580
actually share more of that
extra interest with their

00:11:45.580 --> 00:11:46.790
depositors.

00:11:46.790 --> 00:11:50.340
So you can give higher yielding
checking accounts, I

00:11:50.340 --> 00:11:52.010
guess I could call it.

00:11:52.010 --> 00:12:00.090
Higher interest on deposits.

00:12:00.090 --> 00:12:02.620
So in this situation, the person
who's taking the most

00:12:02.620 --> 00:12:06.240
risk is going to be the most
profitable and they're going

00:12:06.240 --> 00:12:07.690
to give the higher interest
and that might actually

00:12:07.690 --> 00:12:10.910
attract the most deposits,
right?

00:12:10.910 --> 00:12:13.100
If all these guys look the same
to me, and we all know

00:12:13.100 --> 00:12:15.930
it's very hard to know what
banks are actually doing with

00:12:15.930 --> 00:12:18.250
their money, then I'll say,
hey, I'm going to give my

00:12:18.250 --> 00:12:21.410
money to this guy right
over there.

00:12:21.410 --> 00:12:24.090
But then what happens as
soon as things go bad?

00:12:24.090 --> 00:12:26.320
As soon as the world gets a
little bit of difficult, this

00:12:26.320 --> 00:12:31.180
extra risk actually starts
rearing its ugly head and then

00:12:31.180 --> 00:12:32.600
this guy becomes insolvent.

00:12:32.600 --> 00:12:35.040
So this guy's-- let me see if I
can draw his balance sheet.

00:12:35.040 --> 00:12:38.170
His balance sheet will look
something like that.

00:12:38.170 --> 00:12:40.650
He has a bunch of loans out.

00:12:40.650 --> 00:12:43.470
He's got some reserves right
there and then he's got a

00:12:43.470 --> 00:12:49.050
bunch of deposits that you or
I might have given him.

00:12:49.050 --> 00:12:51.710
These are our IOUs to us.

00:12:51.710 --> 00:12:52.840
These are his loans out.

00:12:52.840 --> 00:12:55.210
As soon as things start going
bad, some of these loans are

00:12:55.210 --> 00:12:56.190
going to go bad.

00:12:56.190 --> 00:12:59.440
As soon as one of those loans
go bad, I'm going to hear

00:12:59.440 --> 00:13:02.775
about it on the news, and I'm
going to run over to the bank,

00:13:02.775 --> 00:13:05.400
and I'm going to say, let me get
my money back as quickly

00:13:05.400 --> 00:13:05.990
as possible.

00:13:05.990 --> 00:13:07.950
And let's say I'm the first
one there and I

00:13:07.950 --> 00:13:08.870
get my money back.

00:13:08.870 --> 00:13:10.120
They promised it to me.

00:13:12.460 --> 00:13:15.500
So I withdraw my account right
there, and I get all my money

00:13:15.500 --> 00:13:17.780
back, and I essentially
deplete the reserves.

00:13:17.780 --> 00:13:19.500
Now the next guy's going
to come along.

00:13:19.500 --> 00:13:21.980
This guy was a little bit of
a slower runner than me.

00:13:21.980 --> 00:13:24.680
He comes to the bank, and he
says give me back my reserves.

00:13:24.680 --> 00:13:26.910
And the bank says there
are no reserves.

00:13:26.910 --> 00:13:29.820
And then this guy is mad.

00:13:29.820 --> 00:13:30.850
He has no reserves.

00:13:30.850 --> 00:13:33.920
And, of course, the whole time
this guy's assets are just

00:13:33.920 --> 00:13:35.980
becoming worthless because he's
loaned it to people who

00:13:35.980 --> 00:13:37.440
invested it badly.

00:13:37.440 --> 00:13:39.235
And so you have all of these
people-- when he finds out

00:13:39.235 --> 00:13:41.450
that this guy didn't get his
money, all of these people are

00:13:41.450 --> 00:13:43.780
going to go there and demand
their money and the bank is

00:13:43.780 --> 00:13:46.340
going to be insolvent.

00:13:46.340 --> 00:13:47.490
It's going to be two things.

00:13:47.490 --> 00:13:49.330
It's going to be illiquid
and insolvent.

00:13:49.330 --> 00:13:51.170
And I've explained the
difference, but illiquid might

00:13:51.170 --> 00:13:55.980
mean the loans might be good--
and actually I'll reiterate it

00:13:55.980 --> 00:13:57.980
in this video-- but the loans
might be good, it just doesn't

00:13:57.980 --> 00:13:58.900
have the cash.

00:13:58.900 --> 00:14:01.100
Insolvent means that these loans
aren't good, that the

00:14:01.100 --> 00:14:03.690
value of these loans aren't
equal to these deposits.

00:14:03.690 --> 00:14:06.190
So I'll say that this risky
bank is insolvent.

00:14:09.640 --> 00:14:13.430
Now, you might say these guys
right here deserve it.

00:14:13.430 --> 00:14:17.760
They guys deserve what they got
because they were greedy.

00:14:17.760 --> 00:14:20.310
They invested in this risky
bank that was doing shady

00:14:20.310 --> 00:14:22.010
things just to get a
little bit higher

00:14:22.010 --> 00:14:23.470
interest on their deposits.

00:14:23.470 --> 00:14:25.780
But what happens when
this happens?

00:14:25.780 --> 00:14:30.010
Well, you have these good banks
over here that maybe did

00:14:30.010 --> 00:14:31.370
really conservative lending.

00:14:33.970 --> 00:14:36.110
So this is what their balance
sheets-- maybe both of their

00:14:36.110 --> 00:14:37.050
balance sheets look like this.

00:14:37.050 --> 00:14:39.450
They have some reserves
right there.

00:14:39.450 --> 00:14:44.190
They've made out some good loans
to good entrepreneurs

00:14:44.190 --> 00:14:45.450
and then you have their
depositors.

00:14:49.660 --> 00:14:53.270
But as soon as this guy finds
out about what happened at

00:14:53.270 --> 00:14:55.370
that bank, he's like, things
are really bad.

00:14:55.370 --> 00:14:58.060
Let me just be careful and go
to my bank-- let's say it's

00:14:58.060 --> 00:15:00.490
this depositor right here-- and
take my money out and put

00:15:00.490 --> 00:15:01.800
it into my mattress.

00:15:01.800 --> 00:15:04.500
So he goes to the bank and
he's the first one there.

00:15:04.500 --> 00:15:07.500
He's the fastest runner of
all of the depositors.

00:15:07.500 --> 00:15:08.970
So he claims his deposit.

00:15:08.970 --> 00:15:11.120
So his deposit, he
withdraws, and he

00:15:11.120 --> 00:15:12.370
depletes all of the reserves.

00:15:15.050 --> 00:15:16.750
Now, he wasn't the only
guy who was afraid.

00:15:16.750 --> 00:15:18.610
This guy was afraid as well.

00:15:18.610 --> 00:15:21.330
So he comes running to the bank,
and the bank now says,

00:15:21.330 --> 00:15:25.820
gee, I know I made this promise
to you that you have

00:15:25.820 --> 00:15:30.150
an on-demand account, but that
guy you saw leaving the bank

00:15:30.150 --> 00:15:32.360
with the running shoes on, he
actually just took all of the

00:15:32.360 --> 00:15:36.200
reserves, and I guess I have
to admit now that I lied to

00:15:36.200 --> 00:15:38.130
you, that it isn't completely
on demand.

00:15:38.130 --> 00:15:40.390
I actually loaned out the
rest of your money.

00:15:40.390 --> 00:15:42.620
But I was a good banker.

00:15:42.620 --> 00:15:43.680
These are actually good loans.

00:15:43.680 --> 00:15:45.340
These loans are still worth
what they want.

00:15:45.340 --> 00:15:47.970
If you just wait long enough,
you'll get your money back.

00:15:47.970 --> 00:15:50.280
And, of course, this guy's
going to be not so happy

00:15:50.280 --> 00:15:51.590
because he was lied to.

00:15:51.590 --> 00:15:53.430
And that's not going
to be the only guy.

00:15:53.430 --> 00:15:55.390
Slightly slower runners are
going to come there and also

00:15:55.390 --> 00:15:56.550
demand their money.

00:15:56.550 --> 00:15:58.200
And this is known
as a bank run.

00:16:02.640 --> 00:16:06.950
Where because one maybe bad
apple in the system actually

00:16:06.950 --> 00:16:10.220
is insolvent, everyone becomes
afraid and comes and says,

00:16:10.220 --> 00:16:13.420
give me my money, and because
we have this fractional

00:16:13.420 --> 00:16:15.700
reserve system, the
money isn't there.

00:16:15.700 --> 00:16:19.200
It just will not be there
because 90% of it is lent out.

00:16:19.200 --> 00:16:21.230
And so once this happens there,
then you're going to

00:16:21.230 --> 00:16:24.830
have a run on bank two, and then
everyone is going to take

00:16:24.830 --> 00:16:26.610
their money out of the
banking system.

00:16:26.610 --> 00:16:29.240
So the whole banking system,
this situation right here

00:16:29.240 --> 00:16:32.810
where the assets maybe still
are worth the same or more

00:16:32.810 --> 00:16:35.780
than the liabilities, this guy's
still solvent, but he

00:16:35.780 --> 00:16:38.720
doesn't have any money to pay
these on-demand accounts, so

00:16:38.720 --> 00:16:42.460
this guy is running into
a liquidity problem.

00:16:45.620 --> 00:16:48.460
He doesn't have the reserves,
and that's why we have the

00:16:48.460 --> 00:16:51.920
whole Federal Reserve that's
a lender of last resort.

00:16:51.920 --> 00:16:55.140
This guy can walk to the Federal
Reserve and say, hey,

00:16:55.140 --> 00:16:56.630
I have some people knocking
on my door.

00:16:56.630 --> 00:16:57.590
Give me some loans.

00:16:57.590 --> 00:17:02.870
The Federal Reserve will lend
this guy some money, and, of

00:17:02.870 --> 00:17:08.020
course, then this guy's going
to have another lender to it

00:17:08.020 --> 00:17:10.319
called the Federal Reserve and
he can use this to pay off

00:17:10.319 --> 00:17:11.400
these guys.

00:17:11.400 --> 00:17:13.420
But that leaves another
question: How does the Federal

00:17:13.420 --> 00:17:16.510
Reserve really know the
difference between this guy--

00:17:16.510 --> 00:17:18.650
he might say, OK,
this guy's good.

00:17:18.650 --> 00:17:20.810
Let me just lend to him to make
sure that he doesn't have

00:17:20.810 --> 00:17:22.010
a liquidity problem.

00:17:22.010 --> 00:17:23.940
How does the Federal Reserve
really know the difference

00:17:23.940 --> 00:17:26.750
between this guy and this
guy right here?

00:17:26.750 --> 00:17:28.480
This guy's going to also run
to the Federal Reserve.

00:17:28.480 --> 00:17:31.260
Hey, Federal Reserve, these
loans are still good.

00:17:31.260 --> 00:17:34.940
Just give me a loan so I can pay
these people right there.

00:17:34.940 --> 00:17:36.640
So the Federal Reserve says,
hey, you look good.

00:17:36.640 --> 00:17:40.500
You're a nice clean banker in a
nice, fancy building wearing

00:17:40.500 --> 00:17:42.680
a Rolex on your arm.

00:17:42.680 --> 00:17:43.650
You must be good for it.

00:17:43.650 --> 00:17:46.460
So the Federal Reserve will also
lend this guy, and now

00:17:46.460 --> 00:17:50.170
this guy also owes the Federal
Reserve some money.

00:17:50.170 --> 00:17:52.920
We can already see that in
this fractional reserve

00:17:52.920 --> 00:17:56.750
banking system, you have two
problems. You have an unstable

00:17:56.750 --> 00:18:04.370
equilibrium, where one bad apple
can lead to a run on the

00:18:04.370 --> 00:18:05.620
entire system.

00:18:14.970 --> 00:18:17.590
And I'll talk about what's been
engineered to fix this

00:18:17.590 --> 00:18:20.310
problem: essentially
banking insurance.

00:18:20.310 --> 00:18:22.000
I'll probably wait for that
to the next video.

00:18:22.000 --> 00:18:25.130
So this is the first
problem right here.

00:18:25.130 --> 00:18:29.520
And then the second problem is
that it really becomes hard to

00:18:29.520 --> 00:18:31.710
differentiate between the
good and bad banks.

00:18:37.730 --> 00:18:40.810
I mean, it's not so hard to
differentiate at face-- well,

00:18:40.810 --> 00:18:42.790
it is hard to differentiate at
first and becomes even harder

00:18:42.790 --> 00:18:44.610
to differentiate when you
always have the Federal

00:18:44.610 --> 00:18:48.320
Reserve willing to be a
lender of last resort.

00:18:48.320 --> 00:18:50.500
And when I talk about the
Federal Reserve insurance,

00:18:50.500 --> 00:18:52.740
that makes it even harder to
differentiate, and because

00:18:52.740 --> 00:18:55.345
it's harder to differentiate,
there's a huge incentive for

00:18:55.345 --> 00:18:56.630
banks to take on risk.

00:18:56.630 --> 00:18:59.430
Because when times are good,
they'll make more money than

00:18:59.430 --> 00:19:03.630
everyone else, and then when
times are bad, it's really

00:19:03.630 --> 00:19:06.510
hard, that they'll all
suffer equally.

00:19:06.510 --> 00:19:08.560
So in the video, I'll talk about
some of what has been

00:19:08.560 --> 00:19:10.960
engineered to fix these problems
and see if they

00:19:10.960 --> 00:19:12.770
actually make sense.


Expressing a projection on to a line as a matrix vector prod

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:00.340

00:00:00.340 --> 00:00:03.910
In the last video we saw that
if we had some line that was

00:00:03.910 --> 00:00:09.050
defined as all of the scalar
multiples of some vector-- I'll

00:00:09.050 --> 00:00:11.800
just write it like this.

00:00:11.800 --> 00:00:13.040
The scalar multiples,
obviously,

00:00:13.040 --> 00:00:15.370
are any real number.

00:00:15.370 --> 00:00:18.900
We defined a transformation, and
I didn't speak of it much

00:00:18.900 --> 00:00:21.280
in terms of transformations, but
it was a transformation.

00:00:21.280 --> 00:00:26.220
We defined a projection
onto that line L as a

00:00:26.220 --> 00:00:27.500
transformation.

00:00:27.500 --> 00:00:30.870
In the video, we drew it as
transformations within R2, but

00:00:30.870 --> 00:00:36.110
it could be, in general, a
transformation from Rn to Rn.

00:00:36.110 --> 00:00:44.210
We defined it as, the projection
of x onto L was

00:00:44.210 --> 00:00:49.046
equal to the dot product of x,
with this defining vector.

00:00:49.046 --> 00:00:53.420
x dot this defining vector,
divided by that defining

00:00:53.420 --> 00:00:54.670
vector dotted with itself.

00:00:54.670 --> 00:00:58.160

00:00:58.160 --> 00:01:01.280
All of that times the defining
vector of the line.

00:01:01.280 --> 00:01:03.050
This was our definition.

00:01:03.050 --> 00:01:05.319
A couple of things might have
popped out at you right when

00:01:05.319 --> 00:01:07.490
we first saw this.

00:01:07.490 --> 00:01:10.830
When you dot a vector with
itself, what's that equal to?

00:01:10.830 --> 00:01:15.960
We know that if I take some
vector, and I dot it with

00:01:15.960 --> 00:01:19.100
itself, that is equivalent
to the length

00:01:19.100 --> 00:01:22.480
of the vector squared.

00:01:22.480 --> 00:01:30.610
We can rewrite this as being
equal to x dot v, over the

00:01:30.610 --> 00:01:34.600
length of v squared, all
of that times v.

00:01:34.600 --> 00:01:38.920
Wouldn't it be nice if the
length of v was 1.

00:01:38.920 --> 00:01:43.870
The length of v was
equal to 1.

00:01:43.870 --> 00:01:46.460
If the length of v was 1, or
this is another way of saying

00:01:46.460 --> 00:01:52.670
that, v is a unit vector.

00:01:52.670 --> 00:01:55.040
Our formula for our projection
would just

00:01:55.040 --> 00:01:59.180
simplify to x dot v.

00:01:59.180 --> 00:02:02.980
All of that times, this will
just be some scalar number,

00:02:02.980 --> 00:02:05.180
that times v.

00:02:05.180 --> 00:02:08.660
You say, hey Sal, how do
we know if this is a

00:02:08.660 --> 00:02:10.360
unit vector or not.

00:02:10.360 --> 00:02:14.290
What you can realize is that
any-- Let me draw it this way.

00:02:14.290 --> 00:02:18.280
When I drew it in the previous
video, I just picked

00:02:18.280 --> 00:02:20.720
a line, like that.

00:02:20.720 --> 00:02:23.540
The line can be really defined
to this vector v in the line.

00:02:23.540 --> 00:02:26.150
It can be any of the vectors
that's contained in the line.

00:02:26.150 --> 00:02:28.490
The vector v could
be like that.

00:02:28.490 --> 00:02:30.100
Let's say someone gives
you a vector v that

00:02:30.100 --> 00:02:31.360
isn't a unit vector.

00:02:31.360 --> 00:02:36.050
Let's say that the length
of v is not equal to 1.

00:02:36.050 --> 00:02:39.660
How can you define a line
using sum unit vector.

00:02:39.660 --> 00:02:42.300
You can just normalize v.

00:02:42.300 --> 00:02:46.180
You can define sum unit
vector right here.

00:02:46.180 --> 00:02:48.920
You could define sum
vector right there.

00:02:48.920 --> 00:02:52.340
Let's call it u, and I'll
say it's a unit vector.

00:02:52.340 --> 00:02:59.700
That is equal to 1 over 1 over
the length of v times v.

00:02:59.700 --> 00:03:01.490
I showed you this in the
unit vector video.

00:03:01.490 --> 00:03:03.790
You can construct a unit vector
that goes in the same

00:03:03.790 --> 00:03:07.190
direction as any vector,
essentially just by dividing,

00:03:07.190 --> 00:03:09.070
or I guess multiplying,
that vector

00:03:09.070 --> 00:03:11.160
times 1 over its length.

00:03:11.160 --> 00:03:14.310
In general, we can always
redefine the line.

00:03:14.310 --> 00:03:17.150
All of the possible scalar
multiples of v are going to be

00:03:17.150 --> 00:03:20.880
the same thing as all of the
scalar multiples of our unit

00:03:20.880 --> 00:03:24.760
vector, u, which is just
a scalar multiple of v.

00:03:24.760 --> 00:03:26.860
We can redefine our line.

00:03:26.860 --> 00:03:30.410
If we redefine our line, L,
as being equal to all the

00:03:30.410 --> 00:03:35.360
possible scalar multiples of
our unit vector, where the

00:03:35.360 --> 00:03:39.150
scalars are any members
of the real numbers.

00:03:39.150 --> 00:03:41.930
Our projection definition
simplifies a good bit.

00:03:41.930 --> 00:03:51.270
The projection of x onto L
becomes x dot our unit vector,

00:03:51.270 --> 00:03:57.690
times the unit vector, times
the unit vector itself.

00:03:57.690 --> 00:04:00.890
That case that I did in the
previous video, where I had

00:04:00.890 --> 00:04:02.280
those two vectors.

00:04:02.280 --> 00:04:04.860
Where I said the vector v that
defined the line, I think it

00:04:04.860 --> 00:04:08.660
was vector 2, 1.

00:04:08.660 --> 00:04:16.720
Our vector x was
equal to 2, 3.

00:04:16.720 --> 00:04:18.810
If you want to do this
definition, we just have to

00:04:18.810 --> 00:04:21.720
turn this guy into a
unit vector first.

00:04:21.720 --> 00:04:25.830
The way you turn him into a unit
vector, you figure out

00:04:25.830 --> 00:04:26.680
the magnitude.

00:04:26.680 --> 00:04:30.060
In this case the magnitude
of v is equal to what.

00:04:30.060 --> 00:04:35.270
2 squared plus 1 squared is 1.

00:04:35.270 --> 00:04:37.080
You take the square
root of that.

00:04:37.080 --> 00:04:37.630
Let me just write.

00:04:37.630 --> 00:04:41.320
It's equal to square root of 2
squared plus 1 squared, which

00:04:41.320 --> 00:04:43.370
is equal to the square
root of 5.

00:04:43.370 --> 00:04:47.700
You can define your u-- your
unit vector could just be 1

00:04:47.700 --> 00:04:49.510
over this, times that guy.

00:04:49.510 --> 00:04:54.010
1 over the square root
of 5 times 2, 1.

00:04:54.010 --> 00:04:55.590
You could multiply
it out, or not.

00:04:55.590 --> 00:04:57.200
You could just leave
it in this form.

00:04:57.200 --> 00:05:00.320
You can always, for any vector
v, you can always find a unit

00:05:00.320 --> 00:05:03.870
vector that goes in the same
direction, assuming that we're

00:05:03.870 --> 00:05:05.870
dealing with non-zero vectors.

00:05:05.870 --> 00:05:10.060
You can always reduce anything
like this, to some other

00:05:10.060 --> 00:05:11.180
definition, like this.

00:05:11.180 --> 00:05:13.560
Where this is a unit
vector version of

00:05:13.560 --> 00:05:16.780
your vector v up there.

00:05:16.780 --> 00:05:17.890
I just said that,
look, this is a

00:05:17.890 --> 00:05:19.760
transformation from Rn to Rn.

00:05:19.760 --> 00:05:23.180
The one thing that we're not
sure of, is this a linear

00:05:23.180 --> 00:05:24.390
transformation.

00:05:24.390 --> 00:05:26.590
We can always write
it like this.

00:05:26.590 --> 00:05:28.780
Let's see if this is always
going to be a linear

00:05:28.780 --> 00:05:30.560
transformation.

00:05:30.560 --> 00:05:34.876
There's two conditions for it to
be a linear transformation.

00:05:34.876 --> 00:05:41.260

00:05:41.260 --> 00:05:47.720
Let's see what happens if I take
the projection onto L of

00:05:47.720 --> 00:05:48.040
two vectors.

00:05:48.040 --> 00:05:52.030
Let's say the vector a
plus the vector v.

00:05:52.030 --> 00:05:53.650
If I take the sum of
their vectors.

00:05:53.650 --> 00:05:55.960
If this is a linear
transformation, this should be

00:05:55.960 --> 00:05:58.190
equivalent to taking each
of their projections

00:05:58.190 --> 00:05:59.770
individually, and
then summing.

00:05:59.770 --> 00:06:01.050
Let's see if this is the case.

00:06:01.050 --> 00:06:03.950
This is equal to, by our
definition, we'll use the unit

00:06:03.950 --> 00:06:06.060
vector version, because
it's simpler.

00:06:06.060 --> 00:06:14.410
This is equal to a plus b,
that's our x, dot u.

00:06:14.410 --> 00:06:18.260
And then, all of that times
our unit vector.

00:06:18.260 --> 00:06:20.880
We know that the dot product has
a distributive property,

00:06:20.880 --> 00:06:29.580
so that this is equal to
a dot u plus b dot u.

00:06:29.580 --> 00:06:31.260
These are unit vectors.

00:06:31.260 --> 00:06:33.510
All of that times
the vector u.

00:06:33.510 --> 00:06:34.990
These are just scalar numbers.

00:06:34.990 --> 00:06:37.930
So scalar multiplication has
distributive properties.

00:06:37.930 --> 00:06:44.120
This is equal to a dot u,
times our vector u.

00:06:44.120 --> 00:06:46.200
Remember, this is just going
to be some scalar.

00:06:46.200 --> 00:06:52.670
Plus b dot u times our
unit vector u.

00:06:52.670 --> 00:06:54.980
What is this equal to.

00:06:54.980 --> 00:06:58.070
This right here is equal
to the projection of a.

00:06:58.070 --> 00:07:02.300
This is equal to the projection
of a onto L, by

00:07:02.300 --> 00:07:03.880
definition, right here.

00:07:03.880 --> 00:07:04.910
By this definition.

00:07:04.910 --> 00:07:07.280
If we assume that we're dealing
with the unit vector

00:07:07.280 --> 00:07:09.060
definition for the line.

00:07:09.060 --> 00:07:13.820
This is equal to, this whole
thing right here, is equal to

00:07:13.820 --> 00:07:18.710
plus the projection onto
L of the vector b.

00:07:18.710 --> 00:07:23.210
We see our first condition
for this being a linear

00:07:23.210 --> 00:07:25.150
transformation holds.

00:07:25.150 --> 00:07:28.190
A projection of the sum of the
vectors is equal to the sum of

00:07:28.190 --> 00:07:30.390
the projections of
the vectors.

00:07:30.390 --> 00:07:33.350
Our second condition is that
the projection of a scalar

00:07:33.350 --> 00:07:35.330
multiple should be equal to
a scalar multiple of the

00:07:35.330 --> 00:07:36.710
projection.

00:07:36.710 --> 00:07:39.150
Let me write that down.

00:07:39.150 --> 00:07:44.380
What is the projection onto L
of some scalar multiple of

00:07:44.380 --> 00:07:46.420
some vector a.

00:07:46.420 --> 00:07:53.940
That is equal to ca dot
our unit vector u

00:07:53.940 --> 00:07:57.060
times unit vector u.

00:07:57.060 --> 00:07:58.710
This one is a little bit
more straightforward.

00:07:58.710 --> 00:08:01.060
This is the scalar multiple.

00:08:01.060 --> 00:08:04.410
We see it in our dot product
properties, this is equal to c

00:08:04.410 --> 00:08:09.600
times a dot u, times
the vector u.

00:08:09.600 --> 00:08:12.320
This is just equal to c times,
this right here, is a

00:08:12.320 --> 00:08:14.000
projection of a onto L.

00:08:14.000 --> 00:08:18.320

00:08:18.320 --> 00:08:21.970
We met both of our conditions
for linear transformations.

00:08:21.970 --> 00:08:26.020
We know that our projection
onto a line L in Rn is a

00:08:26.020 --> 00:08:27.480
linear transformation.

00:08:27.480 --> 00:08:30.610
That tells us that we can
represent it as a matrix

00:08:30.610 --> 00:08:31.860
transformation.

00:08:31.860 --> 00:08:35.770

00:08:35.770 --> 00:08:41.220
We know that projection of x
onto L, we already know this

00:08:41.220 --> 00:08:42.440
definition, it can
be rewritten.

00:08:42.440 --> 00:08:43.919
It doesn't hurt to rewrite it.

00:08:43.919 --> 00:08:48.550
As x dot some unit vector
that defines our line.

00:08:48.550 --> 00:08:50.130
Let me draw it with a little
hat to show that

00:08:50.130 --> 00:08:51.680
it is a unit vector.

00:08:51.680 --> 00:08:53.910
Times the unit vector
itself, so that we

00:08:53.910 --> 00:08:56.590
actually get a vector.

00:08:56.590 --> 00:08:59.800
How can I write this as
some matrix product.

00:08:59.800 --> 00:09:01.220
Some matrix vector product.

00:09:01.220 --> 00:09:02.730
I want to write it as
a product of some

00:09:02.730 --> 00:09:05.130
matrix times x.

00:09:05.130 --> 00:09:09.440

00:09:09.440 --> 00:09:11.230
To simplify things, since we're
actually dealing with a

00:09:11.230 --> 00:09:15.920
matrix, let's limit ourselves
to the case of R2.

00:09:15.920 --> 00:09:19.010
I'm assuming that my projection
onto L is going to

00:09:19.010 --> 00:09:22.360
be a mapping from R2 to R2.

00:09:22.360 --> 00:09:23.770
You could do what I'm
doing here with

00:09:23.770 --> 00:09:25.630
an arbitrary dimension.

00:09:25.630 --> 00:09:29.850
If we're doing it in R2, then
our matrix A, right there, is

00:09:29.850 --> 00:09:33.540
going to be a 2 by 2 matrix.

00:09:33.540 --> 00:09:36.200
We've seen in multiple videos
that to figure out the matrix

00:09:36.200 --> 00:09:41.310
A, we just take the identity
matrix that has the standard

00:09:41.310 --> 00:09:43.360
basis vectors as columns.

00:09:43.360 --> 00:09:44.610
0, 1.

00:09:44.610 --> 00:09:46.460
Or 1, 0, and then 0, 1.

00:09:46.460 --> 00:09:48.680
And we apply the transformation
to each of

00:09:48.680 --> 00:09:50.740
these columns.

00:09:50.740 --> 00:09:54.400
We could say that A is going
to be equal to-- its first

00:09:54.400 --> 00:09:58.910
column is going to be equal to
the projection onto L of this

00:09:58.910 --> 00:10:00.200
thing right here.

00:10:00.200 --> 00:10:03.850
We'll do it in this orange
color, right here.

00:10:03.850 --> 00:10:05.050
What is that going to be.

00:10:05.050 --> 00:10:08.300
That is going to
be this dot u.

00:10:08.300 --> 00:10:09.570
Let me write my u.

00:10:09.570 --> 00:10:12.540
My unit vector, let's just
assume that u can be rewritten

00:10:12.540 --> 00:10:17.990
as my unit vector is equal
to sum u1 and u2.

00:10:17.990 --> 00:10:20.100
Just like that.

00:10:20.100 --> 00:10:23.170
I need to take this dot
my unit vector, let

00:10:23.170 --> 00:10:24.620
me write this down.

00:10:24.620 --> 00:10:25.940
Let me write this on the side.

00:10:25.940 --> 00:10:27.560
The first thing I want to do
is figure out what the

00:10:27.560 --> 00:10:32.830
projection-- the projection
onto L, let me

00:10:32.830 --> 00:10:35.270
write it this way.

00:10:35.270 --> 00:10:38.680
We know the projection is just
equal to this dot this times

00:10:38.680 --> 00:10:39.340
that vector.

00:10:39.340 --> 00:10:41.140
Let me write that.

00:10:41.140 --> 00:10:46.660
The vector 1, 0 dot
the unit vector u,

00:10:46.660 --> 00:10:48.970
which is just u1, u2.

00:10:48.970 --> 00:10:53.980
We're going to have that
times my unit vector.

00:10:53.980 --> 00:10:54.770
I'll write it like this.

00:10:54.770 --> 00:10:58.930
Times the vector u1, u2.

00:10:58.930 --> 00:11:00.800
This is going to be my
first column in my

00:11:00.800 --> 00:11:02.580
transformation matrix.

00:11:02.580 --> 00:11:06.190
My second column is going to be
the same thing, but I'm not

00:11:06.190 --> 00:11:07.980
ready to take the projection
of this guy.

00:11:07.980 --> 00:11:10.880
The definition of our projection
is you dot this guy

00:11:10.880 --> 00:11:13.460
with our unit vector.

00:11:13.460 --> 00:11:14.310
So we dot it.

00:11:14.310 --> 00:11:19.570
We're taking the dot
product of 0, 1.

00:11:19.570 --> 00:11:26.190
0, 1 dot my unit vector
dot u1, u2.

00:11:26.190 --> 00:11:28.000
I'm going to multiply
that times my unit

00:11:28.000 --> 00:11:31.830
vector, times u1, u2.

00:11:31.830 --> 00:11:34.670
This seems very complicated, but
it should simplify when we

00:11:34.670 --> 00:11:36.810
actually try to work out our
transformation matrix.

00:11:36.810 --> 00:11:38.520
Let's do it.

00:11:38.520 --> 00:11:40.570
When I dot these two guys,
what do I get.

00:11:40.570 --> 00:11:42.580
Let me write it here.

00:11:42.580 --> 00:11:50.830
My matrix A will become 1 times
u1, plus 0 times u2.

00:11:50.830 --> 00:11:52.720
That's just u1.

00:11:52.720 --> 00:11:55.190
This whole thing just simplifies
to u1, when I take

00:11:55.190 --> 00:11:57.350
the dot product of
these two things.

00:11:57.350 --> 00:12:01.050
Times u1, u2.

00:12:01.050 --> 00:12:03.210
That's going to be
my first column.

00:12:03.210 --> 00:12:06.890
My second column, if I dot these
two guys, I get 0 times

00:12:06.890 --> 00:12:08.680
u1 plus 1 times u2.

00:12:08.680 --> 00:12:16.240
So I'm going to get u2 times
my unit vector, u1, u2.

00:12:16.240 --> 00:12:18.790
If I multiply that out, this
will be equal to what.

00:12:18.790 --> 00:12:20.540
I can just write them
as columns. u1

00:12:20.540 --> 00:12:24.060
times u1 is u1 squared.

00:12:24.060 --> 00:12:29.860
u1 times u2 is u1, u2.

00:12:29.860 --> 00:12:34.680
u2 times u1 is u2 times u1.

00:12:34.680 --> 00:12:38.560
Then, u2 times u2
is u2 squared.

00:12:38.560 --> 00:12:43.110
You give me any unit vector
and I will give you the

00:12:43.110 --> 00:12:45.920
transformation that gives you
any projection of some other

00:12:45.920 --> 00:12:48.430
vector onto the line
defined by that.

00:12:48.430 --> 00:12:50.670
That was a very long
way of saying that.

00:12:50.670 --> 00:12:52.030
Let's go back to what
I did before.

00:12:52.030 --> 00:12:54.880

00:12:54.880 --> 00:12:58.140
Let's say we want to find any
projection onto the line, onto

00:12:58.140 --> 00:13:00.540
the vector, I'll draw it here.

00:13:00.540 --> 00:13:03.580
We'll do the same example that
we did in the last video.

00:13:03.580 --> 00:13:07.710
If I have some vector v
that looks like that.

00:13:07.710 --> 00:13:13.200
We said the vector v was equal
to the vector 2, 1.

00:13:13.200 --> 00:13:14.680
That was my vector v.

00:13:14.680 --> 00:13:16.930
How can we find sum
transformation for the

00:13:16.930 --> 00:13:19.880
projection onto the
line defined by v?

00:13:19.880 --> 00:13:21.940
Onto this line right here.

00:13:21.940 --> 00:13:24.530
The line defined by v.

00:13:24.530 --> 00:13:29.070
What we can first do is convert
v into a unit vector.

00:13:29.070 --> 00:13:32.360
We can convert v into
a unit vector that

00:13:32.360 --> 00:13:34.840
goes in the same direction.

00:13:34.840 --> 00:13:36.530
Some unit vector u.

00:13:36.530 --> 00:13:38.320
We did that already up here.

00:13:38.320 --> 00:13:40.290
Where we essentially just
divided [? bv ?]

00:13:40.290 --> 00:13:42.210
by it's length.

00:13:42.210 --> 00:13:44.840
Let's take v and divide
by it's length.

00:13:44.840 --> 00:13:46.830
The unit vector is this, 1
over the square root of 5

00:13:46.830 --> 00:13:50.800
times our vector v.

00:13:50.800 --> 00:13:55.090
It was 1 over the square
root of 5 times our

00:13:55.090 --> 00:13:56.960
vector v, right there.

00:13:56.960 --> 00:13:59.820
You start with a unit
vector there.

00:13:59.820 --> 00:14:02.610
You just create this matrix,
and then we will have our

00:14:02.610 --> 00:14:04.470
transformation matrix.

00:14:04.470 --> 00:14:07.660
If this is our u, what will
our matrix be equal to.

00:14:07.660 --> 00:14:09.220
This is u.

00:14:09.220 --> 00:14:14.270
Then our matrix would be
equal to u1 squared.

00:14:14.270 --> 00:14:15.220
What is u1 squared.

00:14:15.220 --> 00:14:20.190
Let me rewrite our u a little
bit, not at angles.

00:14:20.190 --> 00:14:24.060
Our vector u, our unit vector
that defines this line, is

00:14:24.060 --> 00:14:29.020
equal to the vector 2 over the
square root of 5 and 1 over

00:14:29.020 --> 00:14:29.910
the square root of 5.

00:14:29.910 --> 00:14:32.730
I just multiplied
out this scalar.

00:14:32.730 --> 00:14:37.870
If we want to construct this
matrix, we get A is equal to

00:14:37.870 --> 00:14:38.870
u1 squared.

00:14:38.870 --> 00:14:39.550
What is this squared?

00:14:39.550 --> 00:14:43.300
It becomes 2 squared 4 over the
square root of 5 squared,

00:14:43.300 --> 00:14:44.700
which is just 5.

00:14:44.700 --> 00:14:48.640
Equals 4 over 5.

00:14:48.640 --> 00:14:50.280
What is u1 times u2?

00:14:50.280 --> 00:14:52.780
2 times 1 over square root of
5 times square root of 5.

00:14:52.780 --> 00:14:55.590
So, 2/5.

00:14:55.590 --> 00:14:56.970
I just multiplied these two.

00:14:56.970 --> 00:14:58.990
What is u2 times u1.

00:14:58.990 --> 00:14:59.600
Same thing.

00:14:59.600 --> 00:15:01.570
Order doesn't matter
when you multiply.

00:15:01.570 --> 00:15:03.760
This will also be 2/5.

00:15:03.760 --> 00:15:04.780
What is u2 squared.

00:15:04.780 --> 00:15:07.270
1 squared over the square root
of 5 squared is just 1/5.

00:15:07.270 --> 00:15:11.170

00:15:11.170 --> 00:15:14.040
Now we can say-- and that's the
neat thing about creating

00:15:14.040 --> 00:15:18.010
these matrices, that the
projection-- Let's say we have

00:15:18.010 --> 00:15:20.230
some, let's say this is the
origin right here, we have

00:15:20.230 --> 00:15:22.540
some other vector
x, right here.

00:15:22.540 --> 00:15:25.220
We can now define our
transformation.

00:15:25.220 --> 00:15:31.970
The projection onto L where
L is equal to any scalar

00:15:31.970 --> 00:15:34.190
multiple of our unit vector u.

00:15:34.190 --> 00:15:37.070
It's right here.

00:15:37.070 --> 00:15:39.030
Is a member of the reals.

00:15:39.030 --> 00:15:40.260
That is our line L.

00:15:40.260 --> 00:15:43.770
The projection onto L
of any vector x is

00:15:43.770 --> 00:15:45.260
equal to this matrix.

00:15:45.260 --> 00:15:54.070
Is equal to the matrix 4, 5,
2/5, 2/5, 1/5 times x.

00:15:54.070 --> 00:15:56.530
Which is a pretty neat result,
at least for me.

00:15:56.530 --> 00:15:59.630
We once again reduced everything
to just a matrix

00:15:59.630 --> 00:16:00.760
multiplication.

00:16:00.760 --> 00:16:03.400
You take this x and you multiply
it by this matrix,

00:16:03.400 --> 00:16:05.470
you're going to get its
projection onto

00:16:05.470 --> 00:16:08.790
the L, onto the line.

00:16:08.790 --> 00:16:13.095
If you take this vector, let's
say a, and you multiply it

00:16:13.095 --> 00:16:15.190
times this matrix right there,
you're going to get its

00:16:15.190 --> 00:16:16.480
projection.

00:16:16.480 --> 00:16:18.420
Its projection onto the line.

00:16:18.420 --> 00:16:21.320
If you could take this vector--
No, it should go

00:16:21.320 --> 00:16:22.010
through the origin.

00:16:22.010 --> 00:16:23.240
I want to draw it in
standard position.

00:16:23.240 --> 00:16:26.660
If you take this vector, right
there, and multiply it times

00:16:26.660 --> 00:16:30.300
this matrix, you're going to get
this vector, right here,

00:16:30.300 --> 00:16:32.750
that is contained in the line.

00:16:32.750 --> 00:16:34.890
When you subtract it from
this, it's orthogonal.

00:16:34.890 --> 00:16:35.640
We know the definition.

00:16:35.640 --> 00:16:37.330
It's kind of the shadow
of that vector.

00:16:37.330 --> 00:16:40.220
Anyway, I think this
is pretty neat.


Introduction to projections

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.630
Let's say I have a line that
goes through the origin.

00:00:03.630 --> 00:00:07.630
I'll draw it in R2, but this
can be extended to an

00:00:07.630 --> 00:00:09.630
arbitrary Rn.

00:00:09.630 --> 00:00:12.410
Let me draw my axes.

00:00:12.410 --> 00:00:15.310
Those are my axes right there,
not perfectly drawn, but you

00:00:15.310 --> 00:00:15.670
get the idea.

00:00:15.670 --> 00:00:19.810
Let me draw a line that goes
through the origin here.

00:00:19.810 --> 00:00:21.600
So that is my line there.

00:00:21.600 --> 00:00:24.830
And we know that a line in any
Rn-- we're doing it in R2--

00:00:24.830 --> 00:00:27.380
can be defined as just all of
the possible scalar multiples

00:00:27.380 --> 00:00:28.550
of some vector.

00:00:28.550 --> 00:00:31.390
So let's say that this is
some vector right here

00:00:31.390 --> 00:00:32.600
that's on the line.

00:00:32.600 --> 00:00:35.120
We can define our line.

00:00:35.120 --> 00:00:38.320
We could say l is equal to
the set of all the scalar

00:00:38.320 --> 00:00:42.700
multiples-- let's say that
that is v, right there.

00:00:42.700 --> 00:00:46.520
So it's all the possible scalar
multiples of our vector

00:00:46.520 --> 00:00:50.870
v where the scalar multiples,
by definition, are just any

00:00:50.870 --> 00:00:53.310
real number.

00:00:53.310 --> 00:00:56.380
So obviously, if you take all
of the possible multiples of

00:00:56.380 --> 00:00:59.760
v, both positive multiples and
negative multiples, and less

00:00:59.760 --> 00:01:04.239
than 1 multiples, fraction
multiples, you'll have a set

00:01:04.239 --> 00:01:08.720
of vectors that will essentially
define or specify

00:01:08.720 --> 00:01:11.340
every point on that line that
goes through the origin.

00:01:11.340 --> 00:01:13.820
And we know, of course, if this
wasn't a line that went

00:01:13.820 --> 00:01:14.720
through the origin,
you would have to

00:01:14.720 --> 00:01:15.910
shift it by some vector.

00:01:15.910 --> 00:01:18.490
It would have to be some
other vector plus cv.

00:01:18.490 --> 00:01:21.450
But anyway, we're starting off
with this line definition that

00:01:21.450 --> 00:01:22.580
goes through the origin.

00:01:22.580 --> 00:01:25.740
What I want to do in this video
is to define the idea of

00:01:25.740 --> 00:01:33.550
a projection onto l of
some other vector x.

00:01:33.550 --> 00:01:36.630
So let me draw my
other vector x.

00:01:36.630 --> 00:01:42.000
Let's say that this right here
is my other vector x.

00:01:42.000 --> 00:01:44.440
Now, a projection, I'm going
to give you just a sense of

00:01:44.440 --> 00:01:47.770
it, and then we'll define it a
little bit more precisely.

00:01:47.770 --> 00:01:49.860
A projection, I always imagine,
is if you had some

00:01:49.860 --> 00:01:52.790
light source that were
perpendicular somehow or

00:01:52.790 --> 00:01:56.370
orthogonal to our line-- so
let's say our light source was

00:01:56.370 --> 00:02:00.220
shining down like this, and
I'm doing that direction

00:02:00.220 --> 00:02:03.950
because that is perpendicular
to my line, I imagine the

00:02:03.950 --> 00:02:08.600
projection of x onto this line
as kind of the shadow of x.

00:02:08.600 --> 00:02:11.485
So if this light was coming
down, I would just draw a

00:02:11.485 --> 00:02:15.630
perpendicular like that, and the
shadow of x onto l would

00:02:15.630 --> 00:02:17.840
be that vector right there.

00:02:17.840 --> 00:02:26.500
So we can view it as the shadow
of x on our line l.

00:02:26.500 --> 00:02:27.550
That's one way to think of it.

00:02:27.550 --> 00:02:29.480
Another way to think of it,
and you can think of it

00:02:29.480 --> 00:02:33.580
however you like, is how much of
x goes in the l direction?

00:02:33.580 --> 00:02:35.610
So the technique would
be the same.

00:02:35.610 --> 00:02:39.570
You would draw a perpendicular
from x to l, and you say, OK

00:02:39.570 --> 00:02:43.740
then how much of l would have to
go in that direction to get

00:02:43.740 --> 00:02:45.330
to my perpendicular?

00:02:45.330 --> 00:02:48.270
Either of those are how I
think of the idea of a

00:02:48.270 --> 00:02:48.790
projection.

00:02:48.790 --> 00:02:52.140
I think the shadow is part of
the motivation for why it's

00:02:52.140 --> 00:02:53.710
even called a projection,
right?

00:02:53.710 --> 00:02:56.720
When you project something,
you're beaming light and

00:02:56.720 --> 00:02:58.870
seeing where the light
hits on a wall, and

00:02:58.870 --> 00:02:59.610
you're doing that here.

00:02:59.610 --> 00:03:01.330
You're beaming light and you're
seeing where that light

00:03:01.330 --> 00:03:03.190
hits on a line in this case.

00:03:03.190 --> 00:03:05.835
But you can't do anything
with this definition.

00:03:05.835 --> 00:03:07.750
This is just kind of an
intuitive sense of what a

00:03:07.750 --> 00:03:08.740
projection is.

00:03:08.740 --> 00:03:10.570
So we need to figure out some
way to calculate this, or a

00:03:10.570 --> 00:03:13.490
more mathematically precise
definition.

00:03:13.490 --> 00:03:16.430
And one thing we can do is,
when I created this

00:03:16.430 --> 00:03:18.360
projection-- let me actually
draw another projection of

00:03:18.360 --> 00:03:21.480
another line or another vector
just so you get the idea.

00:03:21.480 --> 00:03:25.030
If I had some other vector over
here that looked like

00:03:25.030 --> 00:03:27.050
that, the projection of this
onto the line would look

00:03:27.050 --> 00:03:27.730
something like this.

00:03:27.730 --> 00:03:29.440
You would just draw a
perpendicular and its

00:03:29.440 --> 00:03:31.430
projection would be like that.

00:03:31.430 --> 00:03:33.210
But I don't want to talk
about just this case.

00:03:33.210 --> 00:03:35.410
I want to give you the sense
that it's the shadow of any

00:03:35.410 --> 00:03:37.130
vector onto this line.

00:03:37.130 --> 00:03:39.660
So how can we think about it
with our original example?

00:03:39.660 --> 00:03:43.350
In every case, no matter how
I perceive it, I dropped a

00:03:43.350 --> 00:03:45.100
perpendicular down here.

00:03:45.100 --> 00:03:48.270
And so if we construct a vector
right here, we could

00:03:48.270 --> 00:03:50.120
say, hey, that vector is
always going to be

00:03:50.120 --> 00:03:51.840
perpendicular to the line.

00:03:51.840 --> 00:03:52.920
And we can do that.

00:03:52.920 --> 00:03:55.320
I wouldn't have been talking
about it if we couldn't.

00:03:55.320 --> 00:03:58.050
So let me define this vector,
which I've not

00:03:58.050 --> 00:03:59.480
even defined it.

00:03:59.480 --> 00:04:01.450
What is this vector
going to be?

00:04:01.450 --> 00:04:04.410
If this vector-- let me
not use all these.

00:04:07.970 --> 00:04:11.150
We know we want to somehow
get to this blue vector.

00:04:11.150 --> 00:04:13.810
Let me keep it in blue.

00:04:13.810 --> 00:04:19.560
That blue vector is the
projection of x onto l.

00:04:19.560 --> 00:04:21.500
That's what we want to get to.

00:04:21.500 --> 00:04:23.230
Now, one thing we can
look at is this pink

00:04:23.230 --> 00:04:23.920
vector right there.

00:04:23.920 --> 00:04:25.580
What is that pink vector?

00:04:25.580 --> 00:04:30.910
That pink vector that I just
drew, that's the vector x

00:04:30.910 --> 00:04:35.610
minus the projection, minus this
blue vector over here,

00:04:35.610 --> 00:04:40.030
minus the projection
of x onto l, right?

00:04:40.030 --> 00:04:43.120
If you add the projection to
the pink vector, you get x.

00:04:43.120 --> 00:04:46.050
So if you add this blue
projection of x to x minus the

00:04:46.050 --> 00:04:48.480
projection of x, you're, of
course, you going to get x.

00:04:48.480 --> 00:04:51.630
We also know that this pink
vector is orthogonal to the

00:04:51.630 --> 00:04:54.510
line itself, which means it's
orthogonal to every vector on

00:04:54.510 --> 00:04:57.400
the line, which also means that
its dot product is going

00:04:57.400 --> 00:04:57.880
to be zero.

00:04:57.880 --> 00:05:01.330
So let me define the projection
this way.

00:05:01.330 --> 00:05:03.680
The projection, this is going
to be my slightly more

00:05:03.680 --> 00:05:06.300
mathematical definition.

00:05:06.300 --> 00:05:12.270
The projection onto l of some
vector x is going to be some

00:05:12.270 --> 00:05:18.380
vector that's in l, right?

00:05:18.380 --> 00:05:20.390
I drew it right here,
this blue vector.

00:05:20.390 --> 00:05:22.590
I'll trace it with
white right here.

00:05:22.590 --> 00:05:29.070
Some vector in l where, and
this might be a little bit

00:05:29.070 --> 00:05:38.910
unintuitive, where x minus the
projection vector onto l of x

00:05:38.910 --> 00:05:47.120
is orthogonal to my line.

00:05:47.120 --> 00:05:49.850
So I'm saying the projection--
this is my definition.

00:05:49.850 --> 00:05:53.990
I'm defining the projection of x
onto l with some vector in l

00:05:53.990 --> 00:05:57.390
where x minus that projection
is orthogonal to l.

00:05:57.390 --> 00:05:58.490
This is my definition.

00:05:58.490 --> 00:06:02.180
That is a little bit more
precise and I think it makes a

00:06:02.180 --> 00:06:04.940
bit of sense why it connects to
the idea of the shadow or

00:06:04.940 --> 00:06:06.050
projection.

00:06:06.050 --> 00:06:07.420
But how can we deal with this?

00:06:07.420 --> 00:06:08.860
I mean, this is still
just in words.

00:06:08.860 --> 00:06:13.580
How can I actually calculate
the projection of x onto l?

00:06:13.580 --> 00:06:16.620
Well, the key clue here is this
notion that x minus the

00:06:16.620 --> 00:06:20.160
projection of x is
orthogonal to l.

00:06:20.160 --> 00:06:21.930
So let's see if we can
use that somehow.

00:06:21.930 --> 00:06:24.700
So the first thing we need to
realize is, by definition,

00:06:24.700 --> 00:06:30.150
because the projection of x onto
l is some vector in l,

00:06:30.150 --> 00:06:33.660
that means it's some scalar
multiple of v, some scalar

00:06:33.660 --> 00:06:37.510
multiple of our defining vector,
of our v right there.

00:06:37.510 --> 00:06:41.700
So we could also say, look, we
could rewrite our projection

00:06:41.700 --> 00:06:43.640
of x onto l.

00:06:43.640 --> 00:06:46.760
We could write it as some scalar
multiple times our

00:06:46.760 --> 00:06:48.270
vector v, right?

00:06:48.270 --> 00:06:49.000
We can say that.

00:06:49.000 --> 00:06:51.160
This is equivalent to
our projection.

00:06:51.160 --> 00:06:54.120
Now, we also know that x minus
our projection is orthogonal

00:06:54.120 --> 00:06:59.580
to l, so we also know that x
minus our projection-- and I

00:06:59.580 --> 00:07:02.345
just said that I could rewrite
my projection as some multiple

00:07:02.345 --> 00:07:03.990
of this vector right there.

00:07:03.990 --> 00:07:05.140
You could see it the
way I drew it here.

00:07:05.140 --> 00:07:07.790
It almost looks like it's
2 times its vector.

00:07:07.790 --> 00:07:11.330
So we know that x minus our
projection, this is our

00:07:11.330 --> 00:07:16.150
projection right here,
is orthogonal to l.

00:07:16.150 --> 00:07:19.710
Orthogonality, by definition,
means its dot product with any

00:07:19.710 --> 00:07:21.460
vector in l is 0.

00:07:21.460 --> 00:07:23.720
So let's dot it with
some vector in l.

00:07:23.720 --> 00:07:27.060
Or we could dot it with
this vector v.

00:07:27.060 --> 00:07:28.580
That's what we use
to define l.

00:07:28.580 --> 00:07:31.700
So let's dot it with v,
and we know that that

00:07:31.700 --> 00:07:33.380
must be equal to 0.

00:07:33.380 --> 00:07:35.810
We're taking this vector right
here, dotting it with v, and

00:07:35.810 --> 00:07:38.680
we know that this has
to be equal to 0.

00:07:38.680 --> 00:07:41.390
That has to be equal to 0.

00:07:41.390 --> 00:07:44.020
So let's use our properties of
dot products to see if we can

00:07:44.020 --> 00:07:47.910
calculate a particular value of
c, because once we know a

00:07:47.910 --> 00:07:51.130
particular value of c, then we
can just always multiply that

00:07:51.130 --> 00:07:54.910
times the vector v, which we
are given, and we will have

00:07:54.910 --> 00:07:55.760
our projection.

00:07:55.760 --> 00:07:57.720
And then I'll show it to you
with some actual numbers.

00:07:57.720 --> 00:07:59.610
So let's see if we can
calculate a c.

00:07:59.610 --> 00:08:03.200
So if we distribute this c--
oh, sorry, if we distribute

00:08:03.200 --> 00:08:05.670
the v, we know the dot
product exhibits the

00:08:05.670 --> 00:08:06.940
distributive property.

00:08:06.940 --> 00:08:14.470
This expression can be rewritten
as x dot v, right? x

00:08:14.470 --> 00:08:17.725
dot v minus c times v dot v.

00:08:20.450 --> 00:08:21.410
I rearranged things.

00:08:21.410 --> 00:08:26.180
We know that c minus cv dot
v is the same thing.

00:08:26.180 --> 00:08:27.530
We could write it as minus cv.

00:08:27.530 --> 00:08:30.520
This is minus c times v dot v,
and all of this, of course, is

00:08:30.520 --> 00:08:33.260
equal to 0.

00:08:33.260 --> 00:08:36.750
And if we want to solve for c,
let's add cv dot v to both

00:08:36.750 --> 00:08:37.950
sides of the equation.

00:08:37.950 --> 00:08:44.750
And you get x dot v is equal
to c times v dot v.

00:08:44.750 --> 00:08:46.820
Solving for c, let's divide
both sides of this

00:08:46.820 --> 00:08:48.740
equation by v dot v.

00:08:48.740 --> 00:08:51.160
You get-- I'll do it in
a different color.

00:08:51.160 --> 00:09:00.470
c is equal to this: x dot
v divided by v dot v.

00:09:00.470 --> 00:09:01.930
Now, what was c?

00:09:01.930 --> 00:09:04.090
We are saying the projection of
x-- let me write it here.

00:09:04.090 --> 00:09:10.250
The projection of x onto
l is equal to some

00:09:10.250 --> 00:09:11.870
scalar multiple, right?

00:09:11.870 --> 00:09:17.110
We know it's in the line, so
it's some scalar multiple of

00:09:17.110 --> 00:09:21.550
this defining vector,
the vector v.

00:09:21.550 --> 00:09:23.780
And we just figured out
what that scalar

00:09:23.780 --> 00:09:24.730
multiple is going to be.

00:09:24.730 --> 00:09:35.080
It's going to be x dot v over v
dot v, and this, of course,

00:09:35.080 --> 00:09:36.680
is just going to be
a number, right?

00:09:36.680 --> 00:09:37.640
This is a scalar still.

00:09:37.640 --> 00:09:39.170
Even though we have all these
vectors here, when you take

00:09:39.170 --> 00:09:41.460
their dot products, you just end
up with a number, and you

00:09:41.460 --> 00:09:44.010
multiply that number times v.

00:09:44.010 --> 00:09:47.370
You just kind of scale v and
you get your projection.

00:09:47.370 --> 00:09:51.000
So in this case, the way I
drew it up here, my dot

00:09:51.000 --> 00:09:54.240
product should end up with some
scaling factor that's

00:09:54.240 --> 00:09:57.260
close to 2, so that if I start
with a v and I scale it up by

00:09:57.260 --> 00:09:59.700
2, this value would be 2, and
I'd get a projection that

00:09:59.700 --> 00:10:01.210
looks something like that.

00:10:01.210 --> 00:10:03.360
Now, this looks a little
abstract to you, so let's do

00:10:03.360 --> 00:10:05.870
it with some real vectors, and
I think it'll make a little

00:10:05.870 --> 00:10:06.610
bit more sense.

00:10:06.610 --> 00:10:09.450
And nothing I did here
only applies to R2.

00:10:09.450 --> 00:10:12.650
Everything I did here can be
extended to an arbitrarily

00:10:12.650 --> 00:10:16.850
high dimension, so even though
we're doing it in R2, and R2

00:10:16.850 --> 00:10:18.890
and R3 is where we tend to
deal with projections the

00:10:18.890 --> 00:10:21.560
most, this could apply to Rn.

00:10:21.560 --> 00:10:23.760
Let me do this particular
case.

00:10:23.760 --> 00:10:29.940
Let me define my line l to
be the set of all scalar

00:10:29.940 --> 00:10:33.290
multiples of the vector-- I
don't know, let's say the

00:10:33.290 --> 00:10:40.150
vector 2, 1, such that
c is any real number.

00:10:40.150 --> 00:10:46.000
Let me draw my axes here.

00:10:46.000 --> 00:10:48.150
That's my vertical axis.

00:10:48.150 --> 00:10:50.790
This is my horizontal
axis right there.

00:10:50.790 --> 00:10:52.770
And so my line is all the
scalar multiples of the

00:10:52.770 --> 00:10:53.990
vector 2 dot 1.

00:10:53.990 --> 00:10:56.250
And actually, let me just call
my vector 2 dot 1, let me call

00:10:56.250 --> 00:10:58.130
that right there the vector v.

00:10:58.130 --> 00:10:58.950
Let me draw that.

00:10:58.950 --> 00:11:02.570
So I go 1, 2, go up 1.

00:11:02.570 --> 00:11:04.490
That right there
is my vector v.

00:11:07.370 --> 00:11:09.380
And the line is all of
the possible scalar

00:11:09.380 --> 00:11:10.970
multiples of that.

00:11:10.970 --> 00:11:12.010
So let me draw that.

00:11:12.010 --> 00:11:14.020
So all the possible scalar
multiples of that and you just

00:11:14.020 --> 00:11:18.010
keep going in that direction, or
you keep going backwards in

00:11:18.010 --> 00:11:20.040
that direction or anything
in between.

00:11:20.040 --> 00:11:22.990
That's what my line is, all
of the scalar multiples

00:11:22.990 --> 00:11:24.240
of my vector v.

00:11:28.900 --> 00:11:34.680
Now, let's say I have another
vector x, and let's say that x

00:11:34.680 --> 00:11:38.790
is equal to 2, 3.

00:11:38.790 --> 00:11:43.140
Let me draw x. x is 2, and
then you go, 1, 2, 3.

00:11:43.140 --> 00:11:46.540
So x will look like this.

00:11:46.540 --> 00:11:49.110
Vector x will look like that.

00:11:49.110 --> 00:11:51.640
Well, let me draw it a little
bit better than that.

00:11:51.640 --> 00:11:53.780
Vector x will look like that.

00:11:53.780 --> 00:11:55.250
That is vector x.

00:11:55.250 --> 00:11:56.560
But what we want to do
is figure out the

00:11:56.560 --> 00:11:58.420
projection of x onto l.

00:11:58.420 --> 00:12:00.290
We can use this definition
right here.

00:12:00.290 --> 00:12:01.810
So let me write it down.

00:12:01.810 --> 00:12:08.800
The projection of x onto
l is equal to what?

00:12:08.800 --> 00:12:11.850
It's equal to x dot v, right?

00:12:11.850 --> 00:12:14.810
Where v is the defining
vector for our line.

00:12:14.810 --> 00:12:24.750
So it's equal to x, which is
2, 3, dot v, which is 2, 1,

00:12:24.750 --> 00:12:27.800
all of that over v dot v.

00:12:27.800 --> 00:12:36.080
So all of that over 2, 1, dot
2, 1 times our original

00:12:36.080 --> 00:12:37.750
defining vector v.

00:12:37.750 --> 00:12:39.010
So what's our original
defining vector?

00:12:39.010 --> 00:12:41.190
It's this one right
here, 2, 1.

00:12:41.190 --> 00:12:44.410
So times the vector, 2, 1.

00:12:44.410 --> 00:12:46.910
And what does this equal?

00:12:46.910 --> 00:12:51.380
When you take these two dot of
each other, you have 2 times 2

00:12:51.380 --> 00:12:57.240
plus 3 times 1, so 4 plus
3, so you get 7.

00:12:57.240 --> 00:12:59.200
This all simplified to 7.

00:12:59.200 --> 00:13:03.490
And then this, you get 2
times 2 plus 1 times 1,

00:13:03.490 --> 00:13:06.410
so 4 plus 1 is 5.

00:13:06.410 --> 00:13:07.550
So you get 7/5.

00:13:07.550 --> 00:13:08.800
That will all simplified to 5.

00:13:08.800 --> 00:13:10.370
That was a very fast
simplification.

00:13:10.370 --> 00:13:13.270
You might have been daunted
by this strange-looking

00:13:13.270 --> 00:13:15.540
expression, but when you take
dot products, they actually

00:13:15.540 --> 00:13:18.260
tend to simplify very quickly.

00:13:18.260 --> 00:13:20.540
And then you just multiply
that times your defining

00:13:20.540 --> 00:13:22.230
vector for the line.

00:13:22.230 --> 00:13:26.620
So we're scaling it up
by a factor of 7/5.

00:13:26.620 --> 00:13:29.170
So multiply it times
the vector 2, 1,

00:13:29.170 --> 00:13:30.490
and what do you get?

00:13:30.490 --> 00:13:34.730
You get the vector-- let me
do it in a new color.

00:13:34.730 --> 00:13:39.860
You get the vector, 14/5
and the vector 7/5.

00:13:39.860 --> 00:13:41.840
And just so we can visualize
this or plot it a little

00:13:41.840 --> 00:13:43.380
better, let me write
it as decimals.

00:13:43.380 --> 00:13:48.310
14/5 is 2 and 4/5,
which is 2.8.

00:13:48.310 --> 00:13:53.310
And this is 1 and 2/5,
which is 1.4.

00:13:53.310 --> 00:13:56.800
And so the projection of x
onto l is 2.8 and 1.4.

00:13:56.800 --> 00:14:02.740
So 2.8 is right about there,
and I go 1.4 is right about

00:14:02.740 --> 00:14:05.210
there, so the vector is going
to be right about there.

00:14:05.210 --> 00:14:06.655
I haven't even drawn
this too precisely,

00:14:06.655 --> 00:14:07.660
but you get the idea.

00:14:07.660 --> 00:14:09.250
This is the projection.

00:14:09.250 --> 00:14:12.290
Our computation shows
us that this is the

00:14:12.290 --> 00:14:17.710
projection of x onto l.

00:14:17.710 --> 00:14:20.720
If we draw a perpendicular right
there, we see that it's

00:14:20.720 --> 00:14:25.300
consistent with our idea of
this being the shadow of x

00:14:25.300 --> 00:14:27.170
onto our line now.

00:14:27.170 --> 00:14:29.490
Well, now we actually can
calculate projections.

00:14:29.490 --> 00:14:31.110
In the next video, I'll actually
show you how to

00:14:31.110 --> 00:14:34.135
figure out a matrix
representation for this, which

00:14:34.135 --> 00:14:36.360
is essentially a
transformation.


Unit vectors

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:07.100
We covered the idea of vector
length many, many videos ago.

00:00:07.100 --> 00:00:09.780
And I realize that I forgot to
cover an important topic.

00:00:09.780 --> 00:00:12.580
This topic's going to be useful
when we do some types

00:00:12.580 --> 00:00:15.010
of transformation -- actually,
the projections that I'll do

00:00:15.010 --> 00:00:16.300
in the next video.

00:00:16.300 --> 00:00:17.850
The notion that I forgot
to do is the

00:00:17.850 --> 00:00:19.650
notion of a unit vector.

00:00:23.220 --> 00:00:32.176
And all this is, is a vector
that has a length of 1.

00:00:32.176 --> 00:00:35.830
Has length and we've
defined length.

00:00:35.830 --> 00:00:38.610
It has a length of 1.

00:00:38.610 --> 00:00:43.020
So if something is a unit
vector, let's say that u here

00:00:43.020 --> 00:00:48.040
is a unit vector, and
it's a member of Rn.

00:00:51.300 --> 00:00:54.560
Then that means that if we have
u, u looks like this, has

00:00:54.560 --> 00:00:58.920
n components, u2 all
the way to un.

00:00:58.920 --> 00:01:03.160
We know what the length
of this is, right?

00:01:03.160 --> 00:01:07.070
We know that the length of u,
sometimes called the norm of

00:01:07.070 --> 00:01:10.200
u, it's just equal to the square
root of the squared

00:01:10.200 --> 00:01:11.950
sums of all of its components.

00:01:11.950 --> 00:01:14.570
And if you think about it, this
is just an extension of

00:01:14.570 --> 00:01:16.830
the Pythagorean theorem,
to some degree.

00:01:16.830 --> 00:01:21.560
So it's u1 squared plus
u2 squared all

00:01:21.560 --> 00:01:25.250
the way to un squared.

00:01:25.250 --> 00:01:27.780
And it's the square
root of that.

00:01:27.780 --> 00:01:32.250
If this is a unit vector, if
this is a unit vector, so this

00:01:32.250 --> 00:01:40.780
is a unit vector, that implies
that the length of u will be

00:01:40.780 --> 00:01:42.310
equal to 1.

00:01:42.310 --> 00:01:45.430
And that doesn't matter in what
dimension space we are.

00:01:45.430 --> 00:01:47.670
This could be R100
this could be R2.

00:01:47.670 --> 00:01:50.710
For it to have a unit vector in
any of those spaces, their

00:01:50.710 --> 00:01:52.070
length is 1.

00:01:52.070 --> 00:01:53.810
The next obvious question
is, how do you

00:01:53.810 --> 00:01:56.710
construct a unit vector?

00:01:56.710 --> 00:02:00.300
Let's say that I have
some vector, v.

00:02:00.300 --> 00:02:02.030
And let's say it's not
a unit vector.

00:02:02.030 --> 00:02:06.050
It's v1 v2 all the way to vn.

00:02:06.050 --> 00:02:10.360
And I want to turn it into some
vector u that is a unit

00:02:10.360 --> 00:02:13.020
vector, that just goes in
the same direction.

00:02:13.020 --> 00:02:24.520
So u will go in the same
direction, as v, but the

00:02:24.520 --> 00:02:28.130
length of u is going
to be equal to 1.

00:02:28.130 --> 00:02:31.710
How do I construct this
vector u here?

00:02:31.710 --> 00:02:35.450
What I could do is, I could
take the length of v.

00:02:35.450 --> 00:02:37.680
I could find out what the length
of v is, and we know

00:02:37.680 --> 00:02:38.270
how to do that.

00:02:38.270 --> 00:02:42.000
We just apply this definition
of vector length.

00:02:42.000 --> 00:02:44.130
And what happens if I figure out
the length of v, and then

00:02:44.130 --> 00:02:47.610
I multiply the vector
v times that?

00:02:47.610 --> 00:02:53.720
What if I make my u, what if I
say u is equal to, 1 over the

00:02:53.720 --> 00:03:00.210
length of v times v itself?

00:03:00.210 --> 00:03:02.160
What happens here?

00:03:02.160 --> 00:03:04.350
If I take the length
of this thing right

00:03:04.350 --> 00:03:10.970
here, what do I get?

00:03:10.970 --> 00:03:18.580
The length of u is equal to
the length of this scalar.

00:03:18.580 --> 00:03:20.740
Remember this is just
some number, right?

00:03:20.740 --> 00:03:24.070
It's equal to this scalar,
and I'm assuming v

00:03:24.070 --> 00:03:27.210
is a non-zero vector.

00:03:27.210 --> 00:03:33.130
The length of whatever this
scalar number is times v.

00:03:33.130 --> 00:03:35.870
And we know that we can take
this scalar out of the

00:03:35.870 --> 00:03:37.860
formula, we can show that --
I think I've shown it in a

00:03:37.860 --> 00:03:41.730
previous video -- that the
length of c times v is equal

00:03:41.730 --> 00:03:43.300
to c times the length of v.

00:03:43.300 --> 00:03:44.630
Let me write that down.

00:03:44.630 --> 00:03:46.520
And that's essentially what
I'm assuming right here.

00:03:46.520 --> 00:03:50.030
That if I take the length of c
times some vector, v, that is

00:03:50.030 --> 00:03:52.990
equal to c times the
length of v.

00:03:52.990 --> 00:03:55.970
I think we showed this when we
first were introduced to the

00:03:55.970 --> 00:03:56.940
idea of length.

00:03:56.940 --> 00:04:01.390
So we know that this is going
to be equal to 1 over the

00:04:01.390 --> 00:04:05.360
length of vector v -- that's my
c -- times, so this thing

00:04:05.360 --> 00:04:11.270
right here is that thing right
there, times this thing, times

00:04:11.270 --> 00:04:13.710
the length of vector v.

00:04:13.710 --> 00:04:15.000
Well what's this going
to be equal to?

00:04:15.000 --> 00:04:16.800
1 over something times
that something.

00:04:16.800 --> 00:04:19.360
Well this is just going
to be equal to 1.

00:04:19.360 --> 00:04:20.790
So that's all a unit
vector is.

00:04:20.790 --> 00:04:23.180
If you want to find a unit
vector -- or sometimes it's

00:04:23.180 --> 00:04:32.150
called a normalized vector
-- that goes in the same

00:04:32.150 --> 00:04:34.600
direction as some vector v, you
just figure out the length

00:04:34.600 --> 00:04:38.600
of v using the definition
of vector length in Rn.

00:04:38.600 --> 00:04:42.370
And then multiply 1 over that
length times the vector v --

00:04:42.370 --> 00:04:44.220
and this is just a scalar
-- and then you

00:04:44.220 --> 00:04:45.800
get your vector u.

00:04:45.800 --> 00:04:48.860
Let me to do an example, just to
make sure you get the idea.

00:04:55.860 --> 00:05:00.970
Let's say I have some vector
v and it's in R3.

00:05:00.970 --> 00:05:05.380
Let's say it's 1, 2, minus 1.

00:05:05.380 --> 00:05:07.900
What is the length of v?

00:05:07.900 --> 00:05:14.000
The length of v is equal to the
square root of 1 squared

00:05:14.000 --> 00:05:20.490
plus 2 squared plus minus 1
squared, and that is equal to

00:05:20.490 --> 00:05:27.070
the square root of 1 plus 1 plus
4 -- square root of 6.

00:05:27.070 --> 00:05:29.540
So that is the length of v.

00:05:29.540 --> 00:05:32.800
If I want to construct a
normalized vector u that goes

00:05:32.800 --> 00:05:37.130
in the same direction as v, I
can just define my vector u as

00:05:37.130 --> 00:05:41.160
being equal to 1 over the length
of v -- 1 over the

00:05:41.160 --> 00:05:44.130
square root of 6 -- times v.

00:05:44.130 --> 00:05:47.460
So times 1, 2, minus 1.

00:05:47.460 --> 00:05:53.020
Which is equal to 1 over square
root of 6, 2 over

00:05:53.020 --> 00:05:57.620
square root of 6, and minus 1
over the square root of 6.

00:05:57.620 --> 00:06:00.970
And I'll leave it for you to
verify that the length of u is

00:06:00.970 --> 00:06:02.570
going to be equal to 1.

00:06:02.570 --> 00:06:04.420
I'll just throw out one
other idea here that

00:06:04.420 --> 00:06:05.620
you'll often see.

00:06:05.620 --> 00:06:08.270
When something is a unit vector,
instead of using this

00:06:08.270 --> 00:06:11.280
little arrow on top of the
vector, they'll often write a

00:06:11.280 --> 00:06:14.210
unit vector with a little hat
on top of it, like that.

00:06:14.210 --> 00:06:18.010
That signifies that we're
dealing with a unit vector.

00:06:18.010 --> 00:06:20.990
And for those of you've taken
your vector calculus, or have

00:06:20.990 --> 00:06:22.490
done a little bit of
engineering, you're probably

00:06:22.490 --> 00:06:28.100
familiar with the vectors
i, j, and k.

00:06:28.100 --> 00:06:30.450
And the reason why they have
this little hat here is

00:06:30.450 --> 00:06:34.060
because these are all
unit vectors in R3.

00:06:34.060 --> 00:06:37.480
They're members of R3 and
they're all unit vectors.

00:06:37.480 --> 00:06:39.720
These are actually the
basis vectors in R3.

00:06:39.720 --> 00:06:41.410
And for those of you who
have been watching my

00:06:41.410 --> 00:06:44.940
transformation videos, these are
equivalent to the vectors

00:06:44.940 --> 00:06:46.850
e1 -- which I could write
with a hat on it,

00:06:46.850 --> 00:06:49.320
really -- e2, and e3.

00:06:49.320 --> 00:06:52.570
Which are the standard
basis vectors in R3.

00:06:52.570 --> 00:06:54.930
Now that you've been exposed to
it, now I can start to use

00:06:54.930 --> 00:06:58.150
the idea of a unit vector
in future videos.