Preimage and kernel example

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:07.550
Let's say I have some
transformation from R2 to R2.

00:00:07.550 --> 00:00:11.280
And it's essentially just a
multiplication times a matrix.

00:00:11.280 --> 00:00:13.220
And we know that all linear
transformations can be

00:00:13.220 --> 00:00:16.260
expressed as a multiplication
of a matrix, but this one is

00:00:16.260 --> 00:00:25.940
equal to the matrix 1, 3, 2, 6
times whatever vector you give

00:00:25.940 --> 00:00:27.170
me in my domain.

00:00:27.170 --> 00:00:30.590
Times x1, x2.

00:00:30.590 --> 00:00:35.030
Now let's say I have some
subset in my codomain.

00:00:35.030 --> 00:00:36.550
So let me draw this
right here.

00:00:36.550 --> 00:00:39.350
So my domain looks like that.

00:00:39.350 --> 00:00:41.310
It's R2.

00:00:41.310 --> 00:00:44.840
And of course my function or
my transformation maps

00:00:44.840 --> 00:00:48.340
elements of R2 into elements
of its codomain which also

00:00:48.340 --> 00:00:50.100
happens to be R2.

00:00:50.100 --> 00:00:52.310
I could show it mapping into
itself, but for the sake of

00:00:52.310 --> 00:00:55.150
simplicity let's draw
my codomain here.

00:00:55.150 --> 00:00:57.590
And our transformation of course
maps, for any element

00:00:57.590 --> 00:01:01.215
here, the transformation of that
will be an association or

00:01:01.215 --> 00:01:03.560
a mapping into R2.

00:01:03.560 --> 00:01:10.970
Now what if we take some subset
of R2, and let's just

00:01:10.970 --> 00:01:15.650
say it's a set of two vectors,
the zero vector in R2, and the

00:01:15.650 --> 00:01:18.800
vector 1, 2.

00:01:18.800 --> 00:01:20.840
So it's literally, let's
say it's this point.

00:01:20.840 --> 00:01:22.070
Let me do it in a
different color.

00:01:22.070 --> 00:01:26.110
Let's say this is my
zero vector in R2.

00:01:26.110 --> 00:01:26.920
I'm not plotting them.

00:01:26.920 --> 00:01:30.350
I'm just showing that they're
in this, they're in R2.

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That's my zero vector,
and let's say the

00:01:32.430 --> 00:01:35.170
vector 1,2 is here.

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1,2.

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What I want to know is, what are
all of the vectors in my

00:01:40.050 --> 00:01:45.570
domain whose transformations
map to this subset?

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Map to this point?

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So I want to know, I'm
essentially, I want to know

00:01:49.780 --> 00:01:51.840
the preimage of S.

00:01:51.840 --> 00:01:54.730
So the preimage of S, let
me be careful, the

00:01:54.730 --> 00:02:04.450
preimage of S under T.

00:02:04.450 --> 00:02:07.160
And I said be careful because
when you say preimage of

00:02:07.160 --> 00:02:10.360
something without saying under
something else, it implies

00:02:10.360 --> 00:02:10.930
that you're taking.

00:02:10.930 --> 00:02:13.580
Well when you say image of
something, it implies you're

00:02:13.580 --> 00:02:15.520
taking the image of an entire
transformation like I showed

00:02:15.520 --> 00:02:17.370
you, I think, two videos ago.

00:02:17.370 --> 00:02:19.720
But when you're taking the image
or preimage of a set,

00:02:19.720 --> 00:02:23.270
you make sure you say under
what transformation.

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So we want to know the preimage
of this subset of our

00:02:26.630 --> 00:02:28.940
codomain under the
transformation T.

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Let me write this as
T inverse of S.

00:02:32.650 --> 00:02:36.680
And we saw in the last video
this is all of the x, all of

00:02:36.680 --> 00:02:46.240
the elements in our domain where
the transformation of

00:02:46.240 --> 00:02:50.890
those x's is a member of the
subset of our codomain that

00:02:50.890 --> 00:02:53.760
we're trying to find
the preimage of.

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Right?

00:02:54.330 --> 00:02:56.850
Now what is another way
of writing this?

00:02:56.850 --> 00:02:59.170
Well this is, you can write this
as, we're trying to look

00:02:59.170 --> 00:03:06.050
for all of the x's in our domain
such that this, let's

00:03:06.050 --> 00:03:08.930
call this A, that matrix A.

00:03:08.930 --> 00:03:13.600
Such that A times x
is a member of S.

00:03:13.600 --> 00:03:17.300
So that means A times x has to
be equal to this or has to be

00:03:17.300 --> 00:03:18.060
equal to that.

00:03:18.060 --> 00:03:21.810
So that means A times our vector
x has to be equal to

00:03:21.810 --> 00:03:24.030
the zero vector.

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Or A times our vector x has to
be equal to this vector 1,2.

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This is the exact same
statement as

00:03:34.240 --> 00:03:35.020
this one right here.

00:03:35.020 --> 00:03:37.690
I just made it a little bit more
explicit in terms of our

00:03:37.690 --> 00:03:40.210
actual transformation,
A times x.

00:03:40.210 --> 00:03:41.710
And in terms of what
our actual set is.

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Our set is just two vectors.

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So if we want to determine
the preimage of S.

00:03:49.920 --> 00:03:53.050
So we write that the preimage
of S under T.

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That set there.

00:03:54.080 --> 00:03:56.460
We essentially have to just
find all of the x's that

00:03:56.460 --> 00:03:59.780
satisfy these two equations
right there.

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So this equation, so we have
to find all of the x's that

00:04:03.080 --> 00:04:10.280
satisfy, the first one right
here is the matrix 1, 3, 2, 6

00:04:10.280 --> 00:04:16.589
times x1, x2 is equal
to the zero vector.

00:04:16.589 --> 00:04:18.380
That's this equation
right there.

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We need to find all of the
solutions to that.

00:04:20.240 --> 00:04:21.790
And you might already
recognize it.

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All of the solutions to this,
all of the x's that satisfy

00:04:25.540 --> 00:04:27.750
this, is the null space
of this matrix.

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I just thought I'd point
that out on the side.

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Now, that's not the only one.

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We also have to solve
this guy over here.

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I'll do that in blue.

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So the preimage of S under
T is going to be all the

00:04:40.080 --> 00:04:48.280
solutions to this plus all of
the solutions to 1, 3, 2, 6

00:04:48.280 --> 00:04:54.300
times x1, x2 is equal to 1, 2.

00:04:54.300 --> 00:04:57.280
Now we can just solve this
with an augmented matrix.

00:04:57.280 --> 00:05:03.230
So my augmented matrix would
look like 1, 3, 2, 6, 0, 0.

00:05:03.230 --> 00:05:09.160
And here my augmented matrix
would be 1, 3, 2, 6, 1, 2.

00:05:09.160 --> 00:05:11.960
Let's put this in reduced
row echelon form.

00:05:11.960 --> 00:05:15.580
So let's replace our second row
with our second row minus

00:05:15.580 --> 00:05:17.880
2 times the first row.

00:05:17.880 --> 00:05:19.450
So what do we get?

00:05:19.450 --> 00:05:21.930
Our first row stays the same.

00:05:21.930 --> 00:05:24.960
1, 3, 0.

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Let me do them simultaneously.

00:05:27.110 --> 00:05:28.590
Let me solve these systems
in parallel.

00:05:28.590 --> 00:05:31.880
So my first system stays, first
row stays the same.

00:05:31.880 --> 00:05:33.710
1, 3, 1.

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And in both cases, because I
just want to get the left hand

00:05:36.490 --> 00:05:39.660
side of my augmented matrix into
reduced row echelon form,

00:05:39.660 --> 00:05:41.840
I can apply the same
row operation.

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So I'm replacing my second row
with 2 times my first row.

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So 2 minus 2 times 1 is 0.

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6 minus 2 times 3 is 0.

00:05:50.400 --> 00:05:54.650
And of course 0 minus
2 times 0 is 0.

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Here 2 minus 2 times 1 is 0.

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6 minus 2 times 3 is 0.

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And 2 minus 2 times 1 is 0.

00:06:04.950 --> 00:06:07.080
So we get all these zeroes,
and we're actually done.

00:06:07.080 --> 00:06:10.100
We have both of these augmented
matrices in reduced

00:06:10.100 --> 00:06:11.420
row echelon form.

00:06:11.420 --> 00:06:15.330
And how do we go back to solve
all of the x1's and x2's that

00:06:15.330 --> 00:06:16.100
satisfy these?

00:06:16.100 --> 00:06:19.250
Well you recognize that our
first columns right here are

00:06:19.250 --> 00:06:21.080
pivot columns.

00:06:21.080 --> 00:06:24.340
And these are associated with
our variable x1, so we know

00:06:24.340 --> 00:06:26.330
that x1 is a pivot variable.

00:06:26.330 --> 00:06:30.380
And we know that the second
column is a non pivot column.

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Because it has no 1's in it.

00:06:32.180 --> 00:06:35.880
And so it's associated
with x2.

00:06:35.880 --> 00:06:38.380
And since x2 isn't a pivot
column, we know that x2 is a

00:06:38.380 --> 00:06:39.630
free variable.

00:06:42.980 --> 00:06:46.110
Which essentially means we can
set x2 to be anything.

00:06:46.110 --> 00:06:48.700
So let's just set x2.

00:06:48.700 --> 00:06:54.570
x2 is equal to t, where t is
a member of the reals.

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In this case what is x1
going to be equal to?

00:06:58.300 --> 00:07:01.580
So here this top equation
says, let me write it.

00:07:01.580 --> 00:07:04.370
If we just kind of go back to
this world right here, this

00:07:04.370 --> 00:07:12.570
means that x1 plus 3x2 is
equal to this 0 here.

00:07:12.570 --> 00:07:20.000
This top line right here says
x1 plus 3x2 is equal to 1.

00:07:20.000 --> 00:07:25.970
And so if we say x2 is equal to
t, this equation becomes x1

00:07:25.970 --> 00:07:28.650
plus 3t is equal to 0.

00:07:28.650 --> 00:07:32.510
Subtract 3t from both
sides, you get x1 is

00:07:32.510 --> 00:07:34.660
equal to minus 3t.

00:07:34.660 --> 00:07:38.670
And then this equation, you
get x1 is equal to, if we

00:07:38.670 --> 00:07:47.400
substitute x2 with t, is x1
is equal to 1 minus 3t.

00:07:47.400 --> 00:07:49.590
Now if we wanted to write the
solution sets in kind of

00:07:49.590 --> 00:07:50.540
vector notation.

00:07:50.540 --> 00:07:55.420
The solution set for this guy
right here, for this first

00:07:55.420 --> 00:08:03.530
equation right there, is going
to be x1, x2 is equal to what?

00:08:03.530 --> 00:08:12.540
It's going to be equal to, you
have your, well, x2 is just

00:08:12.540 --> 00:08:14.320
going to be equal to t times.

00:08:14.320 --> 00:08:15.670
It's just going to
be equal to t.

00:08:15.670 --> 00:08:16.920
So let me just write it there.

00:08:19.070 --> 00:08:20.620
x2 is just t times 1.

00:08:20.620 --> 00:08:21.470
It's just equal to t.

00:08:21.470 --> 00:08:23.780
I made that definition
up here.

00:08:23.780 --> 00:08:25.340
And then what's x1 equal to?

00:08:25.340 --> 00:08:27.390
It's equal to minus 3 times t.

00:08:27.390 --> 00:08:29.550
If I put a t out here as
a scalar, it's just

00:08:29.550 --> 00:08:32.409
minus 3 times t.

00:08:32.409 --> 00:08:35.500
So this is a solution for
this first equation.

00:08:35.500 --> 00:08:40.280
Some, where t is a member of the
reals, So it's just scalar

00:08:40.280 --> 00:08:43.100
multiples of the vector
minus 3, 1.

00:08:43.100 --> 00:08:45.100
And if we think of this position
vector, this'll be a

00:08:45.100 --> 00:08:46.180
line in R2.

00:08:46.180 --> 00:08:47.590
And I'll draw that
in a second.

00:08:47.590 --> 00:08:49.830
So that's a solution for
this first equation.

00:08:49.830 --> 00:08:55.900
And then the solution for the
second equation, how can we

00:08:55.900 --> 00:08:57.740
set this up?

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It's going to be, make sure you
can see it, It is x1, x2,

00:09:05.630 --> 00:09:07.560
and let's see.

00:09:07.560 --> 00:09:09.670
x2 once again is
just t times 1.

00:09:09.670 --> 00:09:12.770
Let me just write
it like this.

00:09:12.770 --> 00:09:17.310
So it's t times 1.

00:09:17.310 --> 00:09:18.920
That's x2.

00:09:18.920 --> 00:09:20.320
Now what's x1?

00:09:20.320 --> 00:09:24.000
x1 is equal to 1 minus
3 times t.

00:09:24.000 --> 00:09:27.450
So if we do a minus 3 that give
us our minus 3 times t.

00:09:27.450 --> 00:09:29.740
But we need to do
1 minus that.

00:09:29.740 --> 00:09:32.490
Or 1 plus minus 3 times t.

00:09:32.490 --> 00:09:34.660
So what we can do is, we could
say that all the solution set

00:09:34.660 --> 00:09:40.290
here is equal to the vector 1,
here, so now we have 1 plus

00:09:40.290 --> 00:09:42.700
minus 3 times t for x1.

00:09:42.700 --> 00:09:44.330
And here we could say is 0.

00:09:44.330 --> 00:09:46.030
x2 is equal 0 plus t.

00:09:46.030 --> 00:09:47.470
Or x2 is equal to t.

00:09:47.470 --> 00:09:52.510
So this is the solution set
for the second equation.

00:09:52.510 --> 00:09:55.530
So our preimage of S.

00:09:55.530 --> 00:09:59.940
Remember S was just these two
points in our codomain.

00:09:59.940 --> 00:10:05.560
The preimage of S under T is
essentially all of the x's

00:10:05.560 --> 00:10:07.720
that satisfy these
two equations.

00:10:07.720 --> 00:10:09.830
And let's actually
graphs those.

00:10:09.830 --> 00:10:12.190
Let me turn on my graphs.

00:10:12.190 --> 00:10:14.670
That makes it look a little
bit messy, but let me

00:10:14.670 --> 00:10:17.340
graph it down here.

00:10:17.340 --> 00:10:22.635
Let me copy and paste
my two results.

00:10:25.810 --> 00:10:28.750
Then I want to paste it.

00:10:28.750 --> 00:10:31.070
So those are my two results.

00:10:31.070 --> 00:10:33.250
Put my pen back on.

00:10:33.250 --> 00:10:35.530
And now what I can do is I
can graph it right here.

00:10:35.530 --> 00:10:37.040
So let's see.

00:10:37.040 --> 00:10:39.360
The solution set for that first
equation is all the

00:10:39.360 --> 00:10:41.390
multiples of the vector
minus 3, 1.

00:10:41.390 --> 00:10:46.390
So the vector minus 3,
1 looks like this.

00:10:51.940 --> 00:10:53.800
Looks like that.

00:10:53.800 --> 00:10:55.610
That's the vector minus 3, 1.

00:10:55.610 --> 00:10:57.280
But my solution set
is all the scalar

00:10:57.280 --> 00:10:58.770
multiples of minus 3, 1.

00:10:58.770 --> 00:11:00.400
This is a comma right here.

00:11:00.400 --> 00:11:00.690
Right?

00:11:00.690 --> 00:11:03.330
So if I just take all the scalar
multiples of minus 3,

00:11:03.330 --> 00:11:05.240
1, it's going to
look like this.

00:11:05.240 --> 00:11:07.010
So if you take 2 times
it, you're going to

00:11:07.010 --> 00:11:09.670
have minus 6, 2.

00:11:09.670 --> 00:11:10.840
So you're going to
get like that.

00:11:10.840 --> 00:11:16.090
So it's going to be all of
these points right here.

00:11:16.090 --> 00:11:18.000
I wish I could draw a little
bit neater, but I think you

00:11:18.000 --> 00:11:18.735
get the idea.

00:11:18.735 --> 00:11:21.410
It's going to be a
line like that.

00:11:21.410 --> 00:11:24.650
That is that solution
set right there.

00:11:24.650 --> 00:11:27.130
And then what is this solution
set right here?

00:11:27.130 --> 00:11:29.230
It's the vector 1, 0.

00:11:29.230 --> 00:11:33.750
So we go out 1 and 0,
so that's there.

00:11:33.750 --> 00:11:36.590
Plus scalar multiples
of minus 3, 1.

00:11:36.590 --> 00:11:40.310
So plus scalar multiples
of this.

00:11:40.310 --> 00:11:42.900
So if we just had 1 scalar
multiple of minus 3, 1 we'll

00:11:42.900 --> 00:11:46.880
end up right there.

00:11:46.880 --> 00:11:48.740
But we want to put all the
scalar multiples of it,

00:11:48.740 --> 00:11:50.330
because we have this
t right here.

00:11:50.330 --> 00:11:53.620
So we're going to end up with
another line with the same

00:11:53.620 --> 00:11:56.460
slope essentially, that's just
shifted a little bit.

00:11:56.460 --> 00:11:57.825
It's shifted 1 to the right.

00:12:01.350 --> 00:12:03.130
Now what why did we
do all of this?

00:12:03.130 --> 00:12:07.670
Remember what we wanted to find
out is, what were all of

00:12:07.670 --> 00:12:16.190
the vectors, let me turn off the
graph paper, in our domain

00:12:16.190 --> 00:12:21.170
that when we apply the
transformation, map to vectors

00:12:21.170 --> 00:12:24.060
within our subset
of our codomain.

00:12:24.060 --> 00:12:27.720
Map to either 0, 0 or
the vector 1, 2.

00:12:27.720 --> 00:12:30.320
And we've figured all of those
vectors out by solving these

00:12:30.320 --> 00:12:31.230
two equations.

00:12:31.230 --> 00:12:34.630
And we are able to see that
these two lines, so when I

00:12:34.630 --> 00:12:38.710
turn my graph paper on, they
map to the points.

00:12:38.710 --> 00:12:41.430
So when these guys, when you
apply the transformation, I'll

00:12:41.430 --> 00:12:43.530
draw it all on the same graph.

00:12:43.530 --> 00:12:45.760
They map, when you apply the
transformation, to the point

00:12:45.760 --> 00:12:50.520
0, 0 and to the point 1, 2.

00:12:50.520 --> 00:12:51.350
Which was right here.

00:12:51.350 --> 00:12:54.760
So all of these points, when you
apply the transformation,

00:12:54.760 --> 00:12:59.050
and actually all the ones in
the blue, they map to 0, 0.

00:12:59.050 --> 00:13:00.630
Because we solved this
top equation.

00:13:00.630 --> 00:13:03.080
And all the ones in orange,
when you apply the

00:13:03.080 --> 00:13:07.680
transformation, map
to the point 1, 2.

00:13:11.170 --> 00:13:13.370
Now this blue line right here.

00:13:13.370 --> 00:13:16.370
This has a special
name for it.

00:13:16.370 --> 00:13:19.050
And I actually touched on is it
a little bit before, right?

00:13:19.050 --> 00:13:21.420
Everything in this
blue line, right?

00:13:21.420 --> 00:13:24.113
If I call this set right here,
I don't know, let's

00:13:24.113 --> 00:13:25.140
call it B for blue.

00:13:25.140 --> 00:13:26.570
This is the blue line.

00:13:26.570 --> 00:13:29.940
That's this set of vectors
right here.

00:13:29.940 --> 00:13:33.560
Everything there, when I apply
my transformation of those

00:13:33.560 --> 00:13:38.990
blue vectors, or if I take the
image of my blue set under T,

00:13:38.990 --> 00:13:43.410
it all maps to the
zero vector.

00:13:46.950 --> 00:13:49.150
It equals the set of the zero
vector right there.

00:13:49.150 --> 00:13:50.050
We saw that right there.

00:13:50.050 --> 00:13:52.730
And I remember when, well
earlier in the video I pointed

00:13:52.730 --> 00:13:55.110
out that, look , this
set right here.

00:13:55.110 --> 00:13:58.750
This is equivalent to the
null space, right?

00:13:58.750 --> 00:14:04.100
The null space of a matrix is
all the vectors that if you

00:14:04.100 --> 00:14:06.990
multiply it by that
matrix, you get 0.

00:14:06.990 --> 00:14:09.340
So this is a similar
idea here.

00:14:09.340 --> 00:14:11.890
This transformation is
defined by a matrix.

00:14:11.890 --> 00:14:14.040
And we're saying, what are all
of the x's that when you

00:14:14.040 --> 00:14:20.750
transform them you get
the zero vector?

00:14:20.750 --> 00:14:24.810
And so this idea, this blue
thing right here, it's called

00:14:24.810 --> 00:14:30.570
the kernel of T.

00:14:30.570 --> 00:14:34.880
Sometimes it's written just
shorthand K-E-R of T.

00:14:34.880 --> 00:14:38.450
And this literally is all of the
vectors that if you apply

00:14:38.450 --> 00:14:40.630
the transformation, let
me write it this way.

00:14:40.630 --> 00:14:46.850
It's all of the vectors in our
domain, which was R2, such

00:14:46.850 --> 00:14:52.940
that the transformation of those
vectors is equal to the

00:14:52.940 --> 00:14:53.620
zero vector.

00:14:53.620 --> 00:14:56.540
There this is the definition
of the kernel.

00:14:56.540 --> 00:15:03.580
And if the transformation is
equal to some matrix times

00:15:03.580 --> 00:15:05.670
some vector, and we know that
any linear transformation can

00:15:05.670 --> 00:15:12.460
be written as a matrix vector
product, then the kernel of T

00:15:12.460 --> 00:15:16.790
is the same thing as the
null space of A.

00:15:16.790 --> 00:15:18.600
And we saw that earlier
in the video.

00:15:18.600 --> 00:15:22.150
Anyway, hopefully you found
that reasonably useful.


Preimage of a set

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.210
Let's add some transformation
that maps elements in

00:00:04.210 --> 00:00:06.710
set X to set Y.

00:00:06.710 --> 00:00:09.870
We know that we call
X the domain of T.

00:00:09.870 --> 00:00:14.920
So that's my set X and then my
set that I'm mapping into, set

00:00:14.920 --> 00:00:16.270
Y, that's the codomain.

00:00:16.270 --> 00:00:18.750
We know that T is a
transformation that if you

00:00:18.750 --> 00:00:23.510
take any member of X and you
transform it, you'll associate

00:00:23.510 --> 00:00:25.230
it with a member of set Y.

00:00:25.230 --> 00:00:27.430
You'll map it to a
member of set Y.

00:00:27.430 --> 00:00:31.040
That's what the transformation
or the function does.

00:00:31.040 --> 00:00:35.740
Now, if we have some subset of
T, let's call A to be some

00:00:35.740 --> 00:00:37.245
subset of T.

00:00:37.245 --> 00:00:40.090
So let me draw A like that.

00:00:40.090 --> 00:00:43.570
This notation right here
just means subset,

00:00:43.570 --> 00:00:44.730
some subset of T.

00:00:44.730 --> 00:00:50.590
We've defined the notion of an
image of T of A like that,

00:00:50.590 --> 00:00:59.870
which is the image of A, of
our subset A, under T.

00:00:59.870 --> 00:01:03.700
We've defined this to be equal
to the set-- let me write it

00:01:03.700 --> 00:01:09.010
here-- the set of all-- where if
we take each of the members

00:01:09.010 --> 00:01:15.390
of our subset, it's the set of
all of their transformations.

00:01:18.600 --> 00:01:22.060
And, of course, these are going
to be some subset of Y

00:01:22.060 --> 00:01:22.680
right there.

00:01:22.680 --> 00:01:25.140
So we essentially take each
of the members of A.

00:01:25.140 --> 00:01:25.930
This was one of them.

00:01:25.930 --> 00:01:26.960
You find its transformation.

00:01:26.960 --> 00:01:27.900
It's that point.

00:01:27.900 --> 00:01:30.160
You take another member of A.

00:01:30.160 --> 00:01:31.790
This is all set A right here.

00:01:31.790 --> 00:01:32.660
Take another member of A.

00:01:32.660 --> 00:01:33.720
Find its transformation.

00:01:33.720 --> 00:01:34.740
Maybe it's that point.

00:01:34.740 --> 00:01:35.760
You keep doing that.

00:01:35.760 --> 00:01:36.580
Find it's transformation.

00:01:36.580 --> 00:01:37.600
Maybe it's that point.

00:01:37.600 --> 00:01:40.160
And then the set of all of those
transformations, maybe

00:01:40.160 --> 00:01:43.400
it's this blob right here,
we call this the

00:01:43.400 --> 00:01:46.830
image of A under T.

00:01:46.830 --> 00:01:48.630
Now, what if we wanted
to think about

00:01:48.630 --> 00:01:49.770
the opposite problem?

00:01:49.770 --> 00:01:54.340
What if we were to start with
set Y, which is our codomain,

00:01:54.340 --> 00:01:58.570
So that's Y, and we were to
have some subset of Y.

00:01:58.570 --> 00:02:02.980
Let's call our subset of Y S.

00:02:02.980 --> 00:02:07.520
So S is a subset of
our codomain Y.

00:02:07.520 --> 00:02:12.610
And I'm curious about what
subset of X maps into S.

00:02:12.610 --> 00:02:15.230
So I'm curious about this set.

00:02:15.230 --> 00:02:20.170
I'm curious about the set of all
vectors that are members

00:02:20.170 --> 00:02:27.390
of my domain such that they're
mapping or the transformation

00:02:27.390 --> 00:02:35.070
of those vectors ends up
in my subset of S.

00:02:35.070 --> 00:02:40.560
So what I'm saying is, look,
if I take my domain, there

00:02:40.560 --> 00:02:46.530
must be some subset of vectors
right here, where if I take

00:02:46.530 --> 00:02:52.140
any member of this set, it will
map into these guys, and

00:02:52.140 --> 00:02:53.690
that's what I'm defining
right here.

00:02:53.690 --> 00:02:55.140
This is equal to that guy.

00:02:55.140 --> 00:02:58.570
So I'm literally saying what
are all of the members of X

00:02:58.570 --> 00:03:02.570
where those members of
X all map into S?

00:03:02.570 --> 00:03:06.710
Now, I want to make a very
subtle nuance here to point

00:03:06.710 --> 00:03:07.270
out something here.

00:03:07.270 --> 00:03:09.630
I'm not saying that
every point in S

00:03:09.630 --> 00:03:12.190
necessarily gets mapped to.

00:03:12.190 --> 00:03:16.430
For example, maybe there's some
element in S right there

00:03:16.430 --> 00:03:19.800
that no element in X ever
gets mapped to from our

00:03:19.800 --> 00:03:21.140
transformation T.

00:03:21.140 --> 00:03:22.190
That's OK.

00:03:22.190 --> 00:03:25.760
All I'm saying is that
everything in this set maps to

00:03:25.760 --> 00:03:28.460
something within S right here.

00:03:28.460 --> 00:03:34.970
And what we call this set right
here, the notation is

00:03:34.970 --> 00:03:38.000
the inverse T of S, but
this is equal to the

00:03:38.000 --> 00:03:46.540
pre-image of S under T.

00:03:46.540 --> 00:03:47.710
So this is S.

00:03:47.710 --> 00:03:53.530
This is the pre-image
of S under T.

00:03:53.530 --> 00:03:54.190
And that makes sense.

00:03:54.190 --> 00:03:57.980
The image, we go from a subset
of our domain to a subset of

00:03:57.980 --> 00:03:59.050
our codomain.

00:03:59.050 --> 00:04:02.250
Preimage, we go from a subset
of our codomain, and we say

00:04:02.250 --> 00:04:04.920
what subset of our domain
maps into that

00:04:04.920 --> 00:04:06.850
subset of our codomain?

00:04:06.850 --> 00:04:08.720
Now let me ask you an
interesting question, and this

00:04:08.720 --> 00:04:11.170
is kind of for bonus points.

00:04:11.170 --> 00:04:17.850
What is the image of our
pre-image under S?

00:04:23.020 --> 00:04:27.210
So if we take this guy, this
is essentially the image of

00:04:27.210 --> 00:04:30.050
this guy right here, right?

00:04:30.050 --> 00:04:34.455
This part right here is the
pre-image of S right here.

00:04:34.455 --> 00:04:37.530
Now, if we take the image of
this, we're saying if we take

00:04:37.530 --> 00:04:40.520
every member of this, what
vectors do they map into?

00:04:40.520 --> 00:04:42.290
All of them are going to be
within S, so they're going to

00:04:42.290 --> 00:04:45.880
map within S, but they don't
necessarily map to

00:04:45.880 --> 00:04:47.700
everything within S.

00:04:47.700 --> 00:04:51.000
So this is going to be
some subset of S.

00:04:51.000 --> 00:04:54.820
So this right here is going
to be some subset of

00:04:54.820 --> 00:04:56.130
our original S.

00:04:56.130 --> 00:04:58.870
It's not necessarily equal to
S, but it's a subset of it.

00:04:58.870 --> 00:05:00.960
And so this is I think the
motivation for where the

00:05:00.960 --> 00:05:02.330
notation comes from.

00:05:02.330 --> 00:05:04.950
We can construct a subset of S
by taking the image of the

00:05:04.950 --> 00:05:05.900
pre-image of S.

00:05:05.900 --> 00:05:08.950
We can kind of view the image
and the pre-image kind of

00:05:08.950 --> 00:05:11.370
canceling out, and that's why
the inverse notation was

00:05:11.370 --> 00:05:12.720
probably introduced.

00:05:12.720 --> 00:05:14.560
Now this is all very abstract.

00:05:14.560 --> 00:05:16.400
What I'm going to do in the
next video is actually

00:05:16.400 --> 00:05:19.930
calculate or determine
the pre-image of some

00:05:19.930 --> 00:05:21.430
subset of my codomain.


(2^ln x)/x  Antiderivative Example

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.790
I was just looking on the
discussion boards on the Khan

00:00:03.790 --> 00:00:08.150
Academy Facebook page, and Bud
Denny put up this problem,

00:00:08.150 --> 00:00:09.335
asking for it to be solved.

00:00:09.335 --> 00:00:12.010
And it seems like a problem
of general interest.

00:00:12.010 --> 00:00:18.510
If the indefinite integral of
2 to the natural log of x

00:00:18.510 --> 00:00:23.200
over, everything over x, dx.

00:00:23.200 --> 00:00:27.210
And on the message board, Abhi
Khanna also put up a solution,

00:00:27.210 --> 00:00:29.110
and it is the correct solution,
but I thought this was of

00:00:29.110 --> 00:00:32.080
general interest, so I'll
make a quick video on it.

00:00:32.080 --> 00:00:34.420
So the first thing when you see
an integral like this, is you

00:00:34.420 --> 00:00:36.635
say, hey, you know, I have this
natural log of x up in

00:00:36.635 --> 00:00:40.150
the numerator, and
where do I start?

00:00:40.150 --> 00:00:43.150
And the first thing that should
maybe pop out at you, is that

00:00:43.150 --> 00:00:49.370
this is the same thing as the
integral of one over x times 2

00:00:49.370 --> 00:00:53.870
to the natural log of x, dx.

00:00:53.870 --> 00:00:57.580
And so you have an expression
here, or it's kind of part of

00:00:57.580 --> 00:01:00.640
our larger function, and you
have its derivative, right?

00:01:00.640 --> 00:01:03.570
We know that the derivative,
let me write it over here, we

00:01:03.570 --> 00:01:06.260
know that the derivative with
respect to x of the natural

00:01:06.260 --> 00:01:09.870
log of x is equal to 1/x.

00:01:09.870 --> 00:01:13.870
So we have some expression, and
we have its derivative, which

00:01:13.870 --> 00:01:15.980
tells us that we can
use substitution.

00:01:15.980 --> 00:01:18.250
Sometimes you can do in your
head, but this problem, it's

00:01:18.250 --> 00:01:19.690
still not trivial to
do in your head.

00:01:19.690 --> 00:01:21.130
So let's make the substitution.

00:01:21.130 --> 00:01:24.570
Let's substitute this
right here with a u.

00:01:24.570 --> 00:01:26.710
So let's do that.

00:01:26.710 --> 00:01:29.890
So if you define u, and it
doesn't have to be u, it's

00:01:29.890 --> 00:01:32.670
just, that's the convention,
it's called u-substitution, it

00:01:32.670 --> 00:01:35.120
could have been s-substitution
for all we care.

00:01:35.120 --> 00:01:39.760
Let's say u is equal to the
natural log of x, and then du

00:01:39.760 --> 00:01:43.400
dx, the derivative of u with
respect to x, of course

00:01:43.400 --> 00:01:45.220
is equal to 1/x.

00:01:45.220 --> 00:01:49.480
Or, just the differential du,
if we just multiply both sides

00:01:49.480 --> 00:01:53.600
by dx is equal to 1 over x dx.

00:01:53.600 --> 00:01:55.080
So let's make our substitution.

00:01:55.080 --> 00:01:55.990
This is our integral.

00:01:55.990 --> 00:02:00.160
So this will be equal to the
indefinite integral, or the

00:02:00.160 --> 00:02:05.480
antiderivative, of 2 to the
now u, so 2 to the u,

00:02:05.480 --> 00:02:08.450
times 1 over x dx.

00:02:08.450 --> 00:02:10.620
Now what is 1 over x dx?

00:02:10.620 --> 00:02:11.440
That's just du.

00:02:11.440 --> 00:02:13.860
So this term times that
term is just our du.

00:02:13.860 --> 00:02:15.540
Let me do it in a
different color.

00:02:15.540 --> 00:02:20.380
1 over x times dx is
just equal to du.

00:02:20.380 --> 00:02:23.530
That's just equal to that
thing, right there.

00:02:23.530 --> 00:02:26.630
Now, this still doesn't look
like an easy integral, although

00:02:26.630 --> 00:02:29.230
it's gotten simplified
a good bit.

00:02:29.230 --> 00:02:34.000
And to solve, you know,
whenever I see the variable

00:02:34.000 --> 00:02:36.710
that I'm integrating against
in the exponent, you know,

00:02:36.710 --> 00:02:38.570
we don't have any easy
exponent rules here.

00:02:38.570 --> 00:02:42.710
The only thing that I'm
familiar with, where I have my

00:02:42.710 --> 00:02:45.910
x or my variable that I'm
integrating against in

00:02:45.910 --> 00:02:48.430
my exponent, is the
case of e to the x.

00:02:48.430 --> 00:02:51.990
We know that the integral of
e to the x, dx, is equal

00:02:51.990 --> 00:02:55.090
to e to the x plus c.

00:02:55.090 --> 00:02:58.470
So if I could somehow turn this
into some variation of e to the

00:02:58.470 --> 00:03:02.370
x, maybe, or e to the u, maybe
I can make this integral a

00:03:02.370 --> 00:03:03.980
little bit more tractable.

00:03:03.980 --> 00:03:04.600
So let's see.

00:03:04.600 --> 00:03:10.890
How can we redefine
this right here?

00:03:10.890 --> 00:03:14.380
Well, 2, 2 is equal to what?

00:03:14.380 --> 00:03:19.970
2 is the same thing as e to
the natural log of 2, right?

00:03:19.970 --> 00:03:22.850
The natural log of 2 is
the power you have to

00:03:22.850 --> 00:03:26.860
raise e to to get 2.

00:03:26.860 --> 00:03:28.620
So if you raise e to
that power, you're, of

00:03:28.620 --> 00:03:29.410
course going to get 2.

00:03:29.410 --> 00:03:32.030
This is actually the definition
of really, the natural log.

00:03:32.030 --> 00:03:35.490
You raise e to the natural log
of 2, you're going to get 2.

00:03:35.490 --> 00:03:38.180
So let's rewrite this, using
this-- I guess we could call

00:03:38.180 --> 00:03:40.650
this this rewrite or-- I
don't want to call it

00:03:40.650 --> 00:03:41.380
quite a substitution.

00:03:41.380 --> 00:03:44.440
It's just a different way
of writing the number 2.

00:03:44.440 --> 00:03:48.940
So this will be equal to,
instead of writing the number

00:03:48.940 --> 00:03:53.260
2, I could write e to
the natural log of 2.

00:03:53.260 --> 00:03:58.090
And all of that to the u du.

00:03:58.090 --> 00:04:00.120
And now what is this equal to?

00:04:00.120 --> 00:04:02.690
Well, if I take something to an
exponent, and then to another

00:04:02.690 --> 00:04:05.420
exponent, this is the same
thing as taking my base to the

00:04:05.420 --> 00:04:07.250
product of those exponents.

00:04:07.250 --> 00:04:12.810
So this is equal to, let me
switch colors, this is equal

00:04:12.810 --> 00:04:21.620
to the integral of e,
to the u, e to the, let

00:04:21.620 --> 00:04:22.640
me write it this way.

00:04:22.640 --> 00:04:27.660
e to the natural
log of 2 times u.

00:04:27.660 --> 00:04:29.770
I'm just multiplying
these two exponents.

00:04:29.770 --> 00:04:32.190
I raise something to something,
then raise it again, we know

00:04:32.190 --> 00:04:34.170
from our exponent rules,
it's just a product of

00:04:34.170 --> 00:04:35.500
those two exponents.

00:04:35.500 --> 00:04:37.250
du.

00:04:37.250 --> 00:04:39.810
Now, this is just a constant
factor, right here.

00:04:39.810 --> 00:04:42.210
This could be, you know, this
could just be some number.

00:04:42.210 --> 00:04:44.300
We could use a calculator to
figure out what this is.

00:04:44.300 --> 00:04:45.720
We could set this equal to a.

00:04:45.720 --> 00:04:48.050
But we know in general that
the integral, this is pretty

00:04:48.050 --> 00:04:50.710
straightforward, we've
now put it in this form.

00:04:50.710 --> 00:04:56.290
The antiderivative of e
to the au, du, is just

00:04:56.290 --> 00:04:59.750
1 over a e to the au.

00:04:59.750 --> 00:05:02.520
This comes from this definition
up here, and of course plus

00:05:02.520 --> 00:05:03.780
c, and the chain rule.

00:05:03.780 --> 00:05:05.975
If we take the derivative of
this, we take the derivative

00:05:05.975 --> 00:05:08.210
of the inside, which
is just going to be a.

00:05:08.210 --> 00:05:10.510
We multiply that times the
one over a, it cancels

00:05:10.510 --> 00:05:12.920
out, and we're just
left with e to the au.

00:05:12.920 --> 00:05:14.420
So this definitely works out.

00:05:14.420 --> 00:05:18.590
So the antiderivative of this
thing right here is going to be

00:05:18.590 --> 00:05:24.340
equal to 1 over our a, it's
going to be 1 over our constant

00:05:24.340 --> 00:05:29.640
term, 1 over the natural log of
2 times our whole

00:05:29.640 --> 00:05:32.485
expression, e e.

00:05:32.485 --> 00:05:35.030
And I'm going to do something.

00:05:35.030 --> 00:05:38.170
This is just some number times
u, so I can write it as

00:05:38.170 --> 00:05:39.250
u times some number.

00:05:39.250 --> 00:05:42.650
And I'm just doing that to put
in a form that might help us

00:05:42.650 --> 00:05:44.050
simplify it a little bit.

00:05:44.050 --> 00:05:47.270
So it's e to the u times the
natural log of 2, right?

00:05:47.270 --> 00:05:48.780
All I did, is I
swapped this order.

00:05:48.780 --> 00:05:50.640
I could have written this
as e to the natural

00:05:50.640 --> 00:05:53.140
log of 2 times u.

00:05:53.140 --> 00:05:56.590
If this an a, a times u is
the same thing as u times a.

00:05:56.590 --> 00:05:58.530
Plus c.

00:05:58.530 --> 00:06:01.110
So this is our answer, but we
have to kind of reverse

00:06:01.110 --> 00:06:04.340
substitute before we can feel
satisfied that we've taken

00:06:04.340 --> 00:06:07.300
the antiderivative
with respect to x.

00:06:07.300 --> 00:06:09.680
But before I do that, let's
see if I can simplify

00:06:09.680 --> 00:06:10.880
this a little bit.

00:06:10.880 --> 00:06:15.550
What is, if I have, just from
our natural log properties,

00:06:15.550 --> 00:06:18.450
or logarithms, a times
the natural log of b.

00:06:18.450 --> 00:06:20.140
We know this is the same
thing as the natural

00:06:20.140 --> 00:06:21.480
log of b to the a.

00:06:21.480 --> 00:06:22.900
Let me draw a line here.

00:06:22.900 --> 00:06:23.220
Right?

00:06:23.220 --> 00:06:25.130
That this becomes the exponent
on whatever we're taking

00:06:25.130 --> 00:06:26.620
the natural log of.

00:06:26.620 --> 00:06:31.950
So u, let me write this here,
u times the natural log of

00:06:31.950 --> 00:06:35.470
2, is the same thing as the
natural log of 2 to the u.

00:06:35.470 --> 00:06:38.440
So we can rewrite our
antiderivative as being equal

00:06:38.440 --> 00:06:42.020
to 1 over the natural log of 2,
that's just that part here,

00:06:42.020 --> 00:06:46.450
times e to the, this can be
rewritten based on this

00:06:46.450 --> 00:06:51.740
logarithm property, as the
natural log of 2 to the u, and

00:06:51.740 --> 00:06:55.750
of course we still have
our plus c there.

00:06:55.750 --> 00:07:00.500
Now, what is e raised to the
natural log of 2 to the u?

00:07:00.500 --> 00:07:03.260
The natural log of 2 to the u
is the power that you have to

00:07:03.260 --> 00:07:06.960
raise e to to get to
2 to the u, right?

00:07:06.960 --> 00:07:08.040
By definition!

00:07:08.040 --> 00:07:11.540
So if we raise e to that power,
what are we going to get?

00:07:11.540 --> 00:07:13.040
We're going to get 2 to the u.

00:07:13.040 --> 00:07:17.080
So this is going to be equal to
1 over the natural log of 2.

00:07:17.080 --> 00:07:20.350
This simplifies to
just 2 to the u.

00:07:20.350 --> 00:07:21.220
I drew it up here.

00:07:21.220 --> 00:07:24.970
The natural log a I could just
write in general terms, let

00:07:24.970 --> 00:07:27.520
me do it up here, and maybe
I'm beating a dead horse.

00:07:27.520 --> 00:07:30.570
But I can in general write any
number a as being equal to

00:07:30.570 --> 00:07:32.890
e to the natural log of a.

00:07:32.890 --> 00:07:36.230
This is the exponent you have
to raise e to to get a.

00:07:36.230 --> 00:07:38.880
If you raise e to that,
you're going to get a.

00:07:38.880 --> 00:07:42.420
So e to the natural log of 2 to
the u, that's just 2 to the u.

00:07:42.420 --> 00:07:44.900
And then I have my
plus c, of course.

00:07:44.900 --> 00:07:46.450
And now we can
reverse substitute.

00:07:46.450 --> 00:07:48.830
What did we set u equal to?

00:07:48.830 --> 00:07:53.060
We defined u, up here, as equal
to the natural log of x.

00:07:53.060 --> 00:07:55.990
So let's just reverse
substitute right here.

00:07:55.990 --> 00:08:00.570
And so the answer to our
original equation, your answer

00:08:00.570 --> 00:08:03.260
to, let me write it here,
because it's satisfying when

00:08:03.260 --> 00:08:08.950
you see it, to this kind of
fairly convoluted-looking

00:08:08.950 --> 00:08:14.730
antiderivative problem, 2 to
the natural log of x over x dx,

00:08:14.730 --> 00:08:17.815
we now find is equal to, we
just replaced u with natural

00:08:17.815 --> 00:08:21.750
log of x, because that was our
substitution, and 1 over the

00:08:21.750 --> 00:08:29.410
natural log of 2 times 2 to the
natural log of x plus c.

00:08:29.410 --> 00:08:30.100
And we're done.

00:08:30.100 --> 00:08:31.970
This isn't in the denominator,
the way I wrote it might

00:08:31.970 --> 00:08:33.410
look a little ambiguous.

00:08:33.410 --> 00:08:34.110
And we're done!

00:08:34.110 --> 00:08:37.340
And that was a pretty neat
problem, and so thanks

00:08:37.340 --> 00:08:39.900
to Bud for posting that.


im(T): Image of a transformation

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:11.120
Let's say that I have some set
V that is a subspace in Rn.

00:00:11.120 --> 00:00:12.750
And just as a reminder,
what does it mean?

00:00:12.750 --> 00:00:16.960
That's just some set, or some
subset of Rn where if I take

00:00:16.960 --> 00:00:20.680
any two members of that subset--
so let say I take the

00:00:20.680 --> 00:00:24.980
members a and b-- they're both
members my subspace.

00:00:24.980 --> 00:00:28.910
By the fact that this is a
subspace, we then know that

00:00:28.910 --> 00:00:33.400
the addition of these two
vectors, or a plus b, is also

00:00:33.400 --> 00:00:34.770
in my subspace.

00:00:34.770 --> 00:00:40.655
And this is our closure
under addition.

00:00:43.630 --> 00:00:46.180
And by the fact that it's a
subspace, we also know that if

00:00:46.180 --> 00:00:50.010
we multiply any member of our
subspace by a scalar -- so the

00:00:50.010 --> 00:00:52.530
fact that those guys are members
of our subspace -- we

00:00:52.530 --> 00:00:55.340
also know that if I pick one
of them, let's say a, and I

00:00:55.340 --> 00:00:59.500
multiply a by some scalar, that
this is also going to be

00:00:59.500 --> 00:01:01.400
a member of our subspace.

00:01:01.400 --> 00:01:07.190
And we sometimes call this
closure under scalar

00:01:07.190 --> 00:01:08.440
multiplication.

00:01:11.550 --> 00:01:15.360
And then a somewhat redundant
statement is that V, well it

00:01:15.360 --> 00:01:17.030
must contain the zero vector.

00:01:17.030 --> 00:01:19.050
And that's true of
all subspaces.

00:01:19.050 --> 00:01:22.830
V -- let me write it this
way -- the zero vector

00:01:22.830 --> 00:01:24.355
is a member of V.

00:01:24.355 --> 00:01:27.590
And it would be the zero vector
with n components here,

00:01:27.590 --> 00:01:29.820
because V is a subspace of Rn.

00:01:29.820 --> 00:01:32.350
And why I say that's redundant,
because if I say

00:01:32.350 --> 00:01:34.970
that any multiple of these
vectors is also in V, I could

00:01:34.970 --> 00:01:37.300
just set the scalar
to be equal to 0.

00:01:37.300 --> 00:01:39.450
So this statement kind
of takes the

00:01:39.450 --> 00:01:40.630
statement into account.

00:01:40.630 --> 00:01:43.030
But in a lot of textbooks, they
will always write, oh and

00:01:43.030 --> 00:01:45.080
the zero vector has to
be a member of V.

00:01:45.080 --> 00:01:47.430
Although, that's kind of
redundant with the closure

00:01:47.430 --> 00:01:49.020
under scalar multiplication.

00:01:49.020 --> 00:01:50.570
Fair enough.

00:01:50.570 --> 00:01:55.063
Now, let's say that I also have
some transformation T.

00:01:58.120 --> 00:02:07.340
It is a mapping, a function,
from Rn to Rm.

00:02:07.340 --> 00:02:11.340
What I want to understand, in
this video is, I have a

00:02:11.340 --> 00:02:14.140
subspace right here, V.

00:02:14.140 --> 00:02:17.600
I want to understand whether
the transformation of the

00:02:17.600 --> 00:02:22.030
subspace -- and what
did we call that?

00:02:22.030 --> 00:02:29.020
We called that the image of our
subspace, or our subset,

00:02:29.020 --> 00:02:29.840
either way.

00:02:29.840 --> 00:02:35.240
The image of V under T.

00:02:35.240 --> 00:02:37.890
In the last video, just to kind
of help you visualize it.

00:02:37.890 --> 00:02:41.970
How did that work or -- we
had some subset of Rn

00:02:41.970 --> 00:02:42.830
that looked like this.

00:02:42.830 --> 00:02:46.280
It was a triangle that looked
something like that.

00:02:46.280 --> 00:02:49.120
And that was in Rn, this was
actually in R2, it was a

00:02:49.120 --> 00:02:50.850
triangle that looked something
like that.

00:02:50.850 --> 00:02:54.020
And we figured out it's
image under T.

00:02:54.020 --> 00:02:57.280
So we went from R2 to R2.

00:02:57.280 --> 00:02:58.920
and we had our transformation.

00:02:58.920 --> 00:03:01.660
And it ended up looking
something like this.

00:03:01.660 --> 00:03:03.352
If I remember it properly.

00:03:03.352 --> 00:03:06.475
It ended up looking like a --
gee, I don't remember it

00:03:06.475 --> 00:03:08.810
fully, but it was like a
triangle that was skewed like

00:03:08.810 --> 00:03:09.660
this, rotated.

00:03:09.660 --> 00:03:13.670
So it was a -- actually I think
it was more like -- I

00:03:13.670 --> 00:03:14.370
think that's right.

00:03:14.370 --> 00:03:17.880
It was rotated a bit clockwise
like that and it was skewed.

00:03:17.880 --> 00:03:20.530
But the exact particulars
of that last

00:03:20.530 --> 00:03:22.060
video aren't what matter.

00:03:22.060 --> 00:03:24.790
What matters is that you are
able to visualize what an

00:03:24.790 --> 00:03:26.630
image under transformation
means.

00:03:26.630 --> 00:03:30.700
It means you take some subset of
R2, all of the vectors that

00:03:30.700 --> 00:03:32.760
define this triangle
right here.

00:03:32.760 --> 00:03:34.650
That's some subset of R2.

00:03:34.650 --> 00:03:38.010
You transform all of them, and
then you get some subset in

00:03:38.010 --> 00:03:39.260
your codomain.

00:03:42.300 --> 00:03:45.560
You could call this the image,
because the transformation of

00:03:45.560 --> 00:03:50.150
that triangle, or if we call
this s, it's equal to the

00:03:50.150 --> 00:03:51.810
transformation of s.

00:03:51.810 --> 00:03:58.780
Or you could say it's the image
of-- you can just call

00:03:58.780 --> 00:04:00.560
it the set s, but maybe it helps
you to visualize-- call

00:04:00.560 --> 00:04:06.690
it the image of this
triangle under T.

00:04:06.690 --> 00:04:09.310
Or maybe even a neater way of
thinking about it is, this

00:04:09.310 --> 00:04:16.310
triangle-- that skewed, rotated
triangle-- this one is

00:04:16.310 --> 00:04:21.920
the image of this right
triangle under T.

00:04:21.920 --> 00:04:23.210
I think that might make
a little bit of

00:04:23.210 --> 00:04:23.850
visual sense to you.

00:04:23.850 --> 00:04:26.090
And just as a bit of reminder,
in that last video these

00:04:26.090 --> 00:04:28.790
triangles, these weren't
subspaces.

00:04:28.790 --> 00:04:31.900
And just as you could take
scalar multiples of some of

00:04:31.900 --> 00:04:37.220
the vectors that are members of
this triangle, and you'll

00:04:37.220 --> 00:04:38.690
find that they're not going
to be in that triangle.

00:04:38.690 --> 00:04:43.500
So this wasn't a subspace, this
was just a subset of R2.

00:04:43.500 --> 00:04:47.310
All subsets are not subspaces,
but all subspaces are

00:04:47.310 --> 00:04:48.480
definitely subsets.

00:04:48.480 --> 00:04:50.770
Although something can be
a subset of itself.

00:04:50.770 --> 00:04:52.600
I don't want to wander
off too much.

00:04:52.600 --> 00:04:54.150
But this just helps
you visualize what

00:04:54.150 --> 00:04:55.180
we mean by an image.

00:04:55.180 --> 00:05:03.500
It means all of the vectors that
are mapped to, from the

00:05:03.500 --> 00:05:06.590
members of your subset.

00:05:06.590 --> 00:05:09.670
So I want to know whether
the image of V

00:05:09.670 --> 00:05:11.090
under T is a subspace.

00:05:18.170 --> 00:05:20.050
So in order for it to be a
subspace, if I take the

00:05:20.050 --> 00:05:23.740
transformation -- let me
find two members of T.

00:05:23.740 --> 00:05:26.670
Well clearly if I take the
transformation of any members

00:05:26.670 --> 00:05:30.230
of V, I'm getting members
of the image.

00:05:30.230 --> 00:05:30.610
Right?

00:05:30.610 --> 00:05:32.510
So I can write this.

00:05:32.510 --> 00:05:36.520
Clearly the transformation of
a and the transformations of

00:05:36.520 --> 00:05:41.230
b, these are both of members
of our images of V under T.

00:05:41.230 --> 00:05:44.950
These are both members
of that right there.

00:05:44.950 --> 00:05:53.180
So my question to you is what
is the transformation of a

00:05:53.180 --> 00:05:55.410
plus the transformation of b?

00:05:55.410 --> 00:05:57.800
And the way I have written this,
these are two arbitrary

00:05:57.800 --> 00:06:00.660
members of our image
of V under T.

00:06:00.660 --> 00:06:02.710
Or maybe I should call
it T of capital V.

00:06:02.710 --> 00:06:04.710
These are two arbitrary
members.

00:06:04.710 --> 00:06:06.090
So what is this equal to?

00:06:06.090 --> 00:06:08.630
Well, we know from our
properties, our definition of

00:06:08.630 --> 00:06:13.770
linear transformations, the sum
of the transformations of

00:06:13.770 --> 00:06:16.110
two vectors is equal to the
transformation of the sum of

00:06:16.110 --> 00:06:18.690
their of vectors.

00:06:18.690 --> 00:06:24.060
Now, is the transformation
of a plus b, is

00:06:24.060 --> 00:06:28.130
this a member of TV?

00:06:28.130 --> 00:06:30.310
Is it a member of our image?

00:06:30.310 --> 00:06:34.510
Well, a plus b is a member of V,
and the image contains the

00:06:34.510 --> 00:06:37.810
transformation of all
of the members of V.

00:06:37.810 --> 00:06:40.840
So the image contains the
transformation of this guy.

00:06:40.840 --> 00:06:43.600
This guy, a plus b
is a member of V.

00:06:43.600 --> 00:06:45.490
So you're taking a
transformation of a member of

00:06:45.490 --> 00:06:50.060
V which, by definition, is in
your image of V under T.

00:06:50.060 --> 00:06:52.410
So this is definitely true.

00:06:52.410 --> 00:06:55.070
Now, let's ask the
next question.

00:06:55.070 --> 00:07:03.160
If I take a scalar multiple of
some member of my image of V

00:07:03.160 --> 00:07:06.540
under T, or my T of capital
V, right there.

00:07:06.540 --> 00:07:09.400
If I take the sum scalar,
what is this equal to?

00:07:09.400 --> 00:07:12.510
By definition for linear
transformation, this is the

00:07:12.510 --> 00:07:14.680
same thing as a transformation
of the

00:07:14.680 --> 00:07:17.910
scalar times the vector.

00:07:17.910 --> 00:07:22.030
Now is this going to
be a member of our

00:07:22.030 --> 00:07:24.520
image of V under T?

00:07:24.520 --> 00:07:28.790
Well we know that ca is
definitely in V, right?

00:07:28.790 --> 00:07:30.530
That's from the definition
of a subspace.

00:07:30.530 --> 00:07:32.055
This is definitely in V.

00:07:32.055 --> 00:07:35.960
And so, if this is in V, the
transformation of this has to

00:07:35.960 --> 00:07:38.620
be in V's image under T.

00:07:38.620 --> 00:07:41.910
So this is in -- this is
also a member of V.

00:07:41.910 --> 00:07:46.400
And obviously, you can
set this equal to 0.

00:07:46.400 --> 00:07:50.830
The zero vector is a member of
V, so any transformation of --

00:07:50.830 --> 00:07:54.530
if you just put a 0 here, you'll
get the zero vector.

00:07:54.530 --> 00:07:56.920
So the zero vector is definitely
-- I don't care

00:07:56.920 --> 00:07:59.970
what this is, if you multiply
it times 0, you are going to

00:07:59.970 --> 00:08:01.190
get the zero vector.

00:08:01.190 --> 00:08:02.640
So the zero vector
is definitely

00:08:02.640 --> 00:08:05.690
also a member of TV.

00:08:05.690 --> 00:08:12.250
So we come on the result that
T -- the image of V under T,

00:08:12.250 --> 00:08:16.700
is a subspace.

00:08:16.700 --> 00:08:18.500
Which is a useful result
which we will be able

00:08:18.500 --> 00:08:20.370
to use later on.

00:08:20.370 --> 00:08:22.670
But this, I guess, might
naturally lead to the

00:08:22.670 --> 00:08:26.800
question, what if we go --
everything we have been

00:08:26.800 --> 00:08:30.400
dealing with so far have been
subsets, with the case of this

00:08:30.400 --> 00:08:33.659
triangle, or subspaces,
in the case of V.

00:08:33.659 --> 00:08:42.309
But what if I were to take the
image Rn under T, right?

00:08:42.309 --> 00:08:48.550
This is the image
of Rn under T.

00:08:48.550 --> 00:08:50.350
Let's think about
what this means.

00:08:50.350 --> 00:08:55.200
This means, what do we get when
we take any member of Rn,

00:08:55.200 --> 00:08:57.270
what is the set of all
of the vectors?

00:08:57.270 --> 00:08:59.610
Then when we take the
transformation of all of the

00:08:59.610 --> 00:09:01.340
members of Rn, let
me write this.

00:09:01.340 --> 00:09:06.400
This is equal to the set of the
transformation of all of

00:09:06.400 --> 00:09:10.750
the x's, where each x
is a member of Rn.

00:09:10.750 --> 00:09:13.240
So you take each of the members
of Rn and transform

00:09:13.240 --> 00:09:15.020
them, and you create
this new set.

00:09:15.020 --> 00:09:17.910
This is the image
of Rn under T.

00:09:21.385 --> 00:09:24.490
Well, there's a couple of ways
you can think of this.

00:09:24.490 --> 00:09:29.260
Remember when we defined
-- let's see, T is a

00:09:29.260 --> 00:09:33.680
mapping from Rn to Rm.

00:09:33.680 --> 00:09:36.330
We defined this as the domain.

00:09:36.330 --> 00:09:39.715
All of the possible inputs
for our transformation.

00:09:39.715 --> 00:09:43.450
And we define this
as the codomain.

00:09:43.450 --> 00:09:46.900
And remember I told you that
the codomain is essentially

00:09:46.900 --> 00:09:49.190
part of the definition of
the function or of the

00:09:49.190 --> 00:09:52.110
transformation, and it's the
space that we map to.

00:09:52.110 --> 00:09:53.610
It's not necessarily all
of the things that

00:09:53.610 --> 00:09:54.670
we're mapping to.

00:09:54.670 --> 00:09:58.350
For example, the image of Rn
under transformation, maybe

00:09:58.350 --> 00:10:01.320
it's all of Rm or maybe it's
some subset of Rn.

00:10:01.320 --> 00:10:03.190
The way you can think about it,
and I touched on this in

00:10:03.190 --> 00:10:05.910
that first video, is-- and
they'll never, or at least the

00:10:05.910 --> 00:10:09.000
linear algebra books I looked
at, they didn't specify this--

00:10:09.000 --> 00:10:15.070
but you can kind of view
this as the range of T.

00:10:15.070 --> 00:10:24.190
These are the actual members
of Rm that T maps to.

00:10:24.190 --> 00:10:28.210
That if you take the image of
Rn under T, you are actually

00:10:28.210 --> 00:10:35.190
finding-- let's say that
Rm looks like that.

00:10:35.190 --> 00:10:37.310
Obviously it will go
in every direction.

00:10:37.310 --> 00:10:39.780
And let's say that when
you take-- let me

00:10:39.780 --> 00:10:41.050
draw Rn right here.

00:10:44.670 --> 00:10:47.140
And we know that T is a
mapping from Rn to Rm.

00:10:47.140 --> 00:10:51.450
But let's say when you take
every element of Rn and you

00:10:51.450 --> 00:10:56.140
map them into Rm, let's say
you get some subset of Rm,

00:10:56.140 --> 00:10:59.300
let's say you get something
that looks like this.

00:10:59.300 --> 00:11:01.682
So let me see if I can
draw this nicely.

00:11:01.682 --> 00:11:08.270
So you literally map every point
here, and it goes to one

00:11:08.270 --> 00:11:08.860
of these guys.

00:11:08.860 --> 00:11:12.200
Or one of these guys can be
represented as a mapping from

00:11:12.200 --> 00:11:14.760
one of these members
right here.

00:11:14.760 --> 00:11:18.080
So if you map all of them you
get this subset right here.

00:11:18.080 --> 00:11:26.040
This subset is, this is
T the image of Rn, the

00:11:26.040 --> 00:11:28.580
image of Rn under T.

00:11:28.580 --> 00:11:30.460
And in the terminology that
you don't normally see in

00:11:30.460 --> 00:11:31.790
linear algebra a lot,
you can also kind of

00:11:31.790 --> 00:11:33.830
consider it its range.

00:11:33.830 --> 00:11:37.760
The range of T.

00:11:37.760 --> 00:11:40.320
Now, this has a special name.

00:11:40.320 --> 00:11:42.590
This is called -- and I don't
want you to get confused --

00:11:42.590 --> 00:11:45.480
this is called the image of T.

00:11:45.480 --> 00:11:46.880
Image of T.

00:11:46.880 --> 00:11:50.480
This might be a little
confusing, image of T.

00:11:50.480 --> 00:11:57.480
So this is sometimes written
as just im of T.

00:11:57.480 --> 00:11:59.580
Now you are a little confused
here, you are like, before

00:11:59.580 --> 00:12:02.010
when we were talking about
subsets, we would call this

00:12:02.010 --> 00:12:05.770
the image of R subset under T.

00:12:05.770 --> 00:12:07.430
And that is the correct
terminology when you're

00:12:07.430 --> 00:12:08.400
dealing with a subset.

00:12:08.400 --> 00:12:10.920
But when you take, all of
a sudden, the entire n

00:12:10.920 --> 00:12:14.160
dimensional space, and you're
finding that image, we call

00:12:14.160 --> 00:12:17.170
that the image of the actual
transformation.

00:12:17.170 --> 00:12:27.160
So we can also call this set
right here the image of T.

00:12:27.160 --> 00:12:29.980
And now what is the
image of T?

00:12:29.980 --> 00:12:33.020
Well, we know that we can
write any-- and this is

00:12:33.020 --> 00:12:35.990
literally any-- so T is
going from Rn to Rm.

00:12:35.990 --> 00:12:39.900
We can write T of x-- we
can write any linear

00:12:39.900 --> 00:12:44.530
transformation like this-- as
being equal to some matrix,

00:12:44.530 --> 00:12:51.690
some m by n matrix
times a vector.

00:12:51.690 --> 00:12:53.790
And these vectors obviously
are going to be members of

00:12:53.790 --> 00:12:56.810
Rn-- times sum Rn.

00:12:56.810 --> 00:12:58.450
And what is this?

00:12:58.450 --> 00:13:03.310
So what is the image -- let
me write it in a bunch of

00:13:03.310 --> 00:13:12.290
different ways -- what is
the image of Rn under T?

00:13:12.290 --> 00:13:17.180
So we could write that as T --
let me write it this way.

00:13:17.180 --> 00:13:23.780
We could write that as T of Rn,
which is the same thing as

00:13:23.780 --> 00:13:28.130
the image of T.

00:13:28.130 --> 00:13:29.820
Notice we're not saying under
anything else, because now

00:13:29.820 --> 00:13:32.200
were saying the image of the
actual transformation.

00:13:32.200 --> 00:13:39.540
Which we could also write
as the image of T.

00:13:39.540 --> 00:13:42.850
Well what are these equal to?

00:13:42.850 --> 00:13:47.610
This is equal to the set of all
the transformations of x.

00:13:47.610 --> 00:13:53.422
Well all the transformations of
x are going to be Ax where

00:13:53.422 --> 00:13:59.350
x is a member of Rn.

00:13:59.350 --> 00:14:04.420
So x is going to be an n-tuple,
where each element

00:14:04.420 --> 00:14:05.770
has to be a real number.

00:14:05.770 --> 00:14:07.060
So what is this?

00:14:07.060 --> 00:14:12.880
So if we write A-- let
me write my matrix A.

00:14:12.880 --> 00:14:17.680
It's just a bunch of column
vectors, a1, a2.

00:14:17.680 --> 00:14:19.430
It's going to have n
of these, right?

00:14:19.430 --> 00:14:22.000
Because it has n columns.

00:14:22.000 --> 00:14:26.060
And so a times any x is going to
be-- so if I multiply that

00:14:26.060 --> 00:14:28.550
times any x that's
a member of Rn.

00:14:28.550 --> 00:14:32.340
I multiply x1, x2, all
the way to xn.

00:14:32.340 --> 00:14:34.220
We've seen this multiple,
multiple times.

00:14:34.220 --> 00:14:41.390
This is equal to x1-- the scalar
x1, times a1, plus x2

00:14:41.390 --> 00:14:47.900
times a2, all the way
to plus xn times an.

00:14:47.900 --> 00:14:51.200
And we're saying we want the
set of all of these sums of

00:14:51.200 --> 00:14:55.400
these column vectors, where x
can take on any vector in Rn.

00:14:55.400 --> 00:14:59.280
Which means that the elements
of x can take on any real

00:14:59.280 --> 00:15:00.730
scalar values.

00:15:00.730 --> 00:15:04.010
So the set of all of these is
essentially all of the linear

00:15:04.010 --> 00:15:06.480
combinations of the columns
of a, right?

00:15:06.480 --> 00:15:10.380
Because I can set these guys
to be equal to any value.

00:15:10.380 --> 00:15:12.040
So what is that equal to?

00:15:12.040 --> 00:15:14.400
That is equal to, and we
touched on this, or we

00:15:14.400 --> 00:15:17.340
actually talked about this when
we introduced the idea.

00:15:17.340 --> 00:15:30.670
This is equal to the
column space of A.

00:15:30.670 --> 00:15:36.570
Or we just denoted it
sometimes as C of A.

00:15:36.570 --> 00:15:38.520
So that's a pretty
neat result.

00:15:38.520 --> 00:15:42.080
If you take -- it's almost
obvious, I mean it's just I'm

00:15:42.080 --> 00:15:43.950
playing with words a little
bit-- but any linear

00:15:43.950 --> 00:15:46.530
transformation can be
represented as a

00:15:46.530 --> 00:15:48.520
matrix vector product.

00:15:48.520 --> 00:15:52.040
And so the image of any linear
transformation, which means

00:15:52.040 --> 00:15:55.460
the subset of its codomain,
when you map all of the

00:15:55.460 --> 00:15:58.280
elements of its domain into
its codomain, this is the

00:15:58.280 --> 00:16:00.530
image of your transformation.

00:16:00.530 --> 00:16:09.900
This is equivalent to the column
space of the matrix

00:16:09.900 --> 00:16:13.110
that you're transformation
could be represented as.

00:16:13.110 --> 00:16:18.620
And the column space, of course,
is the span of all the

00:16:18.620 --> 00:16:24.790
column vectors of your matrix.

00:16:24.790 --> 00:16:27.880
This is just all of the linear
combinations, or the span, of

00:16:27.880 --> 00:16:30.860
all of your column vectors,
which we do right here.

00:16:30.860 --> 00:16:33.132
Anyway hope you found that a
little interesting, and you

00:16:33.132 --> 00:16:35.710
will be able to use these
results in the future.


Image of a subset under a transformation

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.815
Let's say I have three position
vectors here in R2.

00:00:03.815 --> 00:00:06.180
Let me scroll this over
a little bit.

00:00:06.180 --> 00:00:11.900
Let's say my first position
vector is x0 and it is equal

00:00:11.900 --> 00:00:17.140
to minus 2, minus 2.

00:00:17.140 --> 00:00:20.400
So if I were to graph x0 I would
go minus 2, minus 2.

00:00:20.400 --> 00:00:24.430
x0 looks like that.

00:00:24.430 --> 00:00:29.420
My next position vector I have
is x1 and I'll say that's

00:00:29.420 --> 00:00:33.002
equal to minus 2, 2.

00:00:33.002 --> 00:00:36.130
If I were to graph
here, minus 2, 2.

00:00:36.130 --> 00:00:39.740
That's my next position
vector right there.

00:00:39.740 --> 00:00:41.180
And this is x1.

00:00:41.180 --> 00:00:43.860
And when I say it's a position
vector, they specify a

00:00:43.860 --> 00:00:46.510
specific coordinate in R2.

00:00:46.510 --> 00:00:50.370
Let me draw a third one
just for fun, x2.

00:00:50.370 --> 00:00:58.400
Let's say that that is
equal to 2, minus 2.

00:00:58.400 --> 00:01:02.600
So if I were to draw this, 2
minus 2, it goes right here.

00:01:02.600 --> 00:01:06.860
So that vector right
there is x2.

00:01:06.860 --> 00:01:09.070
Now, what I'm curious
about or, I

00:01:09.070 --> 00:01:10.560
guess, not curious about.

00:01:10.560 --> 00:01:13.570
What I want to do here is define
the line segments that

00:01:13.570 --> 00:01:14.900
connect these points.

00:01:14.900 --> 00:01:17.820
So let's say I have my
first line segment.

00:01:17.820 --> 00:01:22.350
Let me call it L1, or
let me call it L0.

00:01:22.350 --> 00:01:23.860
And I want it to be the
line segment that

00:01:23.860 --> 00:01:25.350
connects x0 to x1.

00:01:25.350 --> 00:01:27.560
How can I construct that?

00:01:27.560 --> 00:01:30.650
So I want to construct this line
segment right here, this

00:01:30.650 --> 00:01:33.240
little-- let me do it in a
different color actually.

00:01:33.240 --> 00:01:35.390
Let me do it in orange, L0.

00:01:35.390 --> 00:01:38.540
So what I want to do is I want
to find the set of all of

00:01:38.540 --> 00:01:41.370
these values right here, all of
the position vectors that

00:01:41.370 --> 00:01:44.490
define points on this
line right there.

00:01:44.490 --> 00:01:48.890
Well we could define it as,
we could start off at x0.

00:01:48.890 --> 00:01:55.310
We could say that orange line is
x0 plus scaled versions of

00:01:55.310 --> 00:01:57.580
the difference of x1 and x0.

00:01:57.580 --> 00:02:00.740
If you take x1 minus x0, you get
this vector right there.

00:02:00.740 --> 00:02:04.530
That's x1 minus x0.

00:02:04.530 --> 00:02:05.470
That orange vector.

00:02:05.470 --> 00:02:07.080
I know I wrote it right there
so it's hard to read.

00:02:07.080 --> 00:02:09.940
But if you just take x1 minus x0
you get that and that make

00:02:09.940 --> 00:02:11.990
sense. x0 plus this orange
vector is equal

00:02:11.990 --> 00:02:13.270
to that blue factor.

00:02:13.270 --> 00:02:16.130
So if you just take different
scaled up versions of this

00:02:16.130 --> 00:02:17.640
guy, you're going to
end up at different

00:02:17.640 --> 00:02:19.810
points in this direction.

00:02:19.810 --> 00:02:23.020
We're starting at x0, maybe
I should do that in green.

00:02:23.020 --> 00:02:26.340
We're starting at x0 and then
we're going to add up scaled

00:02:26.340 --> 00:02:29.540
up versions of this orange
vector, which is just the

00:02:29.540 --> 00:02:31.550
difference of x1 and x0.

00:02:31.550 --> 00:02:32.255
Let me write that.

00:02:32.255 --> 00:02:39.230
So scaled up versions
of x1 minus x0.

00:02:39.230 --> 00:02:40.850
Now, we have to constrict.

00:02:40.850 --> 00:02:43.220
If we just want to be in this
line segment, we have to

00:02:43.220 --> 00:02:44.200
constrict Rt.

00:02:44.200 --> 00:02:47.300
If we said t was a member of
real numbers, if it was any

00:02:47.300 --> 00:02:49.530
real number, then we would
essentially be defining the

00:02:49.530 --> 00:02:53.700
set of this entire vertical
line going up and down to

00:02:53.700 --> 00:02:56.990
infinity in the upwards
direction and

00:02:56.990 --> 00:02:57.810
the downwards direction.

00:02:57.810 --> 00:03:00.210
But we just want to restrict
it to start here

00:03:00.210 --> 00:03:01.110
and then go up here.

00:03:01.110 --> 00:03:04.740
And it doesn't necessarily have
to have any direction.

00:03:04.740 --> 00:03:07.900
We could say this is true, our
little line segment here is

00:03:07.900 --> 00:03:12.010
true, for t-- let me
write it this way.

00:03:12.010 --> 00:03:15.160
t is greater than
or equal to 0.

00:03:15.160 --> 00:03:18.410
So when t equals 0, this term
cancels out and we just have

00:03:18.410 --> 00:03:21.140
this point or this position
right here.

00:03:21.140 --> 00:03:23.410
Let me draw it in green.

00:03:23.410 --> 00:03:25.070
We just have that
position there.

00:03:25.070 --> 00:03:28.140
And then t is going to be
less than or equal to 1.

00:03:28.140 --> 00:03:30.060
What happens when
t is equal to 1?

00:03:30.060 --> 00:03:33.960
When t is equal to 1, this
becomes x1 minus x0.

00:03:33.960 --> 00:03:35.030
You have an x0 there.

00:03:35.030 --> 00:03:37.980
This x0 and that x0 cancel out
and you're just at this point

00:03:37.980 --> 00:03:38.710
right there.

00:03:38.710 --> 00:03:41.890
When t is equal to 1/2, just
to make sure this all make

00:03:41.890 --> 00:03:43.030
sense to you, what happens?

00:03:43.030 --> 00:03:47.080
You have x1 minus x0, which is
this orange vector right here.

00:03:47.080 --> 00:03:49.320
When t is equal to 1/2 you're
essentially scaling that

00:03:49.320 --> 00:03:52.480
orange vector by half and you
end up right at that point,

00:03:52.480 --> 00:03:53.710
which is exactly where
you want to be.

00:03:53.710 --> 00:03:55.290
You want halfway along
that line segment.

00:03:55.290 --> 00:03:57.810
At t is equal to 0.25, you're
going to be here.

00:03:57.810 --> 00:03:59.670
t is equal to 0.75, you're
going to be there.

00:03:59.670 --> 00:04:03.890
So at any value for t being any
real number between 0 and

00:04:03.890 --> 00:04:06.800
1, you're going to end up at all
of the points along that

00:04:06.800 --> 00:04:07.520
line segment.

00:04:07.520 --> 00:04:10.500
So that's our L0.

00:04:10.500 --> 00:04:13.510
It's just a set of vectors.

00:04:13.510 --> 00:04:16.260
Now we can do the same exercise
if we wanted to find

00:04:16.260 --> 00:04:19.700
out the line, the equation
of the line, that goes

00:04:19.700 --> 00:04:23.860
between x1 and x2.

00:04:23.860 --> 00:04:25.970
If we wanted to find the
equation of that line, we

00:04:25.970 --> 00:04:27.220
could call this L1.

00:04:30.330 --> 00:04:41.140
And L1 would be equal to x1 plus
t times x2 minus x1 for 0

00:04:41.140 --> 00:04:43.640
is less than or equal to t is
less than or equal to 1.

00:04:43.640 --> 00:04:44.740
That's L1.

00:04:44.740 --> 00:04:47.140
And then finally, if we want
to make a triangle out of

00:04:47.140 --> 00:04:50.100
this, let's define this
line right here.

00:04:50.100 --> 00:04:52.320
Let's define that as L2.

00:04:52.320 --> 00:04:56.270
L2 would be equal to the set
of all of vectors where you

00:04:56.270 --> 00:04:59.530
start off at x2.

00:04:59.530 --> 00:05:03.860
Set of all of vectors that are
x2 plus some scaled up sum of

00:05:03.860 --> 00:05:05.230
x0 minus x2.

00:05:05.230 --> 00:05:07.650
x0 minus x2 is this
vector right here.

00:05:07.650 --> 00:05:16.020
So x0 minus x2 such that 0 is
less than or equal to t is

00:05:16.020 --> 00:05:18.430
less than or equal to 1.

00:05:18.430 --> 00:05:20.900
And so if you take the
combination, if you were to

00:05:20.900 --> 00:05:27.620
define kind of a super set-- I
could have defined my shape

00:05:27.620 --> 00:05:33.070
as-- let's say it's the union
of all of those guys.

00:05:33.070 --> 00:05:34.310
Well, let me just write it.

00:05:34.310 --> 00:05:37.410
L0, L1, and L2.

00:05:37.410 --> 00:05:40.550
Then you'd have a nice
triangle here.

00:05:40.550 --> 00:05:44.670
If you take the union of all of
these three sets, you get

00:05:44.670 --> 00:05:46.080
that nice triangle there.

00:05:46.080 --> 00:05:48.790
Now, what I want to do in this
video, I think this is all a

00:05:48.790 --> 00:05:49.640
bit of review for you.

00:05:49.640 --> 00:05:51.510
But it's maybe a different way
of looking at things than

00:05:51.510 --> 00:05:53.550
we've done in the past. Is
I want to understand what

00:05:53.550 --> 00:05:59.240
happens to this set right here
when I take a transformation,

00:05:59.240 --> 00:06:01.580
a linear transformation,
of it?

00:06:01.580 --> 00:06:04.630
So let me define a
transformation.

00:06:04.630 --> 00:06:07.100
I'll make it a fairly
straightforward

00:06:07.100 --> 00:06:08.690
transformation.

00:06:08.690 --> 00:06:12.730
Let me define my transformation
of x, of any x,

00:06:12.730 --> 00:06:21.630
to be equal to the matrix
1 minus 1, 2, 0 times

00:06:21.630 --> 00:06:22.970
whatever vector x.

00:06:22.970 --> 00:06:24.960
So times x1, x2.

00:06:24.960 --> 00:06:27.680
And we know that any linear
transformation can actually be

00:06:27.680 --> 00:06:30.010
written as a matrix
and vice versa.

00:06:30.010 --> 00:06:31.730
So you might have said, hey,
you know, you're giving an

00:06:31.730 --> 00:06:33.580
example with the matrix, what
about all those other ways to

00:06:33.580 --> 00:06:34.320
write in your transformation?

00:06:34.320 --> 00:06:36.580
You can write all of
those as a matrix.

00:06:36.580 --> 00:06:39.580
So let's translate-- let's try
to figure out what this is

00:06:39.580 --> 00:06:41.410
going to look like.

00:06:41.410 --> 00:06:45.060
What our triangle is going to
look like when we transform

00:06:45.060 --> 00:06:48.700
every point in it.

00:06:48.700 --> 00:06:52.650
Let me take the transformation
first. The transformation of

00:06:52.650 --> 00:06:59.710
L0 is equal to the
transformation of this thing.

00:06:59.710 --> 00:07:02.050
This is just one of the
particular members.

00:07:02.050 --> 00:07:03.730
For a particular t,
this is one of the

00:07:03.730 --> 00:07:04.830
particular members of L0.

00:07:04.830 --> 00:07:12.900
So it's going to be equal to the
transformation of x0 minus

00:07:12.900 --> 00:07:19.950
the transformation of x1 minus
x0 such that-- sorry.

00:07:19.950 --> 00:07:22.130
Minus t times x1 minus x0.

00:07:22.130 --> 00:07:24.250
That's a lowercase t, not
the transformation.

00:07:24.250 --> 00:07:27.590
Such that 0 is less than or
equal to t is less than or

00:07:27.590 --> 00:07:28.770
equal to 1.

00:07:28.770 --> 00:07:30.280
Let me switch colors.

00:07:30.280 --> 00:07:33.720
This, just by the properties
of linear transformations,

00:07:33.720 --> 00:07:35.920
this is equal to the
transformation-- let me put

00:07:35.920 --> 00:07:41.730
the brackets out-- of x0 minus
the transformation of our

00:07:41.730 --> 00:07:51.000
scalar t times x1 minus x0 for
all t's between 0 and 1.

00:07:51.000 --> 00:07:53.750
That part is getting a little
redundant to keep saying it.

00:07:53.750 --> 00:07:55.540
And then, what does
this equal to?

00:07:55.540 --> 00:07:57.420
If you I take the transformation
of a scaled up

00:07:57.420 --> 00:07:58.710
vector, that's just
the scaled up

00:07:58.710 --> 00:07:59.970
transformation of that vector.

00:07:59.970 --> 00:08:03.290
So this is going to be equal
to this part, the

00:08:03.290 --> 00:08:10.250
transformation of x0 minus t,
our scalar multiple t, times

00:08:10.250 --> 00:08:15.200
the transformation of the
vector x1 minus x0.

00:08:15.200 --> 00:08:17.250
And then let me make sure I
get my parentheses right.

00:08:17.250 --> 00:08:19.730
Such that 0 is less than or
equal to t is less than or

00:08:19.730 --> 00:08:20.590
equal to 1.

00:08:20.590 --> 00:08:23.510
And then the transformation of
the sum of two vectors is

00:08:23.510 --> 00:08:25.370
equal to the sum of their
transformations.

00:08:25.370 --> 00:08:27.070
We've all seen this before.

00:08:27.070 --> 00:08:31.370
So our transformation of our
first line-- this one right

00:08:31.370 --> 00:08:37.669
here-- L0, is equal to the set
where it's the transformation

00:08:37.669 --> 00:08:46.830
of x0 minus t times the
transformation of x1 minus the

00:08:46.830 --> 00:08:49.580
transformation of x0.

00:08:49.580 --> 00:08:52.420
And we've just done our
first line so far.

00:08:52.420 --> 00:08:54.070
I have a parentheses there.

00:08:54.070 --> 00:08:57.820
For 0 is less than t is less
than or equal to 1.

00:08:57.820 --> 00:09:00.010
Now this is a pretty neat
result and it's going to

00:09:00.010 --> 00:09:01.950
simplify our lives a lot.

00:09:01.950 --> 00:09:11.220
The transformation of the line
segment that goes from x0 to

00:09:11.220 --> 00:09:16.420
x1 ends up just being the line
segment that goes from the

00:09:16.420 --> 00:09:20.450
transformation of x0 to the
transformation of x1.

00:09:20.450 --> 00:09:21.170
Let me make this clear.

00:09:21.170 --> 00:09:22.750
What is the transformation
of x0?

00:09:22.750 --> 00:09:24.680
Let's calculate these things.

00:09:24.680 --> 00:09:29.630
So x0 was minus 2, minus 2.

00:09:29.630 --> 00:09:33.030
Let me write out the
transformation of x0.

00:09:33.030 --> 00:09:38.500
So the transformation of x0 is
equal to-- let me write it

00:09:38.500 --> 00:09:40.130
out, so I don't make any
careless mistakes.

00:09:40.130 --> 00:09:45.920
1, 2, minus 1, 0 times minus
2 minus, minus 2.

00:09:45.920 --> 00:09:47.990
And so what would this
be equal to?

00:09:47.990 --> 00:09:52.860
1 minus 2 minus-- so it's
1 minus 2 plus minus

00:09:52.860 --> 00:09:53.760
1 times minus 2.

00:09:53.760 --> 00:09:54.670
That's going to be plus 2.

00:09:54.670 --> 00:09:57.910
So it's minus 2 plus a plus
2, so it's equal to 0.

00:09:57.910 --> 00:10:02.200
And then we have 2 times minus
2, which is minus 4.

00:10:02.200 --> 00:10:03.280
2 times minus 4.

00:10:03.280 --> 00:10:06.000
And then plus 0 times,
so it's minus 4.

00:10:06.000 --> 00:10:08.345
So that's the transformation
of x0.

00:10:08.345 --> 00:10:09.940
Let me graph it.

00:10:09.940 --> 00:10:12.690
So it's 0 minus 4.

00:10:12.690 --> 00:10:15.400
So our x0 vector.

00:10:15.400 --> 00:10:18.260
So this is the transformation
of x0.

00:10:18.260 --> 00:10:23.480
So the transformation associated
this vector with

00:10:23.480 --> 00:10:26.300
this vector down here, the one
that goes straight down.

00:10:26.300 --> 00:10:27.420
Now let me take the
transformation

00:10:27.420 --> 00:10:28.930
of the other guys.

00:10:28.930 --> 00:10:30.480
The transformation of x1.

00:10:30.480 --> 00:10:32.110
I'll just do it right here.

00:10:32.110 --> 00:10:33.660
I'm running out of space.

00:10:33.660 --> 00:10:39.420
The transformation of x1 is
equal to 1, 2, minus 1, 0

00:10:39.420 --> 00:10:42.685
times minus 2, 2.

00:10:46.830 --> 00:10:48.520
So what is this equal to?

00:10:48.520 --> 00:10:53.580
This is equal to 1 times minus
2, plus minus 1 times 2.

00:10:53.580 --> 00:10:55.460
So that's minus 4.

00:10:55.460 --> 00:10:59.520
And then 2 times minus
2 is minus 4.

00:10:59.520 --> 00:11:00.190
Plus 0.

00:11:00.190 --> 00:11:04.180
So minus 4, minus 4.

00:11:04.180 --> 00:11:07.390
So x1 is minus 4, minus 4.

00:11:07.390 --> 00:11:10.020
So our x1 looks like this.

00:11:10.020 --> 00:11:15.290
Our transformation of x1 is this
vector right here in R2.

00:11:15.290 --> 00:11:18.250
Our transformation is
going from R2 to R2.

00:11:18.250 --> 00:11:21.560
So that's why I'm able to draw
them both on this nice

00:11:21.560 --> 00:11:23.010
Cartesian coordinate plane.

00:11:23.010 --> 00:11:24.380
And we have one left.

00:11:24.380 --> 00:11:27.490
Let's take our transformation
of x2.

00:11:27.490 --> 00:11:33.820
So the transformation of x2 is
equal to our transformation

00:11:33.820 --> 00:11:41.760
matrix 1, 2, minus 1,
0 times 2, minus 2.

00:11:41.760 --> 00:11:46.610
And so this will be equal
to 1 times 2 is 2.

00:11:46.610 --> 00:11:48.630
Plus minus 1 times minus 2.

00:11:48.630 --> 00:11:50.940
So it's 2 plus 2 is 4.

00:11:50.940 --> 00:11:53.440
And then have 2 times 2 is 4.

00:11:53.440 --> 00:11:55.770
Plus 0 times minus 2.

00:11:55.770 --> 00:11:58.560
So it's 4, 4.

00:11:58.560 --> 00:12:00.420
So x2 is 4, 4.

00:12:00.420 --> 00:12:02.020
So it's this point right here.

00:12:02.020 --> 00:12:04.130
4, 4 right there.

00:12:07.030 --> 00:12:11.030
So the transformation of x2 is
that vector right there.

00:12:11.030 --> 00:12:14.340
And so, we are able to take the
transformation of each of

00:12:14.340 --> 00:12:16.470
these points of this triangle.

00:12:16.470 --> 00:12:19.110
But who knows what the
transformation does to

00:12:19.110 --> 00:12:21.590
everything in between, to all
of these other-- the actual

00:12:21.590 --> 00:12:23.220
sides of the triangle.

00:12:23.220 --> 00:12:24.940
We're able to do a little
math and we just

00:12:24.940 --> 00:12:25.910
did the first side.

00:12:25.910 --> 00:12:29.230
We just did L0 right there
we found, just using the

00:12:29.230 --> 00:12:31.510
properties of a linear
transformation, the definition

00:12:31.510 --> 00:12:34.070
of a linear transformation
actually, we were able to find

00:12:34.070 --> 00:12:39.480
that the transformation of L0 of
this vertical line here, it

00:12:39.480 --> 00:12:43.170
just ends up becoming the line
where we can start off at the

00:12:43.170 --> 00:12:44.550
transformation of x0.

00:12:47.140 --> 00:12:49.780
The point specified by this
vector right here.

00:12:49.780 --> 00:12:52.540
And to that I add up scaled
multiples of the

00:12:52.540 --> 00:12:56.040
transformation of x1 minus
the transformation of x0.

00:12:56.040 --> 00:13:00.560
What is this, the transformation
of x1 minus the

00:13:00.560 --> 00:13:02.110
transformation of x0?

00:13:02.110 --> 00:13:05.450
The transformation of x1 is just
this vector right here.

00:13:05.450 --> 00:13:07.880
The transformation of x0
is just that vector.

00:13:07.880 --> 00:13:10.800
So this whole term right here is
just this vector minus that

00:13:10.800 --> 00:13:14.170
vectors or it's this
vector right there.

00:13:14.170 --> 00:13:15.920
It's just that vector
right there.

00:13:15.920 --> 00:13:18.960
And so, what we essentially
have, we've defined the same

00:13:18.960 --> 00:13:21.430
way that we did in the first
part of this video.

00:13:21.430 --> 00:13:24.900
This is just the same thing as
the line segment that connects

00:13:24.900 --> 00:13:27.870
the point defined here and
the point defined there.

00:13:27.870 --> 00:13:30.390
We took the difference of the
two and we have scaled up

00:13:30.390 --> 00:13:33.930
versions of that between
t is equal to 0 and 1.

00:13:33.930 --> 00:13:36.480
So the transformation of L0
really just became the

00:13:36.480 --> 00:13:39.040
transformation-- is just
the line between the

00:13:39.040 --> 00:13:42.260
transformations of both of the
endpoints, which is a pretty

00:13:42.260 --> 00:13:43.210
neat result.

00:13:43.210 --> 00:13:45.530
It makes our lives simple.

00:13:45.530 --> 00:13:47.910
We can do the exact same logic
to say, you know what?

00:13:47.910 --> 00:13:52.880
What's going to be the
transformation of L1?

00:13:52.880 --> 00:13:57.130
Well L1 was between the
points x1 and x2.

00:13:57.130 --> 00:13:59.940
It was between that point
and that point.

00:13:59.940 --> 00:14:02.060
That was L1.

00:14:02.060 --> 00:14:04.020
So using the same logic,
we can do the

00:14:04.020 --> 00:14:05.160
math all over again.

00:14:05.160 --> 00:14:06.540
But it applies to any line.

00:14:06.540 --> 00:14:08.500
I did it all abstractly here.

00:14:08.500 --> 00:14:13.030
The transformation of L1 is
going to be the line that

00:14:13.030 --> 00:14:15.420
connects the transformation
of the two endpoints.

00:14:15.420 --> 00:14:16.830
So it's going to be the
line that connects the

00:14:16.830 --> 00:14:19.930
transformation of x1 and the
transformation of x2.

00:14:22.880 --> 00:14:27.630
Let me make this right here, is
the transformation of L1.

00:14:27.630 --> 00:14:32.850
This right here is the
transformation of L0.

00:14:32.850 --> 00:14:36.190
And then finally, what's the
transformation of L2?

00:14:38.750 --> 00:14:43.330
L2 connects the points
x2 and x0.

00:14:43.330 --> 00:14:45.760
So that is L2 right there.

00:14:45.760 --> 00:14:48.590
So the transformation of it
using the same math that we've

00:14:48.590 --> 00:14:51.140
done before is really just
the line connecting the

00:14:51.140 --> 00:14:53.540
transformations of
those two points.

00:14:53.540 --> 00:14:56.950
So the transformation of L2 is
going to be equal to the line

00:14:56.950 --> 00:14:59.410
that connects the transformation
of x2 to the

00:14:59.410 --> 00:15:00.920
transformation of x0.

00:15:00.920 --> 00:15:02.570
So it's going to be that
line right there.

00:15:05.390 --> 00:15:11.120
So this is the transformation
of L2, or if we defined our

00:15:11.120 --> 00:15:14.060
whole shape or our whole
triangle as the set of all of

00:15:14.060 --> 00:15:16.160
these, the transformation
of that.

00:15:16.160 --> 00:15:20.750
So the transformation of
our whole shape is

00:15:20.750 --> 00:15:22.880
now this skewed triangle.

00:15:22.880 --> 00:15:24.900
I think you're now getting a
sense of why this might be

00:15:24.900 --> 00:15:28.020
useful in things like computer
graphics or game development.

00:15:28.020 --> 00:15:29.920
Because when you look at things
from different angles,

00:15:29.920 --> 00:15:32.160
you start to skew them
and whatever else.

00:15:32.160 --> 00:15:34.920
But taking this transformation,
we were able

00:15:34.920 --> 00:15:41.280
to turn this set of vectors or
the positions-- or I guess

00:15:41.280 --> 00:15:44.860
this shape, which is specified
by this set of

00:15:44.860 --> 00:15:45.660
vectors right here.

00:15:45.660 --> 00:15:48.945
We were able to change it into
this shape in R2 specified by

00:15:48.945 --> 00:15:50.970
a different set of vectors.

00:15:50.970 --> 00:15:55.610
But the whole takeaway from this
video is, you don't have

00:15:55.610 --> 00:16:00.280
to individually figure out,
gee, what does this point

00:16:00.280 --> 00:16:02.540
right here translate
into over here?

00:16:02.540 --> 00:16:04.450
All you have to say is what
were my endpoints?

00:16:04.450 --> 00:16:06.450
Figure out their transformations
and then

00:16:06.450 --> 00:16:08.140
connect the dots in
the same order.

00:16:08.140 --> 00:16:11.470
That's what is essentially the
big takeaway from here.

00:16:11.470 --> 00:16:15.350
And this idea of when you
transform one set into another

00:16:15.350 --> 00:16:17.690
set, they have some terminology

00:16:17.690 --> 00:16:19.100
associated with it.

00:16:19.100 --> 00:16:21.760
So let's say the transformation
of L0.

00:16:21.760 --> 00:16:25.010
L0 was the set of vectors
that specified

00:16:25.010 --> 00:16:27.240
this line right here.

00:16:27.240 --> 00:16:37.000
The transformation of L0, which
is this set, is a set of

00:16:37.000 --> 00:16:38.190
vectors-- sorry.

00:16:38.190 --> 00:16:42.000
L0 was this set.

00:16:42.000 --> 00:16:46.720
It was the set of vectors in our
co-domain that specified

00:16:46.720 --> 00:16:48.000
these points.

00:16:48.000 --> 00:16:58.780
This is called the image
of L0 under T.

00:16:58.780 --> 00:17:01.490
And it kind of makes sense.

00:17:01.490 --> 00:17:02.750
Why do they call it the image?

00:17:02.750 --> 00:17:08.859
Because t is taking this thing
right here, this L0, and kind

00:17:08.859 --> 00:17:11.780
of distorting it or creating
a new image

00:17:11.780 --> 00:17:13.810
of it in the co-domain.

00:17:13.810 --> 00:17:16.540
It's taking a set in the domain
and creating a new

00:17:16.540 --> 00:17:19.939
image of it in the co-domain
right there.

00:17:22.589 --> 00:17:27.250
We could say that t of the
transformation of our entire

00:17:27.250 --> 00:17:31.400
shape, I defined our entire
shape up here as this whole

00:17:31.400 --> 00:17:32.920
triangle right there.

00:17:32.920 --> 00:17:36.440
That's the image and that
turned into this purple

00:17:36.440 --> 00:17:37.630
triangle here.

00:17:37.630 --> 00:17:43.710
That's the image of s under T.

00:17:43.710 --> 00:17:45.060
So hopefully you found that
pretty interesting.

00:17:45.060 --> 00:17:47.080
This is actually a super useful
takeaway if you ever

00:17:47.080 --> 00:17:50.740
want to become a 3D programmer
of some type.

00:17:50.740 --> 00:17:54.580
In then next video, we'll
explore what happens when s is

00:17:54.580 --> 00:17:57.590
no longer just a subset
of our domain.

00:17:57.590 --> 00:18:01.120
Everything we've been dealing
so far-- L0, L1, and L2, or

00:18:01.120 --> 00:18:04.380
our entire triangle, these
were all subsets or Rn.

00:18:04.380 --> 00:18:06.040
In the next video we'll talk
about what happens when you

00:18:06.040 --> 00:18:08.660
take the transformation
of all of Rn.