Determining whether a transformation is onto

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:04.000
Let's say I have a linear
transformation T that's a

00:00:04.000 --> 00:00:10.490
mapping between Rn and Rm.

00:00:10.490 --> 00:00:13.640
We know that we can represent
this linear transformation as

00:00:13.640 --> 00:00:14.870
a matrix product.

00:00:14.870 --> 00:00:18.920
So we can say that T of x, so
the transformation T-- let me

00:00:18.920 --> 00:00:21.360
write it a little higher to save
space-- so we can write

00:00:21.360 --> 00:00:25.500
that the transformation T
applied to some vector x is

00:00:25.500 --> 00:00:30.440
equal to some matrix times x.

00:00:30.440 --> 00:00:35.640
And this matrix, since it's a
mapping from Rn to Rm, this is

00:00:35.640 --> 00:00:40.470
going to be a m by n matrix.

00:00:40.470 --> 00:00:43.270
Because each of these entries
are going have n components,

00:00:43.270 --> 00:00:46.790
they're going to be members of
Rn, so this guy has to have n

00:00:46.790 --> 00:00:50.870
columns in order to be able
to have this matrix vector

00:00:50.870 --> 00:00:52.910
product well defined.

00:00:52.910 --> 00:00:55.260
So we can go back to what we've
been talking about now.

00:00:55.260 --> 00:00:57.130
In the last couple of videos,
we've have been talking about

00:00:57.130 --> 00:01:00.130
invertibility of functions,
and we can just as easily

00:01:00.130 --> 00:01:01.850
apply it to this transformation,
because

00:01:01.850 --> 00:01:04.190
transformations are
just functions.

00:01:04.190 --> 00:01:07.060
We just use the word
transformations when we start

00:01:07.060 --> 00:01:11.120
talking about maps between the
vector spaces or between sets

00:01:11.120 --> 00:01:13.220
of vectors, but they are
essentially the same thing.

00:01:13.220 --> 00:01:14.630
Everything we've done
the last two videos

00:01:14.630 --> 00:01:15.530
have been very general.

00:01:15.530 --> 00:01:19.660
I never told you what the set
of our domain and the set of

00:01:19.660 --> 00:01:21.540
our co-domains were
made up of.

00:01:21.540 --> 00:01:24.240
Now we're dealing with vectors,
so we can apply the

00:01:24.240 --> 00:01:25.640
same ideas.

00:01:25.640 --> 00:01:32.220
So T is a mapping right
here from Rn to Rm.

00:01:34.830 --> 00:01:40.470
So if you take some vector here,
x, T will map it to some

00:01:40.470 --> 00:01:42.870
other vector in Rm.

00:01:42.870 --> 00:01:43.940
Call that Ax.

00:01:43.940 --> 00:01:46.550
If we take this matrix vector
product right there and this

00:01:46.550 --> 00:01:49.450
is the mapping T right there.

00:01:49.450 --> 00:01:51.840
So let's ask the same questions
about T that we've

00:01:51.840 --> 00:01:54.000
been asking in general
about functions.

00:01:54.000 --> 00:01:55.250
Is T invertible?

00:02:03.830 --> 00:02:06.390
And we learned in the last
video that there's two

00:02:06.390 --> 00:02:08.320
conditions for invertibility.

00:02:08.320 --> 00:02:16.430
T has to be onto, or the other
way, the other word was

00:02:16.430 --> 00:02:17.680
surjective.

00:02:20.710 --> 00:02:22.720
That's one condition
for invertibility.

00:02:22.720 --> 00:02:24.950
And then T also has
to be 1 to 1.

00:02:31.340 --> 00:02:38.390
And the fancy word for that was
injective, right there.

00:02:38.390 --> 00:02:40.850
So in this video, I'm going to
just focus on this first one.

00:02:40.850 --> 00:02:43.400
So I'm not going to prove to you
whether T is invertibile.

00:02:43.400 --> 00:02:46.820
We will at least be able to try
to figure out whether T is

00:02:46.820 --> 00:02:49.480
onto, or whether it's
surjective.

00:02:49.480 --> 00:02:54.590
So just as a reminder, what does
onto or surjective mean?

00:02:54.590 --> 00:02:57.840
It means that you take any
element in Rm, you take any

00:02:57.840 --> 00:03:01.420
element in your co-domain, so
you give me any element in

00:03:01.420 --> 00:03:04.870
your co-domain-- let's call that
element b; it's going to

00:03:04.870 --> 00:03:10.930
be a vector-- the fact, if T is
surjective or if T is onto,

00:03:10.930 --> 00:03:14.710
that means that any b that you
pick in our co-domain, there's

00:03:14.710 --> 00:03:18.510
always going to be some vector,
at least one vector,

00:03:18.510 --> 00:03:19.570
in our domain.

00:03:19.570 --> 00:03:21.410
That if you apply the
transformation to it you're

00:03:21.410 --> 00:03:22.700
going to get b.

00:03:22.700 --> 00:03:27.070
Or another way to think about
it is the image of our

00:03:27.070 --> 00:03:29.610
transformation is all of Rm.

00:03:29.610 --> 00:03:32.040
All of these guys
can be reached.

00:03:32.040 --> 00:03:34.230
So let's think about
what that means.

00:03:34.230 --> 00:03:36.820
So we know that the
transformation is Ax.

00:03:36.820 --> 00:03:38.820
It's some matrix A.

00:03:38.820 --> 00:03:41.410
So the transformation of x--
I'll just rewrite it-- is

00:03:41.410 --> 00:03:44.970
equal to some matrix A-- that's
an m by n matrix--

00:03:44.970 --> 00:03:48.660
times the vector x.

00:03:48.660 --> 00:03:59.530
Now, if T has to be onto, that
means that Ax, this matrix

00:03:59.530 --> 00:04:06.680
vector product, has to be equal
to any member of our

00:04:06.680 --> 00:04:10.140
co-domain can be reached by
multiplying our matrix A by

00:04:10.140 --> 00:04:12.270
some member of our domain.

00:04:12.270 --> 00:04:13.760
So what's another way of
thinking about it?

00:04:13.760 --> 00:04:19.920
Another way of thinking about
it is for any b, so onto,

00:04:19.920 --> 00:04:27.930
implies that for any vector b
that is a member of Rm-- so

00:04:27.930 --> 00:04:40.372
any b here-- there exists at
least one solution to a times

00:04:40.372 --> 00:04:43.490
x is equal to b.

00:04:43.490 --> 00:04:52.600
Where, of course, x is, the
vector x is, a member of Rn.

00:04:52.600 --> 00:04:54.820
It's just another way of saying
exactly what I've been

00:04:54.820 --> 00:04:56.540
saying in the first part
of this video.

00:04:56.540 --> 00:05:01.160
You give me any b in this set,
and then there has to be, if

00:05:01.160 --> 00:05:06.130
we assume that T is onto, or for
T to be onto, there has to

00:05:06.130 --> 00:05:09.570
be at least one solution
to Ax is equal to b.

00:05:09.570 --> 00:05:12.430
There has to be at least one x
out here, that if I multiply

00:05:12.430 --> 00:05:14.720
it by A, I get to my b.

00:05:14.720 --> 00:05:17.290
And this has to be true for
every, maybe I should write,

00:05:17.290 --> 00:05:19.300
for every instead of any.

00:05:19.300 --> 00:05:20.170
But they are the same idea.

00:05:20.170 --> 00:05:23.290
But, for every b in Rm, we have
to be able to find at

00:05:23.290 --> 00:05:25.900
least one x that makes
this true.

00:05:25.900 --> 00:05:26.890
So what does that mean?

00:05:26.890 --> 00:05:37.680
That means that A times x has
to be equal to-- you can

00:05:37.680 --> 00:05:45.670
construct any member of Rm by
taking a product of A and x,

00:05:45.670 --> 00:05:49.640
where x is a member of Rn,
where x is a member here.

00:05:49.640 --> 00:05:50.520
Now what is this?

00:05:50.520 --> 00:05:54.310
If x is an arbitrary member of
Rn, Let me write like this.

00:05:54.310 --> 00:05:56.190
We know that matrix A would
look like this.

00:05:56.190 --> 00:05:57.970
It'll be a bunch of
column vectors.

00:05:57.970 --> 00:06:03.300
a1, a2 and it has n columns,
so it looks like that.

00:06:03.300 --> 00:06:06.950
That's what our matrix
A looks like.

00:06:06.950 --> 00:06:10.000
So we're saying if you take this
product, you have to be

00:06:10.000 --> 00:06:13.360
able to reach any guy,
any member of Rm.

00:06:13.360 --> 00:06:15.710
Well, what does this product
look like if where taking this

00:06:15.710 --> 00:06:17.280
product instead of writing
an x there, I could

00:06:17.280 --> 00:06:19.070
write x like this.

00:06:19.070 --> 00:06:23.480
x1, x2, all the way to xn.

00:06:23.480 --> 00:06:27.770
So this product is going to be
x1 times the first column

00:06:27.770 --> 00:06:38.710
vector of A, plus x2 times the
second column vector of A, all

00:06:38.710 --> 00:06:44.340
the way to plus xn times the
nth column vector of A.

00:06:44.340 --> 00:06:45.900
That's what this product is.

00:06:45.900 --> 00:06:50.320
And in order for T to be onto,
this combination has to be

00:06:50.320 --> 00:06:53.040
equal to any vector in Rm.

00:06:53.040 --> 00:06:54.130
Well, what does this mean?

00:06:54.130 --> 00:06:56.430
These are just linear
combinations of the column

00:06:56.430 --> 00:06:57.680
vectors of A.

00:07:01.200 --> 00:07:08.730
So another way to say that is
for T to be onto, So for T to

00:07:08.730 --> 00:07:16.410
be surjective or onto, the
column vectors of A have to

00:07:16.410 --> 00:07:22.120
span Rm, have to span
our co-domain.

00:07:22.120 --> 00:07:23.940
They have to span
this right here.

00:07:23.940 --> 00:07:26.550
You have to be able to get any
vector here with a linear

00:07:26.550 --> 00:07:28.800
combination of these guys.

00:07:28.800 --> 00:07:29.420
Right?

00:07:29.420 --> 00:07:31.780
And the linear combination is
set up because the weights are

00:07:31.780 --> 00:07:35.230
just arbitrary members
of the real numbers.

00:07:35.230 --> 00:07:38.070
This vector is just a bunch
of arbitrary real numbers.

00:07:38.070 --> 00:07:53.390
So for T to be onto, the span of
a1, a2, all the way to an,

00:07:53.390 --> 00:07:59.270
has to be equal to Rm, has to
be equal to your co-domain.

00:07:59.270 --> 00:08:02.100
That just means that you can
achieve any vector in your

00:08:02.100 --> 00:08:04.190
co-domain with the linear
combinations of the column

00:08:04.190 --> 00:08:05.270
vectors of this.

00:08:05.270 --> 00:08:08.590
What's the span of a matrices'
column vectors?

00:08:08.590 --> 00:08:12.450
By definition that is the
matrices' column space.

00:08:12.450 --> 00:08:15.910
So we could say, that means that
the span of these guys

00:08:15.910 --> 00:08:18.725
have to be Rn, or another way
is that the column space of

00:08:18.725 --> 00:08:24.950
A-- let me switch colors-- the
column space of A has to be

00:08:24.950 --> 00:08:30.330
equal to Rm.

00:08:30.330 --> 00:08:33.220
So how do we know if
a vector's column

00:08:33.220 --> 00:08:37.059
space is equal to Rm?

00:08:37.059 --> 00:08:42.340
So here, maybe it might be
instructive to think about,

00:08:42.340 --> 00:08:46.090
when can we not find solutions
to the equation

00:08:46.090 --> 00:08:49.960
Ax is equal to b?

00:08:49.960 --> 00:08:52.710
So whenever you're faced
with this type of an

00:08:52.710 --> 00:08:54.270
equation, what do we do?

00:08:54.270 --> 00:08:57.680
We can set up an augmented
matrix that looks like this,

00:08:57.680 --> 00:09:00.340
where you put A on this side,
and then you put the vector b

00:09:00.340 --> 00:09:02.770
on the right-hand side.

00:09:02.770 --> 00:09:05.340
And then you, essentially,
perform a bunch of row

00:09:05.340 --> 00:09:05.870
operations.

00:09:05.870 --> 00:09:09.780
You have to perform the entire
rows in both cases, on both

00:09:09.780 --> 00:09:11.800
sides, and we've done
this multiple times.

00:09:11.800 --> 00:09:14.610
Your goal is to get the
left-hand side into reduced

00:09:14.610 --> 00:09:16.850
row echelon form.

00:09:16.850 --> 00:09:20.330
So what you want to do is,
eventually, get your augmented

00:09:20.330 --> 00:09:22.240
matrix to look like this.

00:09:22.240 --> 00:09:26.190
Where the left-handed side is--
let me define R, capital

00:09:26.190 --> 00:09:30.460
R, to be the reduced row
echelon form of A.

00:09:30.460 --> 00:09:31.820
And we've done many
videos on that.

00:09:31.820 --> 00:09:36.160
That's just, you have a matrix
where you have your pivots and

00:09:36.160 --> 00:09:40.680
the pivot will be the only
non-zero entry in its column.

00:09:40.680 --> 00:09:43.310
But not every column has
to have a pivot in it.

00:09:43.310 --> 00:09:48.140
Maybe, you have a free column
or non-pivot column and then

00:09:48.140 --> 00:09:49.570
they could have a
bunch of 0's.

00:09:49.570 --> 00:09:50.650
And maybe this has a pivot.

00:09:50.650 --> 00:09:53.000
This would have to be 0 if
there is a pivot there.

00:09:53.000 --> 00:09:56.000
These would have to be 0,
and so on and so forth.

00:09:56.000 --> 00:09:57.910
And maybe the next pivot
is right there.

00:09:57.910 --> 00:10:00.440
These would have to be 0,
and you get the idea.

00:10:00.440 --> 00:10:03.230
You could have some columns
the don't have pivots, but

00:10:03.230 --> 00:10:05.880
whenever you do have a pivot,
they have to be the only

00:10:05.880 --> 00:10:07.600
non-zero entry in
their column.

00:10:07.600 --> 00:10:09.776
This is reduced row
echelon form.

00:10:12.630 --> 00:10:15.900
So, what we do with any matrix
is we keep performing those

00:10:15.900 --> 00:10:18.100
row operations so that we
eventually get it into a

00:10:18.100 --> 00:10:20.160
reduced row echelon form.

00:10:20.160 --> 00:10:23.090
And as we do that we are
performing those same

00:10:23.090 --> 00:10:24.680
operations on the
right-hand side.

00:10:24.680 --> 00:10:26.415
We're performing on
the entire row of

00:10:26.415 --> 00:10:28.340
these augmented matrices.

00:10:28.340 --> 00:10:32.020
So this vector b right here, I
guess I could write it as a

00:10:32.020 --> 00:10:34.910
vector, it's eventually going
to be some other vector c

00:10:34.910 --> 00:10:36.680
right here.

00:10:36.680 --> 00:10:39.030
You know, if this is 1, 2, 3,
maybe I will perform a bunch

00:10:39.030 --> 00:10:41.260
of operations and this
will be 3, 2, 1, or

00:10:41.260 --> 00:10:42.630
something of that nature.

00:10:42.630 --> 00:10:52.316
Now, when does this not
have a solution?

00:10:55.840 --> 00:10:57.290
We reviewed this early on.

00:10:57.290 --> 00:11:00.070
The only time where you don't
have a solution, remember,

00:11:00.070 --> 00:11:05.950
there are three cases, where
you have many solutions.

00:11:09.040 --> 00:11:12.160
and that's the situation where
you have free variables.

00:11:12.160 --> 00:11:13.990
We've talked about
that before.

00:11:13.990 --> 00:11:20.030
You have the case where you have
only one unique solution,

00:11:20.030 --> 00:11:20.770
that's the other case.

00:11:20.770 --> 00:11:22.000
And then you have
the final case

00:11:22.000 --> 00:11:23.250
where you have no solutions.

00:11:26.660 --> 00:11:28.680
And when do you have
no solutions?

00:11:28.680 --> 00:11:31.360
What has to happen to
have no solution?

00:11:31.360 --> 00:11:34.010
To have no solutions, when you
perform these row operations,

00:11:34.010 --> 00:11:37.220
you have to eventually get to
a point where your matrix

00:11:37.220 --> 00:11:39.680
looks something like this.

00:11:39.680 --> 00:11:41.360
I don't know what all this stuff
looks like maybe there's

00:11:41.360 --> 00:11:43.670
a 1 here, a bunch of stuff.

00:11:43.670 --> 00:11:45.390
There is a 1 here and a 0.

00:11:45.390 --> 00:11:48.610
But if you have a whole row, at
least one whole row of 0's,

00:11:48.610 --> 00:11:51.130
you just have a bunch
of 0's like that.

00:11:51.130 --> 00:11:53.080
And over here you have something
that is non-zero.

00:11:58.590 --> 00:12:03.860
This is the only time that
you have no solution.

00:12:03.860 --> 00:12:06.810
So let's remember what
we're even talking

00:12:06.810 --> 00:12:07.700
about this stuff for.

00:12:07.700 --> 00:12:12.880
We are saying that our
transformation is onto, if its

00:12:12.880 --> 00:12:16.750
column vectors or if its column
space is Rm, if its

00:12:16.750 --> 00:12:19.160
column vectors span Rm.

00:12:19.160 --> 00:12:21.110
And what I'm trying to figure
out is how do I know

00:12:21.110 --> 00:12:23.080
that it spans Rm?

00:12:23.080 --> 00:12:27.270
So, essentially, for it to span
Rm, you can give me any b

00:12:27.270 --> 00:12:30.850
here, any b that's a member of
Rm, and I should be able to

00:12:30.850 --> 00:12:31.970
get a solution.

00:12:31.970 --> 00:12:33.820
So we asked ourselves the
question, when do

00:12:33.820 --> 00:12:35.070
we not get a solution?

00:12:38.070 --> 00:12:40.690
Well, we definitely don't get a
solution if we have a bunch

00:12:40.690 --> 00:12:41.980
of 0's in a row.

00:12:41.980 --> 00:12:43.560
And then we have something
non-zero here.

00:12:43.560 --> 00:12:46.300
That's definitely not going
to be a solution.

00:12:46.300 --> 00:12:51.300
Now there's the other case where
we have a bunch of 0's.

00:12:51.300 --> 00:12:53.200
So there's the other case
where we do have some

00:12:53.200 --> 00:12:57.920
solutions, where it's only
valid for certain b's.

00:12:57.920 --> 00:13:01.140
So that's the case-- let
me draw it like this.

00:13:01.140 --> 00:13:02.250
Let me start it like this.

00:13:02.250 --> 00:13:10.010
So let's say I have my matrix
A, and I have my b1, b2, let

00:13:10.010 --> 00:13:13.620
me write it all the way,
and then you have bm.

00:13:13.620 --> 00:13:15.960
Remember, this is
a member of Rm.

00:13:15.960 --> 00:13:19.050
And we do our reduced row
echelon form with this

00:13:19.050 --> 00:13:22.460
augmented matrix, and
A goes to its

00:13:22.460 --> 00:13:24.850
reduced row echelon form.

00:13:24.850 --> 00:13:27.950
Let's say it's reduced row
echelon form has a row of 0's

00:13:27.950 --> 00:13:29.600
at the end of it.

00:13:29.600 --> 00:13:31.530
So it just has a row
of 0's right there.

00:13:31.530 --> 00:13:33.420
Everything else looks
like your standard

00:13:33.420 --> 00:13:35.050
stuff, 1's and 0's.

00:13:35.050 --> 00:13:36.885
But the last row, let's say
it's a bunch of 0's.

00:13:36.885 --> 00:13:40.000
And when we perform the row
operations here on this

00:13:40.000 --> 00:13:44.600
generalized member of Rm, this
last row has some function.

00:13:44.600 --> 00:13:53.010
Maybe it just looks like 2b1
plus 3b2-- I'm just writing a

00:13:53.010 --> 00:13:56.770
particular case, it won't always
be this-- minus b3.

00:13:56.770 --> 00:13:59.090
And it will essentially
be some function

00:13:59.090 --> 00:14:01.660
of all of the b's.

00:14:01.660 --> 00:14:04.180
So let me write it this way.

00:14:04.180 --> 00:14:06.150
I'm writing a particular case
in here, maybe I shouldn't

00:14:06.150 --> 00:14:07.050
have written a particular
case.

00:14:07.050 --> 00:14:14.500
This will be some function of
b1, b2, all the way to bm.

00:14:14.500 --> 00:14:17.090
Now, clearly if this
is non-zero, we

00:14:17.090 --> 00:14:18.630
don't have a solution.

00:14:18.630 --> 00:14:23.170
And so if we don't have a
solution for some cases of b,

00:14:23.170 --> 00:14:26.410
then we are definitely
not spanning Rm.

00:14:26.410 --> 00:14:27.810
So, let me write that down.

00:14:27.810 --> 00:14:45.110
If we don't have a solution for
some cases of b, then we

00:14:45.110 --> 00:14:52.180
don't span Rm.

00:14:52.180 --> 00:14:54.840
I don't know if I'm overstating
something that is

00:14:54.840 --> 00:14:56.600
maybe obvious to you, but I
really want to make sure you

00:14:56.600 --> 00:14:57.950
understand this.

00:14:57.950 --> 00:15:01.680
Anytime you just want to solve
the equation Ax is equal to

00:15:01.680 --> 00:15:04.350
b-- and remember, we want to
make sure that this can be

00:15:04.350 --> 00:15:07.440
true for any b we chose-- what
we could do is we just set up

00:15:07.440 --> 00:15:09.510
this augmented matrix like this,
and we perform a bunch

00:15:09.510 --> 00:15:13.120
of row operations until we get
A, we get this matrix A to

00:15:13.120 --> 00:15:14.770
reduced row echelon form.

00:15:14.770 --> 00:15:17.460
As we do this, the right-hand
side is going to be a bunch of

00:15:17.460 --> 00:15:18.380
functions of b.

00:15:18.380 --> 00:15:21.310
So maybe the first
row is b1 minus

00:15:21.310 --> 00:15:23.600
b2, plus b4, or something.

00:15:23.600 --> 00:15:24.850
And then the next row will
be something like that.

00:15:24.850 --> 00:15:29.770
We've seen examples of this in
the past. And if you end up by

00:15:29.770 --> 00:15:33.670
doing the reduced row echelon
form with a row of 0's here,

00:15:33.670 --> 00:15:36.640
the only way that you're going
to have a solution is if your

00:15:36.640 --> 00:15:41.080
vector b, if its entries satisfy
this function here on

00:15:41.080 --> 00:15:43.800
the right, so that this
thing equals 0.

00:15:43.800 --> 00:15:48.160
So it's only going to be
true for certain b's.

00:15:48.160 --> 00:15:51.680
And if this only has a solution
for the certain b's

00:15:51.680 --> 00:15:55.280
that make this equal to 0, then
we definitely are not

00:15:55.280 --> 00:15:58.010
spanning all of Rm.

00:15:58.010 --> 00:16:00.660
Let me do it visually.

00:16:00.660 --> 00:16:06.030
So if that is Rm, and if we
put-- if this is only 0 for a

00:16:06.030 --> 00:16:10.870
couple of b's, for, let's say
for some handful of b's, then

00:16:10.870 --> 00:16:14.250
these are the only guys that we
can reach by multiplying A

00:16:14.250 --> 00:16:17.030
times some vector in Rm.

00:16:17.030 --> 00:16:19.840
And we definitely won't
be spanning all of Rm.

00:16:19.840 --> 00:16:22.650
In order to span all of Rm, when
we put this in a reduced

00:16:22.650 --> 00:16:25.550
row echelon form, we have to
always find a solution.

00:16:25.550 --> 00:16:28.180
And the only way we are always
going be finding a solution is

00:16:28.180 --> 00:16:30.430
if we don't run into this
condition where we

00:16:30.430 --> 00:16:31.780
have a row of 0's.

00:16:31.780 --> 00:16:33.460
Because when you have a row 0's,
then you have to put the

00:16:33.460 --> 00:16:36.190
constraint that this right-hand
side has to be 0.

00:16:36.190 --> 00:16:39.070
So what's the only reduced row
echelon form where you don't

00:16:39.070 --> 00:16:41.020
have a row of 0's at the end?

00:16:41.020 --> 00:16:44.560
Well, any row in reduced row
echelon form either has to

00:16:44.560 --> 00:16:49.590
have all 0's or it has to have
a pivot entry in every row.

00:16:49.590 --> 00:16:58.450
So the only way that you span--
so T is onto, if and

00:16:58.450 --> 00:17:06.170
only if, it's the column space
of its transformation vector

00:17:06.170 --> 00:17:07.800
is equal to Rm.

00:17:07.800 --> 00:17:10.859
Its column vectors
span all of Rm.

00:17:10.859 --> 00:17:13.930
And the only way that that's
happening is if the reduced

00:17:13.930 --> 00:17:33.970
row echelon form of A has a
pivot entry in every row.

00:17:33.970 --> 00:17:35.330
And how many rows
does it have?

00:17:35.330 --> 00:17:36.830
This is an m by n matrix.

00:17:36.830 --> 00:17:40.730
It has m rows and n columns.

00:17:40.730 --> 00:17:42.710
So it has a pivot entry
in every row.

00:17:42.710 --> 00:17:52.500
That means that it has to have
m pivot entries, right there.

00:17:52.500 --> 00:17:54.640
Now, what's another way of
thinking about that?

00:17:54.640 --> 00:17:58.240
Remember, earlier on, several
videos ago, we thought about

00:17:58.240 --> 00:18:01.270
how do you figure out-- and this
might confuse things a

00:18:01.270 --> 00:18:07.020
little bit-- how do you
figure out the basis

00:18:07.020 --> 00:18:10.480
for your column space?

00:18:10.480 --> 00:18:18.550
So the basis for the column
space of a matrix, and this is

00:18:18.550 --> 00:18:19.330
a bit of review.

00:18:19.330 --> 00:18:23.450
What we did is we say, look, you
take your matrix, and you

00:18:23.450 --> 00:18:30.980
put it in reduced row echelon
form of your matrix, and then,

00:18:30.980 --> 00:18:34.390
essentially-- let me draw it a
little bit different heree--

00:18:34.390 --> 00:18:36.160
well, you put it in reduced
row echelon form.

00:18:36.160 --> 00:18:38.360
So let's say that's just reduced
row echelon form.

00:18:38.360 --> 00:18:43.100
And you look at which columns
have pivot entries.

00:18:43.100 --> 00:18:46.360
And the corresponding columns in
your original matrices, in

00:18:46.360 --> 00:18:48.620
your original matrix,
forms the basis

00:18:48.620 --> 00:18:49.560
for your column space.

00:18:49.560 --> 00:18:51.640
So let me draw that out.

00:18:51.640 --> 00:18:53.400
So, I'll do a particular
instance.

00:18:53.400 --> 00:18:58.240
So, let's say that it has its
column vectors a1, a2,

00:18:58.240 --> 00:19:00.020
all the way to an.

00:19:00.020 --> 00:19:01.480
That's what A looks like.

00:19:01.480 --> 00:19:04.510
And when you put it in reduced
row echelon form, let's say

00:19:04.510 --> 00:19:09.370
that this column right here
has a pivot entry.

00:19:09.370 --> 00:19:10.370
That column has a pivot entry.

00:19:10.370 --> 00:19:11.420
Let's say that this
one doesn't.

00:19:11.420 --> 00:19:13.110
Let's say this is a 2 there.

00:19:13.110 --> 00:19:15.030
I'm just picking particular
numbers.

00:19:19.880 --> 00:19:21.300
Let's say there's a 3.

00:19:21.300 --> 00:19:23.990
Let's say all of these are
non-pivot entries, and then

00:19:23.990 --> 00:19:26.800
our last entry n is
a pivot entry.

00:19:26.800 --> 00:19:30.180
So it just has a bunch of 0's,
and then a 1 like that.

00:19:30.180 --> 00:19:32.590
How do you determine what are
the basis vectors for the

00:19:32.590 --> 00:19:33.800
column space?

00:19:33.800 --> 00:19:36.000
Well, obviously, the column
space is everything that's

00:19:36.000 --> 00:19:37.290
spanned by all these guys.

00:19:37.290 --> 00:19:38.840
But, what's the minimum
set you need to

00:19:38.840 --> 00:19:40.280
have that same span?

00:19:40.280 --> 00:19:42.610
Well, you look at which one
has a corresponding pivot

00:19:42.610 --> 00:19:44.230
entries or pivot columns.

00:19:44.230 --> 00:19:47.430
And you say, I have a pivot
column here, and I have a

00:19:47.430 --> 00:19:48.890
pivot column there.

00:19:48.890 --> 00:19:51.700
So the basis for my column space
must be this column in

00:19:51.700 --> 00:19:55.240
my original matrix, and that
column in my original matrix.

00:19:55.240 --> 00:20:01.380
And then we said, well, how do
you define the dimension of

00:20:01.380 --> 00:20:02.220
it's column space?

00:20:02.220 --> 00:20:05.510
And you just essentially count
the number of vectors you need

00:20:05.510 --> 00:20:10.340
for your basis, and we call
that the rank of A.

00:20:10.340 --> 00:20:11.180
This is all review.

00:20:11.180 --> 00:20:17.510
The rank of A was equal to the
dimension of the column space

00:20:17.510 --> 00:20:27.400
of A, which is equal to number
of basis vectors for the

00:20:27.400 --> 00:20:29.140
column space.

00:20:29.140 --> 00:20:30.330
And this is how you
determine it.

00:20:30.330 --> 00:20:33.340
You essentially figure out how
many pivot columns you have,

00:20:33.340 --> 00:20:35.290
the number of pivot columns
you have is the number of

00:20:35.290 --> 00:20:37.510
basis vectors you have,
and so that's going to

00:20:37.510 --> 00:20:38.840
be the rank of A.

00:20:38.840 --> 00:20:40.870
Now, the whole reason why I
talked about this is we just

00:20:40.870 --> 00:20:49.070
said that our transformation T
is onto, if and only if, its

00:20:49.070 --> 00:20:52.850
column space is Rm, which is
the case if it has a pivot

00:20:52.850 --> 00:20:55.690
entry in every row in its
reduced row echelon form.

00:20:55.690 --> 00:21:01.490
Or, since it has m rows, it has
to have m pivot entries.

00:21:01.490 --> 00:21:05.100
So for every row, you have a
pivot entry, but every pivot

00:21:05.100 --> 00:21:08.050
entry corresponds to
a pivot column.

00:21:11.030 --> 00:21:15.360
So if you have m pivot entries,
you also have m pivot

00:21:15.360 --> 00:21:20.960
columns, which means that if you
were to do this exercise

00:21:20.960 --> 00:21:25.240
right here, you would have m
basis vectors for your column

00:21:25.240 --> 00:21:27.540
space, or that you would
have a rank of m.

00:21:30.820 --> 00:21:33.780
So this whole video was
just a big long way of

00:21:33.780 --> 00:21:35.150
saying that T is onto.

00:21:39.976 --> 00:21:45.940
And another way of saying that
is that if you have your

00:21:45.940 --> 00:21:50.030
domain here, which was Rm, and
you have your co-domain here,

00:21:50.030 --> 00:21:56.930
that is Rm, that every member
of Rm can be reached by T by

00:21:56.930 --> 00:21:59.720
some member of Rn.

00:21:59.720 --> 00:22:02.650
Any guy here, there's always at
least one guy here, that if

00:22:02.650 --> 00:22:04.400
you apply T to it, you
get right there.

00:22:04.400 --> 00:22:05.800
There might be more than one.

00:22:05.800 --> 00:22:07.560
We're not even talking
about 1 to 1 yet.

00:22:07.560 --> 00:22:20.840
So we say that T is onto, if and
only if, the rank of its

00:22:20.840 --> 00:22:26.620
transformation matrix,
A, is equal to m.

00:22:26.620 --> 00:22:31.140
So that was the big takeaway
of this video.

00:22:31.140 --> 00:22:33.080
Let's just actually do an
example, because sometimes

00:22:33.080 --> 00:22:36.070
when you do things really
abstract it seems a little bit

00:22:36.070 --> 00:22:39.180
confusing, when you see
something particular.

00:22:39.180 --> 00:22:42.960
Let me define some
transformation S.

00:22:42.960 --> 00:22:48.870
Let's say the transformation S
is a mapping from R2 to R3.

00:22:48.870 --> 00:22:52.620
And let's say that S applied to
some vector x is equal to

00:22:52.620 --> 00:23:00.020
the matrix 1, 2, 3, 4, 5,
6, times the vector x.

00:23:00.020 --> 00:23:03.100
So this is clearly
a 3 by 2 matrix.

00:23:03.100 --> 00:23:10.290
Let's see if S is onto.

00:23:10.290 --> 00:23:12.970
Well, based on what we just did,
we just have to go and

00:23:12.970 --> 00:23:14.870
put this guy in reduced
row echelon form.

00:23:14.870 --> 00:23:16.660
So let's do that.

00:23:16.660 --> 00:23:20.150
So if you put this guy into
reduced row echelon form, so

00:23:20.150 --> 00:23:25.760
let's keep 1, 2, 3, 4, 5, 6.

00:23:25.760 --> 00:23:29.640
Now, let's keep the first row
the same, that's 1, 2.

00:23:29.640 --> 00:23:33.470
Let's replace the second row
with the second row minus 3

00:23:33.470 --> 00:23:34.540
times the first row.

00:23:34.540 --> 00:23:36.400
Actually, let's replace it with
3 times the first row

00:23:36.400 --> 00:23:37.350
minus the second row.

00:23:37.350 --> 00:23:39.800
So, 3 times 1 minus 3 is 0.

00:23:39.800 --> 00:23:44.900
3 times 2 minus 4, that's
6 minus 4, is 2.

00:23:44.900 --> 00:23:48.780
Now let's replace the third
row with 5 times the first

00:23:48.780 --> 00:23:49.780
row, minus the third row.

00:23:49.780 --> 00:23:52.610
So 5 times 1 minus 5 is 0.

00:23:52.610 --> 00:23:56.850
5 times 2 is 10, minus 6 is 4.

00:23:56.850 --> 00:24:02.080
Now, let's see if we can
get a 1 right here.

00:24:02.080 --> 00:24:05.650
So I'm going to keep my
middle row the same.

00:24:05.650 --> 00:24:09.730
Or, actually, let's just divide
the middle row by 2, or

00:24:09.730 --> 00:24:11.923
multiply it by 1/2 So, you
get 0, 1 and then you

00:24:11.923 --> 00:24:14.910
have a 0, 4, 1, 2.

00:24:14.910 --> 00:24:18.170
And now let's try to make these
0, get it into reduced

00:24:18.170 --> 00:24:19.020
row echelon form.

00:24:19.020 --> 00:24:21.660
So I'm going to keep my middle
row the same, 0, 1.

00:24:21.660 --> 00:24:24.460
And let's replace the top row
with the top row minus 2 times

00:24:24.460 --> 00:24:25.250
the second row.

00:24:25.250 --> 00:24:27.490
So 1 minus 2 times 0 is 1.

00:24:27.490 --> 00:24:30.170
2 minus 2, times 1 is 0.

00:24:30.170 --> 00:24:33.470
Now, let's replace this last row
with the last row minus 4

00:24:33.470 --> 00:24:35.500
times this row.

00:24:35.500 --> 00:24:38.080
So we get 0 minus 4,
times that, is 0.

00:24:38.080 --> 00:24:44.560
4 minus 4, times 1, is 0.

00:24:44.560 --> 00:24:49.840
So notice, we have a
row with 0's here.

00:24:49.840 --> 00:24:58.940
We have two pivot entries or two
rows with pivot entries,

00:24:58.940 --> 00:25:04.460
and we also have two pivot
columns right there.

00:25:04.460 --> 00:25:14.390
So the rank of this guy right
here, of 1, 2, 3, 4, 5, 6.

00:25:14.390 --> 00:25:18.290
The rank of that is equal
to 2, which is not

00:25:18.290 --> 00:25:20.090
equal to our co-domain.

00:25:20.090 --> 00:25:21.875
It is not equal to R3.

00:25:21.875 --> 00:25:33.180
It's not equal to R3, so S is
not onto, or not surjective.

00:25:33.180 --> 00:25:35.250
It's one of the two conditions
for invertibility.

00:25:35.250 --> 00:25:43.400
So, we definitely know that
S is not invertible.

00:25:43.400 --> 00:25:44.400
Hopefully that's helpful.

00:25:44.400 --> 00:25:46.420
Now, in the next video we're
going to focus on the second

00:25:46.420 --> 00:25:50.830
condition for invertibility,
and that's being 1 to 1.


Relating invertibility to being onto and one-to-one

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.660
A couple of videos ago, we
learned that a function that

00:00:03.660 --> 00:00:13.550
is a mapping from the set x to
the set y is invertible if and

00:00:13.550 --> 00:00:19.070
only if-- and I'll write that as
a if with two f's --if and

00:00:19.070 --> 00:00:24.160
only if for every y-- so
let me write this down.

00:00:24.160 --> 00:00:26.480
I'll do this in yellow.

00:00:26.480 --> 00:00:35.610
For every y that is a member of
our co-domain, there exists

00:00:35.610 --> 00:00:46.260
a unique-- and I'll make that a
little bit bold --a unique x

00:00:46.260 --> 00:00:56.730
that is a member of our domain,
such that f of this x

00:00:56.730 --> 00:00:58.580
is equal to this y.

00:00:58.580 --> 00:01:03.320
So that's just saying that if
I take my domain right here,

00:01:03.320 --> 00:01:09.410
that's x, and then I take a
co-domain here, that is y, we

00:01:09.410 --> 00:01:11.290
say that the function
f is invertible.

00:01:15.280 --> 00:01:16.740
And we know what invertibility
means.

00:01:16.740 --> 00:01:19.337
It means that there's this
other function called the

00:01:19.337 --> 00:01:22.410
inverse that can essentially,
if you take that in

00:01:22.410 --> 00:01:25.145
composition with f, it's like
taking the identity on x.

00:01:25.145 --> 00:01:27.360
Or if you take f in composition
with it, it's like

00:01:27.360 --> 00:01:28.850
taking the identity on y.

00:01:28.850 --> 00:01:30.130
And we've done that multiple
times so I

00:01:30.130 --> 00:01:31.330
won't repeat that there.

00:01:31.330 --> 00:01:34.360
We know what invertibility
means, but we learned that

00:01:34.360 --> 00:01:38.290
it's invertible, if and only if,
for every y here, so you

00:01:38.290 --> 00:01:42.320
take any y here, any y that's
a member of your co-domain,

00:01:42.320 --> 00:01:45.740
there exists a unique x,
such that f of x is

00:01:45.740 --> 00:01:46.990
equal to that y.

00:01:49.270 --> 00:01:51.908
Let me write it this way.

00:01:51.908 --> 00:01:56.370
If this is an x, let's say
that's an x not, f of x not

00:01:56.370 --> 00:01:58.020
would be equal to y.

00:01:58.020 --> 00:02:01.390
So this y would be equal
to f of x not.

00:02:01.390 --> 00:02:02.450
You apply the function here.

00:02:02.450 --> 00:02:04.860
It's going to map it
to this point here.

00:02:04.860 --> 00:02:07.670
It wouldn't be invertible
if you had this.

00:02:07.670 --> 00:02:11.870
If you had two members
of x mapping here.

00:02:11.870 --> 00:02:15.400
That would break invertibility
if you had this situation,

00:02:15.400 --> 00:02:18.270
because then you wouldn't have
the unique condition.

00:02:18.270 --> 00:02:20.980
You have to have a unique x
that maps to this thing.

00:02:20.980 --> 00:02:24.050
And what I just drew here, this
other pink mapping, we

00:02:24.050 --> 00:02:27.750
don't have a unique x that maps
to y, we have two x's

00:02:27.750 --> 00:02:29.040
that map to y.

00:02:29.040 --> 00:02:31.960
Now, based on what I just told
you on that last video, what

00:02:31.960 --> 00:02:33.342
does this mean?

00:02:33.342 --> 00:02:37.110
If we have a unique x
that maps to each y?

00:02:37.110 --> 00:02:39.910
That means that we have to have
a one-to-one mapping.

00:02:39.910 --> 00:02:42.200
That f has to be one-to-one.

00:02:42.200 --> 00:02:43.820
Let me write that.

00:02:43.820 --> 00:02:48.470
Another way of saying
this, is that f is

00:02:48.470 --> 00:02:52.220
one-to-one, or injective.

00:02:55.500 --> 00:02:59.210
If we have two guys mapping to
the same y, that would break

00:02:59.210 --> 00:03:00.030
down this condition.

00:03:00.030 --> 00:03:02.170
We wouldn't be one-to-one and
we couldn't say that there

00:03:02.170 --> 00:03:09.440
exists a unique x solution to
this equation right here.

00:03:09.440 --> 00:03:13.290
Now, the other part of this is
that for every y -- you could

00:03:13.290 --> 00:03:17.560
pick any y here and there
exists a unique x

00:03:17.560 --> 00:03:19.840
that maps to that.

00:03:19.840 --> 00:03:21.540
There cannot be some y here.

00:03:21.540 --> 00:03:24.640
Let's say that there's some y
here, and no one maps to that.

00:03:28.580 --> 00:03:33.810
If that's the case, then we
don't have our conditions for

00:03:33.810 --> 00:03:35.070
invertibility.

00:03:35.070 --> 00:03:39.380
So that would be
not invertible.

00:03:39.380 --> 00:03:43.550
Everything in y, every element
of y, has to be mapped to.

00:03:43.550 --> 00:03:45.510
All of these guys have
to be mapped to.

00:03:45.510 --> 00:03:49.110
And they can only be mapped to
by one of the elements of x.

00:03:49.110 --> 00:03:52.940
Everything here has to be mapped
to by a unique guy.

00:03:52.940 --> 00:03:56.100
Now, in the last video, what did
we call it went a function

00:03:56.100 --> 00:04:00.350
maps to every element
of your co-domain?

00:04:00.350 --> 00:04:03.940
So this every here, what is
another way of saying that?

00:04:03.940 --> 00:04:07.220
That a function maps to every
element of your co-domain?

00:04:07.220 --> 00:04:09.940
On the last video I explained
that notion is called a

00:04:09.940 --> 00:04:19.730
surjective or onto function.

00:04:19.730 --> 00:04:22.150
So the whole reason why I'm
doing this video is because I

00:04:22.150 --> 00:04:25.360
really just want to restate
this condition for

00:04:25.360 --> 00:04:30.280
invertibility using the
vocabulary that I introduced

00:04:30.280 --> 00:04:32.020
to you in the last video.

00:04:32.020 --> 00:04:36.800
So given that the statement for
every y that's a member of

00:04:36.800 --> 00:04:40.370
our co-domain, there exists
a x that maps to it.

00:04:40.370 --> 00:04:42.220
We could just say that
f is surjective.

00:04:49.510 --> 00:04:51.450
If we just say that f is
surjective, that means that

00:04:51.450 --> 00:04:53.070
everything here is mapped to.

00:04:53.070 --> 00:04:56.440
But it could be mapped to, maybe
this person right here,

00:04:56.440 --> 00:04:58.090
could be mapped to
by more than one.

00:04:58.090 --> 00:05:02.150
Surjective doesn't, by itself,
make the condition that

00:05:02.150 --> 00:05:04.830
there's a unique mapping
from a member of x to

00:05:04.830 --> 00:05:05.895
that element of y.

00:05:05.895 --> 00:05:09.130
So in order to get that, in
order to satisfy the unique

00:05:09.130 --> 00:05:12.200
condition of this condition for
invertibility, we have to

00:05:12.200 --> 00:05:15.665
say that f is also injective.

00:05:19.500 --> 00:05:22.130
And obviously, maybe the less
formal terms for either of

00:05:22.130 --> 00:05:25.950
these, you call this onto,
and you could call this

00:05:25.950 --> 00:05:28.700
one-to-one.

00:05:28.700 --> 00:05:31.280
So using the terminology that we
learned in the last video,

00:05:31.280 --> 00:05:34.340
we can restate this condition
for invertibility.

00:05:34.340 --> 00:05:38.290
We can say that a function that
is a mapping from the

00:05:38.290 --> 00:05:48.820
domain x to the co-domain y is
invertible, if and only if --

00:05:48.820 --> 00:06:03.320
I'll write it out -- f is both
surjective and injective.

00:06:06.230 --> 00:06:09.530
Or we could have said, that f is
invertible, if and only if,

00:06:09.530 --> 00:06:18.200
f is onto and one-to-one.

00:06:18.200 --> 00:06:21.110
And these are really just fancy
ways of saying for every

00:06:21.110 --> 00:06:26.290
y in our co-domain, there's a
unique x that f maps to it.

00:06:26.290 --> 00:06:30.850
There isn't more than one and
every y does get mapped to.


Surjective (onto) and injective (one-to-one) functions

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.080
In this video I want to
introduce you to some

00:00:03.080 --> 00:00:06.460
terminology that will be useful
in our discussion of

00:00:06.460 --> 00:00:08.140
functions and invertibility.

00:00:08.140 --> 00:00:09.970
And this is, in general,
terminology that you'll

00:00:09.970 --> 00:00:12.476
probably see in your
mathematical careers.

00:00:12.476 --> 00:00:17.160
So let's say I have a function
f, and it is a mapping from

00:00:17.160 --> 00:00:21.420
the set x to the set y.

00:00:21.420 --> 00:00:24.630
We've drawn this diagram many
times, but it never hurts to

00:00:24.630 --> 00:00:25.310
draw it again.

00:00:25.310 --> 00:00:29.490
So that is my set
x or my domain.

00:00:29.490 --> 00:00:33.750
And then this is the set y over
here, or the co-domain.

00:00:33.750 --> 00:00:37.370
Remember the co-domain is the
set that you're mapping to.

00:00:37.370 --> 00:00:39.430
You don't necessarily have to
map to every element of the

00:00:39.430 --> 00:00:40.940
set, or none of the elements
of the set.

00:00:40.940 --> 00:00:43.240
This is just all of the
elements, the set that you

00:00:43.240 --> 00:00:46.820
might map elements in
your co-domain to.

00:00:46.820 --> 00:00:48.670
So let's see.

00:00:48.670 --> 00:00:53.280
If I have some element there, f
will map it to some element

00:00:53.280 --> 00:00:54.970
in y in my co-domain.

00:00:57.700 --> 00:01:02.490
So the first idea, or term, I
want to introduce you to, is

00:01:02.490 --> 00:01:04.720
the idea of a function
being surjective.

00:01:09.280 --> 00:01:10.960
And sometimes this
is called onto.

00:01:14.340 --> 00:01:20.590
And a function is surjective or
onto, if for every element

00:01:20.590 --> 00:01:24.750
in your co-domain-- so let me
write it this way, if for

00:01:24.750 --> 00:01:34.560
every, let's say y, that is a
member of my co-domain, there

00:01:34.560 --> 00:01:39.250
exists-- that's the little
shorthand notation for exists

00:01:39.250 --> 00:01:48.510
--there exists at least
one x that's a

00:01:48.510 --> 00:01:51.090
member of x, such that.

00:01:51.090 --> 00:01:52.560
And I can write such
that, like that.

00:01:52.560 --> 00:01:54.570
Actually, let me just
write the word out.

00:01:54.570 --> 00:02:03.710
Such that f of x
is equal to y.

00:02:03.710 --> 00:02:07.320
So it's essentially saying, you
can pick any y here, and

00:02:07.320 --> 00:02:11.050
every y here is being mapped
to by at least one of

00:02:11.050 --> 00:02:12.790
the x's over here.

00:02:12.790 --> 00:02:15.660
So, for example, actually let
me draw a simpler example

00:02:15.660 --> 00:02:17.270
instead of drawing
these blurbs.

00:02:17.270 --> 00:02:20.130
Let's say that I have
a set y that

00:02:20.130 --> 00:02:21.990
literally looks like this.

00:02:21.990 --> 00:02:25.540
Let's say that a set y-- I'll
draw it very --and let's say

00:02:25.540 --> 00:02:27.170
it has four elements.

00:02:27.170 --> 00:02:32.660
It has the elements
a, b, c, and d.

00:02:32.660 --> 00:02:34.760
This is my set y right there.

00:02:34.760 --> 00:02:40.020
And let's say my set
x looks like that.

00:02:40.020 --> 00:02:47.190
And let's say it has the
elements 1, 2, 3, and 4.

00:02:47.190 --> 00:02:52.140
Now, in order for my function f
to be surjective or onto, it

00:02:52.140 --> 00:02:54.750
means that every one of these
guys have to be able

00:02:54.750 --> 00:02:56.550
to be mapped to.

00:02:56.550 --> 00:02:59.560
So what does that mean?

00:02:59.560 --> 00:03:00.880
If every one of these
guys, let me

00:03:00.880 --> 00:03:02.140
just draw some examples.

00:03:02.140 --> 00:03:05.510
Let's say that this
guy maps to that.

00:03:05.510 --> 00:03:07.980
Let's say that this
guy maps to that.

00:03:07.980 --> 00:03:10.500
Let's say that this
guy maps to that.

00:03:10.500 --> 00:03:13.270
And let's say, let me draw a
fifth one right here, let's

00:03:13.270 --> 00:03:17.440
say that both of these guys
right here map to d.

00:03:17.440 --> 00:03:20.650
So f of 4 is d and
f of 5 is d.

00:03:20.650 --> 00:03:23.110
This is an example of a
surjective function.

00:03:23.110 --> 00:03:27.270
So these are the mappings
of f right here.

00:03:27.270 --> 00:03:33.990
This function right here
is onto or surjective.

00:03:33.990 --> 00:03:34.740
Why is that?

00:03:34.740 --> 00:03:38.120
Because every element here
is being mapped to.

00:03:38.120 --> 00:03:40.580
Now, let me give you an example
of a function that is

00:03:40.580 --> 00:03:41.870
not surjective.

00:03:41.870 --> 00:03:44.000
Let me add some more
elements to y.

00:03:44.000 --> 00:03:48.690
Let's say element y has another
element here called e.

00:03:48.690 --> 00:03:50.980
Now, all of a sudden, this
is not surjective.

00:03:56.440 --> 00:03:57.360
And why is that?

00:03:57.360 --> 00:04:00.380
Because there's some element
in y that is not

00:04:00.380 --> 00:04:02.860
being mapped to.

00:04:02.860 --> 00:04:06.770
If I tell you that f is a
surjective function, it means

00:04:06.770 --> 00:04:10.400
if you take, essentially, if you
map all of these values,

00:04:10.400 --> 00:04:13.220
everything here is being mapped
to by at least one

00:04:13.220 --> 00:04:14.220
element here.

00:04:14.220 --> 00:04:16.839
So you could have it, everything
could be kind of a

00:04:16.839 --> 00:04:17.870
one-to-one mapping.

00:04:17.870 --> 00:04:20.190
And I'll define that a little
bit better in the future.

00:04:20.190 --> 00:04:23.630
So it could just be like
that, and like that.

00:04:23.630 --> 00:04:25.920
And you could even have, it's
at least one, so you could

00:04:25.920 --> 00:04:29.480
even have two things in here
mapping to one thing in here.

00:04:29.480 --> 00:04:32.230
But the main requirement
is that everything here

00:04:32.230 --> 00:04:34.090
does get mapped to.

00:04:34.090 --> 00:04:38.840
Another way to think about it,
is that if you take the image.

00:04:38.840 --> 00:04:41.600
So surjective function--
let me write this here.

00:04:47.480 --> 00:04:52.020
Let me write it this way --so if
I say that f is surjective

00:04:52.020 --> 00:04:57.100
or onto, these are equivalent
terms, that means that the

00:04:57.100 --> 00:04:59.560
image of f.

00:04:59.560 --> 00:05:03.660
Remember the image was, all
of the values that f

00:05:03.660 --> 00:05:05.360
actually maps to.

00:05:05.360 --> 00:05:08.125
So that means that the image
of f is equal to y.

00:05:11.930 --> 00:05:14.460
Now, we learned before, that
your image doesn't have to

00:05:14.460 --> 00:05:15.930
equal your co-domain.

00:05:15.930 --> 00:05:19.680
But if you have a surjective
or an onto function, your

00:05:19.680 --> 00:05:21.590
image is going to equal
your co-domain.

00:05:21.590 --> 00:05:23.990
Everything in your co-domain
gets mapped to.

00:05:23.990 --> 00:05:27.450
Actually, another word
for image is range.

00:05:27.450 --> 00:05:33.030
You could also say that your
range of f is equal to y.

00:05:33.030 --> 00:05:35.070
Remember the difference-- and
I drew this distinction when

00:05:35.070 --> 00:05:37.270
we first talked about functions
--the distinction

00:05:37.270 --> 00:05:40.560
between a co-domain and a range,
a co-domain is the set

00:05:40.560 --> 00:05:41.680
that you can map to.

00:05:41.680 --> 00:05:43.360
You don't have to map
to everything.

00:05:43.360 --> 00:05:49.510
The range is a subset of
your co-domain that you

00:05:49.510 --> 00:05:50.970
actually do map to.

00:05:50.970 --> 00:05:52.840
If you were to evaluate the
function at all of these

00:05:52.840 --> 00:05:56.130
points, the points that you
actually map to is your range.

00:05:56.130 --> 00:05:57.540
And that's also called
your image.

00:05:57.540 --> 00:06:00.340
And the word image
is used more in a

00:06:00.340 --> 00:06:02.440
linear algebra context.

00:06:02.440 --> 00:06:05.600
But if your image or your
range is equal to your

00:06:05.600 --> 00:06:08.140
co-domain, if everything in your
co-domain does get mapped

00:06:08.140 --> 00:06:10.730
to, then you're dealing
with a surjective

00:06:10.730 --> 00:06:13.230
function or an onto function.

00:06:13.230 --> 00:06:17.490
Now, the next term I want to
introduce you to is the idea

00:06:17.490 --> 00:06:18.905
of an injective function.

00:06:28.250 --> 00:06:34.820
And this is sometimes called
a one-to-one function.

00:06:34.820 --> 00:06:39.950
So let me draw my domain
and co-domain again.

00:06:39.950 --> 00:06:43.250
So let's say that that
is my domain

00:06:43.250 --> 00:06:45.930
and this is my co-domain.

00:06:45.930 --> 00:06:48.520
So this is x and this is y.

00:06:48.520 --> 00:06:55.960
If I say that f is injective
or one-to-one, that implies

00:06:55.960 --> 00:07:00.410
that for every value that is
mapped to-- so let me write it

00:07:00.410 --> 00:07:04.800
this way --for every value that
is mapped to-- so let's

00:07:04.800 --> 00:07:08.020
say, I'll say it a couple of
different ways --there is at

00:07:08.020 --> 00:07:10.320
most one x that maps to it.

00:07:15.030 --> 00:07:18.320
Or another way to say it is that
for any y that's a member

00:07:18.320 --> 00:07:24.550
of y-- let me write it this
way --for any y that is a

00:07:24.550 --> 00:07:31.850
member y, there is at most one--
let me write most in

00:07:31.850 --> 00:07:40.840
capital --at most one x, such
that f of x is equal to y.

00:07:40.840 --> 00:07:42.830
There might be no x's
that map to it.

00:07:42.830 --> 00:07:46.270
So for example, you could have
a little member of y right

00:07:46.270 --> 00:07:48.720
here that just never
gets mapped to.

00:07:48.720 --> 00:07:50.820
Everyone else in y gets mapped
to, but that guy

00:07:50.820 --> 00:07:51.740
never gets mapped to.

00:07:51.740 --> 00:07:54.020
So this would be a case
where we don't have

00:07:54.020 --> 00:07:55.660
a surjective function.

00:07:55.660 --> 00:07:59.640
This is not onto because this
guy, he's a member of the

00:07:59.640 --> 00:08:02.580
co-domain, but he's not
a member of the

00:08:02.580 --> 00:08:03.610
image or the range.

00:08:03.610 --> 00:08:05.140
He doesn't get mapped to.

00:08:05.140 --> 00:08:09.400
But this would still be an
injective function as long as

00:08:09.400 --> 00:08:12.830
every x gets mapped
to a unique y.

00:08:12.830 --> 00:08:16.250
Now, how can a function not be
injective or one-to-one?

00:08:16.250 --> 00:08:19.210
And I think you get the idea
when someone says one-to-one.

00:08:19.210 --> 00:08:24.680
Well, if two x's here get mapped
to the same y, or three

00:08:24.680 --> 00:08:28.110
get mapped to the same y, this
would mean that we're not

00:08:28.110 --> 00:08:36.254
dealing with an injective or
a one-to-one function.

00:08:40.169 --> 00:08:41.179
So that's all it means.

00:08:41.179 --> 00:08:42.750
Let me draw another
example here.

00:08:45.270 --> 00:08:47.400
Let's actually go back to
this example right here.

00:08:47.400 --> 00:08:50.080
When I added this e here, we
said this is not surjective

00:08:50.080 --> 00:08:52.640
anymore because every one
of these guys is not

00:08:52.640 --> 00:08:54.100
being mapped to.

00:08:54.100 --> 00:08:56.170
Is this an injective function?

00:08:56.170 --> 00:09:01.310
Well, no, because I have f of 5
and f of 4 both mapped to d.

00:09:01.310 --> 00:09:04.090
So this is what breaks its
one-to-one-ness or its

00:09:04.090 --> 00:09:05.490
injectiveness.

00:09:05.490 --> 00:09:07.910
This is what breaks it's
surjectiveness.

00:09:07.910 --> 00:09:10.840
Now if I wanted to make this a
surjective and an injective

00:09:10.840 --> 00:09:14.300
function, I would delete that
mapping and I would change f

00:09:14.300 --> 00:09:15.920
of 5 to be e.

00:09:15.920 --> 00:09:17.580
Now everything is one-to-one.

00:09:17.580 --> 00:09:21.890
I don't have the mapping from
two elements of x, going to

00:09:21.890 --> 00:09:23.750
the same element of y anymore.

00:09:23.750 --> 00:09:26.340
And everything in y now
gets mapped to.

00:09:26.340 --> 00:09:29.630
So this is both onto
and one-to-one.


Proof: Invertibility implies a unique solution to f(x)=y

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:04.450
I've got a function f and it's
a mapping from the set

00:00:04.450 --> 00:00:08.340
x to the set y.

00:00:08.340 --> 00:00:11.180
And let's just say for the sake
of argument, let's say

00:00:11.180 --> 00:00:12.430
that f is invertible.

00:00:17.760 --> 00:00:21.960
What I want to know is what
does this imply about this

00:00:21.960 --> 00:00:23.460
equation right here.

00:00:23.460 --> 00:00:27.320
The equation f of
x is equal to y.

00:00:27.320 --> 00:00:31.580
I want to know that for every
y that's a member of our

00:00:31.580 --> 00:00:32.500
co-domains.

00:00:32.500 --> 00:00:38.420
So for every y, let me write
this down, for every y that's

00:00:38.420 --> 00:00:46.830
a member of my co-domain is
there a unique, in all caps

00:00:46.830 --> 00:00:54.730
UNIQUE, solution x that's
a member of our domain.

00:00:54.730 --> 00:00:58.030
Such that, and I could write
such that well, I'll just

00:00:58.030 --> 00:00:58.990
write it out.

00:00:58.990 --> 00:01:01.140
I was going to write the mathy
way, but I think it's nicer to

00:01:01.140 --> 00:01:03.510
write in the actual
words sometimes.

00:01:03.510 --> 00:01:08.110
Such that f of x
is equal to y.

00:01:08.110 --> 00:01:09.810
So if we just, let me
just draw everything

00:01:09.810 --> 00:01:11.410
out a little bit.

00:01:11.410 --> 00:01:14.640
We have our set x right here.

00:01:14.640 --> 00:01:15.900
This is x.

00:01:15.900 --> 00:01:19.180
We have our co-domain y here.

00:01:19.180 --> 00:01:22.420
We know that f, if you take some
point here, let's call

00:01:22.420 --> 00:01:27.450
that a, it's a member of x and
you apply the function f to

00:01:27.450 --> 00:01:31.210
it, it will map you to some
element in set y.

00:01:31.210 --> 00:01:33.870
So that's f of a right there.

00:01:33.870 --> 00:01:37.550
This is, so far, what
this tells us.

00:01:37.550 --> 00:01:41.560
Now I want to look at this
equation here, and I want to

00:01:41.560 --> 00:01:46.280
know that if I can pick any y
in this set any lower-case y

00:01:46.280 --> 00:01:47.590
in this set y.

00:01:47.590 --> 00:01:51.490
So let's say I pick
something here.

00:01:51.490 --> 00:01:54.150
Let's say that's b.

00:01:54.150 --> 00:01:57.370
I want to know is there a
unique solution to the

00:01:57.370 --> 00:02:01.690
equation f of x is equal to b.

00:02:01.690 --> 00:02:06.920
Is there a unique solution?

00:02:06.920 --> 00:02:10.110
So one, I guess, you have to
think, is there a solution.

00:02:10.110 --> 00:02:13.600
So is there a solution to
saying, look is there some x

00:02:13.600 --> 00:02:17.210
here that if I apply the
transformation f to it that I

00:02:17.210 --> 00:02:21.140
get there, and I also want
to know is it unique.

00:02:21.140 --> 00:02:23.800
For example, if this is the only
one that is unique, but

00:02:23.800 --> 00:02:25.750
it's not unique if there's some
other guy, if there is

00:02:25.750 --> 00:02:26.690
more than one, solution.

00:02:26.690 --> 00:02:29.680
If there is some other guy
in x that if I apply the

00:02:29.680 --> 00:02:32.080
transformation I also go to b.

00:02:32.080 --> 00:02:35.795
This would make it non-unique,
not unique.

00:02:38.590 --> 00:02:41.000
So what I want to concern
ourselves with in this video

00:02:41.000 --> 00:02:45.780
is somehow, is invertibility
related to the idea of a

00:02:45.780 --> 00:02:50.500
unique solution to this for
any y in our co-domain.

00:02:50.500 --> 00:02:53.820
So let's just work through our
definitions of invertibility

00:02:53.820 --> 00:02:57.310
and see if we can get anywhere
constructive.

00:02:57.310 --> 00:03:03.940
So by definition, f is
invertible implies that there

00:03:03.940 --> 00:03:08.340
exists this little backward
looking, three-looking thing,

00:03:08.340 --> 00:03:09.580
this means there exists.

00:03:09.580 --> 00:03:12.670
I think it's nice to be exposed
sometimes to the mathy

00:03:12.670 --> 00:03:14.380
notations Let me just
write that.

00:03:18.140 --> 00:03:26.070
That means there exists some
function, let's call it f

00:03:26.070 --> 00:03:39.910
inverse, that's a mapping from
y to x, such that, and

00:03:39.910 --> 00:03:43.530
actually the colons are also the
shorthand for such that,

00:03:43.530 --> 00:03:44.780
but I'll write it out.

00:03:48.855 --> 00:03:57.090
Such that the composition of f
inverse with f is equal to the

00:03:57.090 --> 00:03:59.130
identity on x.

00:03:59.130 --> 00:04:01.910
So, essentially, it's saying
look, if I apply f to

00:04:01.910 --> 00:04:05.060
something in x then I apply f
inverse to that, I'm going to

00:04:05.060 --> 00:04:08.160
get back to that point which is,
essentially, equivalent.

00:04:08.160 --> 00:04:10.560
Or it isn't just essentially
equivalent, it is equivalent

00:04:10.560 --> 00:04:14.850
to just applying the identity
function, So that's i x.

00:04:14.850 --> 00:04:16.769
So you just get what
you put into it.

00:04:16.769 --> 00:04:19.722
Such that this inverse function,
the composition of

00:04:19.722 --> 00:04:21.320
the inverse with the function,
is equal to

00:04:21.320 --> 00:04:22.730
the identity function.

00:04:22.730 --> 00:04:27.650
And that the composition of the
function with the inverse

00:04:27.650 --> 00:04:30.540
function is equal to the
identity function on y.

00:04:30.540 --> 00:04:35.800
So if you started y and you
apply the inverse, then you

00:04:35.800 --> 00:04:38.260
apply the function to that,
you're going to end up back at

00:04:38.260 --> 00:04:39.490
y at that same point.

00:04:39.490 --> 00:04:41.880
And that's equivalent
to just applying

00:04:41.880 --> 00:04:44.350
the identity function.

00:04:44.350 --> 00:04:46.860
So this is what invertibility
tells me, this is how I

00:04:46.860 --> 00:04:49.990
defined invertibility
in the last video.

00:04:49.990 --> 00:04:53.400
Now we concerned ourselves-- so
we are concerned with this

00:04:53.400 --> 00:04:54.400
equation up here.

00:04:54.400 --> 00:04:56.650
We're concerned with the
equation, I will write it in

00:04:56.650 --> 00:05:01.920
pink, f of x is equal to y.

00:05:01.920 --> 00:05:05.930
And what we want to know for
any y, or any lower-case

00:05:05.930 --> 00:05:09.790
cursive y in our big set
y, is there a unique

00:05:09.790 --> 00:05:12.050
x solution to this.

00:05:12.050 --> 00:05:14.760
So, what we can do is we know
that f is invertible.

00:05:14.760 --> 00:05:16.600
I told you that from
the get-go.

00:05:16.600 --> 00:05:20.930
So given that f is invertible,
we know that there is this f

00:05:20.930 --> 00:05:26.240
inverse function, and I can
apply that f inverse function.

00:05:26.240 --> 00:05:29.880
It's a mapping from y to x.

00:05:29.880 --> 00:05:34.240
So, I can apply it to
any element in y.

00:05:34.240 --> 00:05:38.280
So for any y, let's say that
this is my y right there.

00:05:38.280 --> 00:05:41.670
So I can apply my f
inverse to that y.

00:05:41.670 --> 00:05:44.520
And I'm going to go over here
and, of course, y is

00:05:44.520 --> 00:05:47.260
equal to f of x.

00:05:47.260 --> 00:05:49.470
These are the exact
same points.

00:05:49.470 --> 00:05:52.560
So let's apply our f inverse
function to this.

00:05:52.560 --> 00:05:54.790
So if I apply the f inverse
function to both sides of the

00:05:54.790 --> 00:05:59.850
equation, this right here's an
element in y, and this is the

00:05:59.850 --> 00:06:00.850
same element in y.

00:06:00.850 --> 00:06:01.100
Right?

00:06:01.100 --> 00:06:02.570
They are the same element.

00:06:02.570 --> 00:06:06.630
Now, if I apply the mapping, the
inverse mapping, to both

00:06:06.630 --> 00:06:10.740
of that, that's going to take
me to some element in x.

00:06:10.740 --> 00:06:13.070
So let's do that.

00:06:13.070 --> 00:06:19.990
So, if I take the inverse
function on both sides of this

00:06:19.990 --> 00:06:24.150
equation, where some element
over here in y, and I'm taking

00:06:24.150 --> 00:06:28.650
the inverse function to get to
some element in x, what's this

00:06:28.650 --> 00:06:30.330
going to be equal to?

00:06:30.330 --> 00:06:32.940
Well, on the right-hand side,
we could just write the f

00:06:32.940 --> 00:06:34.440
inverse of y.

00:06:34.440 --> 00:06:36.850
That's going to be some
element over here.

00:06:36.850 --> 00:06:39.150
But what does the left-hand
side of this equation

00:06:39.150 --> 00:06:40.470
translate to?

00:06:40.470 --> 00:06:42.810
The definition of this inverse
function is that when you take

00:06:42.810 --> 00:06:45.520
the composition with f, you're
going to end up with the

00:06:45.520 --> 00:06:46.410
identity function.

00:06:46.410 --> 00:06:48.820
This is going to be equivalent
to-- let me write it this way.

00:06:48.820 --> 00:06:51.330
This is equal to the composition
of f inverse with

00:06:51.330 --> 00:06:56.790
f of x, which is equivalent to
the identity function being

00:06:56.790 --> 00:06:58.120
applied to x.

00:06:58.120 --> 00:07:01.090
And then the identity function
being applied to x is what?

00:07:01.090 --> 00:07:01.970
That's just x.

00:07:01.970 --> 00:07:05.310
This thing right here
just reduces to x.

00:07:05.310 --> 00:07:06.750
This reduces to x.

00:07:06.750 --> 00:07:10.040
So, we started with the idea
that f is invertible.

00:07:10.040 --> 00:07:12.340
We use the definition of
invertibility that there

00:07:12.340 --> 00:07:15.760
exists this inverse function
right there.

00:07:15.760 --> 00:07:18.140
And then we essentially apply
the inverse function to both

00:07:18.140 --> 00:07:21.590
sides of this equation and say,
look you give me any y,

00:07:21.590 --> 00:07:26.340
any lower-case cursive y in this
set y, and I will find

00:07:26.340 --> 00:07:28.100
you a unique x.

00:07:28.100 --> 00:07:33.660
This is the only x that
satisfies this equation.

00:07:33.660 --> 00:07:35.760
Remember how do I know
it's the only x?

00:07:35.760 --> 00:07:38.130
Because this is the only
possible inverse function.

00:07:42.690 --> 00:07:46.740
Only one inverse function
which is true.

00:07:46.740 --> 00:07:48.550
I proved that to you
in the last video.

00:07:48.550 --> 00:07:50.880
That if f is invertible,
it only has one

00:07:50.880 --> 00:07:52.050
unique inverse function.

00:07:52.050 --> 00:07:55.030
We tried before to have maybe
two inverse functions, but we

00:07:55.030 --> 00:07:57.140
saw they have to be
the same thing.

00:07:57.140 --> 00:07:59.590
So since we only have one
inverse function and it

00:07:59.590 --> 00:08:03.940
applies to anything in this big
upper-case set y, we know

00:08:03.940 --> 00:08:04.800
we have a solution.

00:08:04.800 --> 00:08:06.630
And because it's only one
inverse function, and

00:08:06.630 --> 00:08:10.980
functions only map to one value
in this case, then we

00:08:10.980 --> 00:08:13.470
know this is a unique
solution.

00:08:13.470 --> 00:08:14.680
So let's write this down.

00:08:14.680 --> 00:08:18.660
So we've established that if f
is invertible, I'll do this in

00:08:18.660 --> 00:08:35.409
orange, then the equation f of
x is equal to y for all.

00:08:35.409 --> 00:08:37.470
That little v that it looks
like it's filled up with

00:08:37.470 --> 00:08:44.660
something, for all y, the member
of our set y, has a

00:08:44.660 --> 00:08:45.910
unique solution.

00:08:56.160 --> 00:09:00.370
And that unique solution, if you
really care about it, is

00:09:00.370 --> 00:09:02.710
going to be the inverse
function applied to y.

00:09:02.710 --> 00:09:05.440
It might seem like a bit of a
no brainer, but you can see

00:09:05.440 --> 00:09:07.436
you have to be a little bit
precise about it in order to

00:09:07.436 --> 00:09:08.460
get to the point you want.

00:09:08.460 --> 00:09:10.740
Let's see if the opposite
is true.

00:09:10.740 --> 00:09:14.680
Let's see if we assume-- let's
see if we start from the

00:09:14.680 --> 00:09:23.010
assumption, that for all y that
is a member of our set Y,

00:09:23.010 --> 00:09:29.530
that the solution, that the
equation f of x, is equal to y

00:09:29.530 --> 00:09:36.490
has a unique solution.

00:09:36.490 --> 00:09:39.440
Let's assume this and see if it
can get us the other way.

00:09:39.440 --> 00:09:43.690
If given this, we can
prove invertibility.

00:09:43.690 --> 00:09:45.060
So let's think about
the first way.

00:09:45.060 --> 00:09:49.650
So we're saying that for any y--
let me draw my sets again.

00:09:52.350 --> 00:09:59.210
So this is my set X and this
is my set Y right there.

00:09:59.210 --> 00:10:01.440
Now we're working for the
assumption that you can pick

00:10:01.440 --> 00:10:06.670
any element in Y right here,
and then the equation right

00:10:06.670 --> 00:10:09.550
here has a unique solution.

00:10:09.550 --> 00:10:11.700
Let's call that unique
solution.

00:10:11.700 --> 00:10:13.220
Well, we could call
it whatever.

00:10:13.220 --> 00:10:14.610
But a unique solution x.

00:10:14.610 --> 00:10:17.130
So you can pick any point here,
and I've given you,

00:10:17.130 --> 00:10:21.810
we're assuming now, that, look,
you pick a point in Y, I

00:10:21.810 --> 00:10:24.880
can find you some point
in X such that f of

00:10:24.880 --> 00:10:26.360
x is equal to y.

00:10:26.360 --> 00:10:28.040
And not only can I find
that for you,

00:10:28.040 --> 00:10:30.580
that is a unique solution.

00:10:30.580 --> 00:10:34.390
So given that, let me define
a new function.

00:10:34.390 --> 00:10:37.450
Let me define the function s.

00:10:37.450 --> 00:10:40.890
The function s is a mapping
from y to x.

00:10:40.890 --> 00:10:47.740
It's a mapping from y to x and s
of, let's say s of y, where,

00:10:47.740 --> 00:10:52.420
of course, y is a member
of our set capital-Y.

00:10:52.420 --> 00:11:09.540
s of y is equal to the unique
solution in x to f of x is

00:11:09.540 --> 00:11:11.310
equal to y.

00:11:11.310 --> 00:11:13.060
Now, you're saying, hey, Sal, it
looks a little convoluted.

00:11:13.060 --> 00:11:14.010
But think about it.

00:11:14.010 --> 00:11:16.980
This is a completely valid
function definition.

00:11:16.980 --> 00:11:17.470
Right?

00:11:17.470 --> 00:11:20.520
We're starting with the idea
that you give me any y here.

00:11:20.520 --> 00:11:25.260
You give me any member of this
set, and I can always find you

00:11:25.260 --> 00:11:28.010
a unique solution to
this equation.

00:11:28.010 --> 00:11:31.450
Well OK, so that means that any
guy here can be associated

00:11:31.450 --> 00:11:34.280
with a unique solution in the
set X, where the unique

00:11:34.280 --> 00:11:35.610
solution is the unique
solution to

00:11:35.610 --> 00:11:36.920
this equation here.

00:11:36.920 --> 00:11:40.160
So, why don't I just define a
function that says, look I'm

00:11:40.160 --> 00:11:43.380
going to associate every member
y with its unique

00:11:43.380 --> 00:11:46.210
solution to f of x
is equal to y.

00:11:46.210 --> 00:11:48.760
That's how I'm defining this
function right here.

00:11:48.760 --> 00:11:50.430
And, of course, this is
a completely valid

00:11:50.430 --> 00:11:51.920
mapping from y to x.

00:11:51.920 --> 00:11:55.630
And we know that this only has
one legitimate value because

00:11:55.630 --> 00:11:59.830
this, any value y, any
lower-case value y, in this

00:11:59.830 --> 00:12:03.430
set has a unique solution
to f of x is equal to y.

00:12:03.430 --> 00:12:05.430
So this can only equal
one value.

00:12:05.430 --> 00:12:07.680
So it's well defined.

00:12:07.680 --> 00:12:15.670
So let's apply, let's take some
element here, let me do a

00:12:15.670 --> 00:12:18.050
good color, let's
say this is b.

00:12:18.050 --> 00:12:20.520
And b is a member of y.

00:12:20.520 --> 00:12:23.970
So let's find, so let's just map
it using our new function

00:12:23.970 --> 00:12:24.450
right here.

00:12:24.450 --> 00:12:29.650
So let's take it, and map it,
and this is s of b right here.

00:12:29.650 --> 00:12:35.390
s of b which is a member of x
Now, we know that s of b is a

00:12:35.390 --> 00:12:37.450
unique solution by definition.

00:12:37.450 --> 00:12:39.650
I know it seems a little
circular, but it's not.

00:12:39.650 --> 00:12:42.600
We know that s of
b is a solution.

00:12:42.600 --> 00:12:56.720
So we know that s of b is the
unique solution to f of x is

00:12:56.720 --> 00:12:58.530
equal to b.

00:12:58.530 --> 00:13:01.670
Well, if this is the case, if
this is good, we just got this

00:13:01.670 --> 00:13:03.640
because this is what
this function does.

00:13:03.640 --> 00:13:08.350
It maps every y to the unique
solution to this equation.

00:13:08.350 --> 00:13:10.910
Because we said that every
y has a unique solution.

00:13:10.910 --> 00:13:12.810
So, if this is the case,
then what happens if I

00:13:12.810 --> 00:13:17.150
take f of s of b.

00:13:17.150 --> 00:13:19.190
Well, I just said this is the
unique solution to this.

00:13:19.190 --> 00:13:23.250
So if I put this guy in here,
what am I going to get?

00:13:23.250 --> 00:13:25.410
I'm going to get b.

00:13:25.410 --> 00:13:31.810
Or other way of saying this is
that the composition of f with

00:13:31.810 --> 00:13:37.730
s applied to b is equal to b.

00:13:37.730 --> 00:13:41.440
Or another way to say it, is
that we take the composition

00:13:41.440 --> 00:13:46.290
of f with s, this is the same
thing, because if I apply s to

00:13:46.290 --> 00:13:49.090
b, and then I apply f back to
that, that's the composition,

00:13:49.090 --> 00:13:50.170
I just get back to b.

00:13:50.170 --> 00:13:51.100
That's what's happening here.

00:13:51.100 --> 00:13:55.370
So this is the same thing as
the identity function on y

00:13:55.370 --> 00:13:57.730
being applied to b.

00:13:57.730 --> 00:13:58.400
So it's equal to b.

00:13:58.400 --> 00:14:02.520
So we can say that the
composition-- we can say that

00:14:02.520 --> 00:14:06.670
there exists, and we know that
this function exists, or that

00:14:06.670 --> 00:14:08.700
we can always construct this.

00:14:08.700 --> 00:14:11.450
So we already know
that this exists.

00:14:11.450 --> 00:14:13.650
This existed by me constructing
it, but I've,

00:14:13.650 --> 00:14:15.960
hopefully, shown you that
this is well defined.

00:14:15.960 --> 00:14:20.100
That from our assumption that
this always has a unique

00:14:20.100 --> 00:14:23.710
solution in x for any y here, I
can define this in a fairly

00:14:23.710 --> 00:14:24.330
reasonable way.

00:14:24.330 --> 00:14:25.700
So it definitely exists.

00:14:25.700 --> 00:14:29.230
And not only does it exist,
but we know that the

00:14:29.230 --> 00:14:33.020
composition of f with this
function that I just

00:14:33.020 --> 00:14:42.710
constructed here is equal to
the identity function on y.

00:14:42.710 --> 00:14:46.460
Now let's do another
little experiment.

00:14:46.460 --> 00:14:50.240
Let's take a particular-- let
me just draw sets again.

00:14:52.850 --> 00:14:54.190
This is our set X.

00:14:54.190 --> 00:14:58.790
Let me take some member
of set X, call it a.

00:14:58.790 --> 00:15:02.880
Let me take my set
Y right there.

00:15:02.880 --> 00:15:05.350
And so we can apply the function
to a and we'll get a

00:15:05.350 --> 00:15:07.210
member of set Y.

00:15:07.210 --> 00:15:08.960
Let's call that right there.

00:15:08.960 --> 00:15:15.880
Let's call that f of
a right there.

00:15:15.880 --> 00:15:20.120
Now, if I apply my magic
function here that always, I

00:15:20.120 --> 00:15:23.170
can give you any member of set Y
and I'll give you the unique

00:15:23.170 --> 00:15:25.800
solution in X to
this equation.

00:15:25.800 --> 00:15:27.400
So, let me apply that to this.

00:15:27.400 --> 00:15:30.290
Let me apply s to this.

00:15:30.290 --> 00:15:33.440
So, if I apply s to this it'll
give me the unique solution.

00:15:33.440 --> 00:15:35.580
So, let me write this down.

00:15:35.580 --> 00:15:44.050
So if I apply s to this, I'm
going to apply s to this-- and

00:15:44.050 --> 00:15:46.050
maybe I shouldn't point it back
at that, I don't want to

00:15:46.050 --> 00:15:48.400
imply that it necessarily
points back at that.

00:15:48.400 --> 00:15:52.490
So, let me apply s
that, s to this.

00:15:52.490 --> 00:15:54.200
So what is this going
to point to?

00:15:54.200 --> 00:15:57.040
What is that point going
to be right there?

00:15:57.040 --> 00:16:03.790
So that's going to be s of this
point, which is f of a,

00:16:03.790 --> 00:16:06.840
which we know is the
unique solution.

00:16:06.840 --> 00:16:15.570
So, this is equal to the
unique solution to the

00:16:15.570 --> 00:16:23.670
equation f of x is equal
to this y right here.

00:16:23.670 --> 00:16:26.360
Or this y right here is
just called f of a.

00:16:26.360 --> 00:16:27.240
Right?

00:16:27.240 --> 00:16:31.820
Remember, the mapping s just
mapped you from any member of

00:16:31.820 --> 00:16:35.010
a to the unique solution
to the equation f of

00:16:35.010 --> 00:16:36.600
x is equal to that.

00:16:36.600 --> 00:16:41.650
So this is the mapping from f of
a to the unique, so this s

00:16:41.650 --> 00:16:44.340
of f of a is going to be a
mapping to the-- or this right

00:16:44.340 --> 00:16:46.360
here, is going to be the
unique solution to the

00:16:46.360 --> 00:16:49.880
equation f of x is equal
to this member of y.

00:16:49.880 --> 00:16:51.740
And what's the remember
y I called?

00:16:51.740 --> 00:16:55.760
It's called f of a.

00:16:55.760 --> 00:17:00.120
Well, you could go say this in
a very convoluted way, but if

00:17:00.120 --> 00:17:02.880
I were to just, before you
learned any linear algebra, if

00:17:02.880 --> 00:17:06.470
I said look if I have the
equation f of x is

00:17:06.470 --> 00:17:08.020
equal to f of a.

00:17:08.020 --> 00:17:11.160
What is the unique solution
to this equation?

00:17:11.160 --> 00:17:13.290
What does x equal?

00:17:13.290 --> 00:17:14.693
x would have to be equal to a.

00:17:17.510 --> 00:17:21.030
So the unique solution to the
equation f of x is equal to f

00:17:21.030 --> 00:17:23.810
of a is equal to a.

00:17:23.810 --> 00:17:31.590
And we know that there's only
one solution to that because

00:17:31.590 --> 00:17:34.170
that was one of our starting
assumptions.

00:17:34.170 --> 00:17:35.660
So this thing is equal to a.

00:17:35.660 --> 00:17:42.080
Or we could write s of
f of a is equal to a.

00:17:42.080 --> 00:17:47.640
Or that the composition of s
with f is equal or applied to

00:17:47.640 --> 00:17:50.350
a is equal to a.

00:17:50.350 --> 00:17:54.920
Or that the composition of s
with f is just the identity

00:17:54.920 --> 00:17:57.320
function on the set x.

00:17:57.320 --> 00:17:57.570
Right?

00:17:57.570 --> 00:18:00.390
This is a mapping right
here from x to x.

00:18:00.390 --> 00:18:04.630
So we could write that the
composition of s with f is the

00:18:04.630 --> 00:18:06.700
identity on x.

00:18:06.700 --> 00:18:09.480
So, what have we done so far?

00:18:09.480 --> 00:18:13.970
We started with the idea that
you pick any y in our set

00:18:13.970 --> 00:18:18.290
capital-Y here and we're going
to have a unique solution x

00:18:18.290 --> 00:18:19.550
such that this is true.

00:18:19.550 --> 00:18:21.490
Such that f of x
is equal to y.

00:18:21.490 --> 00:18:23.560
That's what the assumption
we started off with.

00:18:23.560 --> 00:18:27.080
We constructed this function
s that immediately maps any

00:18:27.080 --> 00:18:30.810
member here with its unique
solution to this equation.

00:18:30.810 --> 00:18:31.540
Fair enough.

00:18:31.540 --> 00:18:34.040
Now from that, we said this
definitely exists.

00:18:34.040 --> 00:18:35.850
Not only does it exist, but
we figured out that the

00:18:35.850 --> 00:18:39.310
composition of f with our
constructed function is equal

00:18:39.310 --> 00:18:42.530
to the identity on the set y.

00:18:42.530 --> 00:18:46.290
And then we also learned that
s-- the composition of s with

00:18:46.290 --> 00:18:50.110
f is the identity
function on x.

00:18:50.110 --> 00:18:51.180
Let me write this.

00:18:51.180 --> 00:18:52.940
So we learned this, and we
also learned that the

00:18:52.940 --> 00:18:58.380
composition f with s is equal
to the identity on y.

00:18:58.380 --> 00:19:01.040
And s clearly exists because I
constructed it, and we know

00:19:01.040 --> 00:19:05.010
it's well defined because every
y-- for every y here

00:19:05.010 --> 00:19:07.300
there is a solution to this.

00:19:07.300 --> 00:19:13.720
So given that I was able to find
for my function f, I was

00:19:13.720 --> 00:19:17.760
able to find a function that
these two things are true.

00:19:17.760 --> 00:19:19.580
This is by definition.

00:19:19.580 --> 00:19:20.830
What it means to
be invertible.

00:19:24.320 --> 00:19:26.840
Remember, so this means
that f is invertible.

00:19:26.840 --> 00:19:29.510
Remember f being invertible,
in order for f to be

00:19:29.510 --> 00:19:34.780
invertible, that means there
must exist some function from,

00:19:34.780 --> 00:19:39.250
so if f is a mapping from to x
to y, invertibility means that

00:19:39.250 --> 00:19:43.720
there must be some function f
inverse that is a mapping from

00:19:43.720 --> 00:19:51.500
y to x such that, so I can write
there exists a function,

00:19:51.500 --> 00:19:58.610
the inverse function composed
with our function should be

00:19:58.610 --> 00:20:03.240
equal to the identity on x.

00:20:03.240 --> 00:20:07.070
And the inverse and the function
in the composition of

00:20:07.070 --> 00:20:09.300
the function, with the inverse
function, should be the

00:20:09.300 --> 00:20:11.340
identity on y.

00:20:11.340 --> 00:20:14.370
Well, we just found
a function.

00:20:14.370 --> 00:20:16.530
It exists, and that
function is s.

00:20:16.530 --> 00:20:17.950
Where both of these
things are true.

00:20:17.950 --> 00:20:21.200
We can say that s is
equal to f inverse.

00:20:21.200 --> 00:20:23.910
So f is definitely invertible.

00:20:23.910 --> 00:20:25.380
So, hopefully, you found
this satisfying.

00:20:25.380 --> 00:20:29.220
This proof is very subtle and
very nuanced because we keep

00:20:29.220 --> 00:20:32.340
bouncing between our
sets X and Y.

00:20:32.340 --> 00:20:36.760
But what we've shown is that if
f, in the beginning part of

00:20:36.760 --> 00:20:40.290
this video, we show that if f
is invertible then there is

00:20:40.290 --> 00:20:45.450
for any y a unique solution to
the equation f of x equals y.

00:20:45.450 --> 00:20:47.560
And the second part of the
video, we showed that the

00:20:47.560 --> 00:20:54.650
other way is true, That if-- let
me put it this way, that

00:20:54.650 --> 00:21:04.840
if for all Y, a member of
capital Y, there is a unique

00:21:04.840 --> 00:21:15.870
solution to f of x is equal to
x, then f is invertible.

00:21:15.870 --> 00:21:20.260
So the fact that both of these
assumptions imply each other,

00:21:20.260 --> 00:21:24.120
we can write our final
conclusion of the video.

00:21:24.120 --> 00:21:29.540
That f being invertible, if f,
which is a mapping from x to y

00:21:29.540 --> 00:21:35.770
is invertible, this is true if
and only if, and we could

00:21:35.770 --> 00:21:38.540
write that either as
a two-way arrow.

00:21:38.540 --> 00:21:43.000
Or we could write if for if and
only if so both of these

00:21:43.000 --> 00:21:44.380
statements imply each other.

00:21:44.380 --> 00:21:49.900
If and only if for all y, for
every y that is a member of

00:21:49.900 --> 00:21:59.300
our set y there exists a
unique-- I can actually write

00:21:59.300 --> 00:22:07.720
that like that means there's
exists a unique x for the-- or

00:22:07.720 --> 00:22:17.970
let me write this way there
exists a unique solution to

00:22:17.970 --> 00:22:27.050
the equation f of
x is equal to y.

00:22:27.050 --> 00:22:29.880
So that was our big takeaway
in this video.

00:22:29.880 --> 00:22:32.440
That invertibility of a function
implies there's a

00:22:32.440 --> 00:22:36.630
unique solution to this equation
for any y that's in

00:22:36.630 --> 00:22:39.790
the co-domain of our function.