Using matrices to transform a 4D vector

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:01.590
- [Trainer] We've already
thought a lot about

00:00:01.590 --> 00:00:05.890
two by two transformation
matrices as being able to map

00:00:05.890 --> 00:00:08.720
any point in the coordinate
plane to any other point

00:00:08.720 --> 00:00:10.420
or any two-dimensional vector

00:00:10.420 --> 00:00:12.690
to any other two-dimensional vector.

00:00:12.690 --> 00:00:16.160
What we're going to do in
this video is generalize a bit

00:00:16.160 --> 00:00:18.450
and realize that the same principles

00:00:18.450 --> 00:00:22.230
can be used for n dimensional spaces.

00:00:22.230 --> 00:00:24.430
Now I know that sounds a little bit fancy

00:00:24.430 --> 00:00:26.470
and it really is on some level,

00:00:26.470 --> 00:00:28.830
but it's really the same ideas.

00:00:28.830 --> 00:00:31.420
So for example, let's extend what we know

00:00:31.420 --> 00:00:32.590
about two dimensions,

00:00:32.590 --> 00:00:35.660
let's extend it to say four dimensions.

00:00:35.660 --> 00:00:39.340
So let's write a
four-dimensional vector here

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and it is hard to visualize
in four dimensions,

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so don't be hard on yourself
if you have trouble.

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Two dimensions not too hard,
three dimensions not too hard,

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four dimensions a little bit hard for us.

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Maybe we have to think about
time as the fourth dimension

00:00:52.380 --> 00:00:54.690
but in matrix world or in vector world,

00:00:54.690 --> 00:00:56.250
it's pretty easy to represent them

00:00:56.250 --> 00:00:58.250
as hard a it is to visualize.

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So a four-dimensional vector,
we'll just have four numbers.

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Negative one, let's see, negative three,

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I'm just making these up
randomly, negative five and one.

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This is a four-dimensional vector,

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and we could view it
as being a weighted sum

00:01:13.140 --> 00:01:16.610
of the unit vectors in
the different dimensions

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of four-dimensional space.

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I guess you could say it.

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You could say that this
is the same thing as,

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actually let me color code a little bit.

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This would be equal to
negative one times the one,

00:01:27.370 --> 00:01:32.370
zero, zero, zero vector
plus negative three,

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plus negative three times the
zero, one, zero, zero vector

00:01:39.820 --> 00:01:44.820
plus negative five, plus
negative five times the zero,

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zero, one, zero vector.

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I think you see where this is going.

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And then last but not least,

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plus one times the zero,
zero, zero, one vector.

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Now when I write it this way,

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you might immediately start realizing,

00:02:02.067 --> 00:02:05.330
"Oh I think I know how to
do transformations here."

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For example, if I were to give
you the transformation matrix

00:02:09.610 --> 00:02:11.160
and this would be a transformation matrix

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for four dimensions.

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This is gonna be a four-by-four matrix.

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So I'm gonna write some
random numbers here.

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One, zero, negative three,
negative one, two, zero,

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negative three, one, three,
two, zero, two, three,

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negative one, zero and three.

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So my question to you is,
what would be the mapping

00:02:35.230 --> 00:02:38.370
of this four-dimensional
vector if we were to apply

00:02:38.370 --> 00:02:43.050
this transformation to
four-dimensional space?

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What would be the result?

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Pause this video and think about it.

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Well, it's completely
analogous to what we did

00:02:49.620 --> 00:02:53.250
in the two-by-two world
in two-dimensional space.

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We thought about, all
right, instead of the one,

00:02:55.500 --> 00:03:00.023
zero, zero, zero vector, we're
now going to use this vector.

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Instead of the zero,
one, zero, zero vector,

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we're now going to use this vector.

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Instead of this one in
that blue-green color,

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we're now going to use this one.

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And last but not least instead of that,

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I guess we could say
same in colored vector,

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we're now going to be using this one.

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So another way to think about it is,

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the mapping of this vector,
let me write it this way.

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Let me make a little line here

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so we can separate things a little bit

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but we could write, all
right, a little bit smaller,

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hopefully you can see this.

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So this is our original
vector, negative five, one,

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but we wanna do the prime.

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What does it get mapped to
under this transformation?

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Well, this is going to be negative one,

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instead of this unit
vector right over here,

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it's gonna be negative one
of this one right over here.

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So it's negative one times
all of this business,

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one, two, three, and three.

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And then we could have just
instead of plus negative three

00:03:57.670 --> 00:04:02.260
I can just write in minus three
times all of this business,

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zero, zero, two, negative one.

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And then we have minus five
times all of this business,

00:04:13.690 --> 00:04:16.270
negative three, negative three

00:04:16.270 --> 00:04:20.350
and then we get zero, zero, and then,

00:04:20.350 --> 00:04:23.130
that definitely gets a little
bit more work involved,

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the more dimensions we have,
plus one times this business.

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So plus one times negative
one, one, two, three.

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And so what's this going to be equal to?

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So actually this could be a
good time to pause the video too

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and have a go at it.

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All right, so this is
going to be this first one,

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I just make all of these negatives.

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So negative one, negative
two, negative three,

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negative three, and to
that, I'm going to add,

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let's see if I multiply all
of those times negative three,

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I'm going to get zero, zero, negative six

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and positive three.

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And then if I multiply all
of these times negative five,

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I am going to get 15, 15, zero and zero.

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And then if I multiply
all of these times one,

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well, I just get those things again.

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So that's going to be negative
one, one, two, and three

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and we are in the home stretch.

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So now we can just add everything together

00:05:32.230 --> 00:05:34.000
the corresponding terms.

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And so this is going to
be negative one plus zero,

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plus 15, plus negative one.

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So that's gonna be the
same thing as 15 minus two

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which is going to be 13,

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then negative two plus
zero, plus 15, plus one,

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so that's gonna be 16
minus two which is 14,

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then we have negative
three plus negative six

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which is negative nine and
then we add two to that,

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so that is negative seven.

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And then negative three
plus three is zero,

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plus zero is zero, plus three
is three, and we are done.

00:06:07.840 --> 00:06:12.600
We have found the mapping of
this four-dimensional vector

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based on a four-by-four
transformation matrix.

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Very cool.


Using matrices to transform the plane: Composing matrices

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:01.220
- [Instructor] So what I have here

00:00:01.220 --> 00:00:03.060
is two different transformation matrices,

00:00:03.060 --> 00:00:04.880
and what we're going to
think about in this video

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is can we construct a new matrix

00:00:07.530 --> 00:00:10.550
that's based on the composition
of these transformations?

00:00:10.550 --> 00:00:12.380
Or a simpler way of saying that,

00:00:12.380 --> 00:00:14.720
a new transformation
that's based on applying

00:00:14.720 --> 00:00:16.640
one of these transformations first,

00:00:16.640 --> 00:00:19.490
and then the other one right after that.

00:00:19.490 --> 00:00:22.070
So first, let's just
review what's going on.

00:00:22.070 --> 00:00:27.020
If we have some random vector here, a, b,

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we know that we could view this

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as a times the one, zero vector,

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the unit vector in the x direction,

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plus b times the zero, one vector,

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the unit vector that goes
in the vertical direction.

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Now, if you were to apply
this transformational

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capital A here,

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it tells you instead of using
one, zero and zero, one,

00:00:48.790 --> 00:00:51.310
use these two columns instead.

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So if you were to apply
the transformation here,

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I guess we could call it a,
b prime, that is going to be,

00:00:59.240 --> 00:01:02.200
if you apply capital A
transformation matrix,

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it's going to be a times, not one, zero,

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you use zero, five instead.

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And then plus b times not zero, one,

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you use two, negative one instead.

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So that's just a little bit of review.

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But what we're gonna think
about in this video is,

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what would be the transformation
matrix for the composition?

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And I could write that as
B of A right over here.

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And you might recognize
this from function notation,

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where essentially it's saying

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you would apply the function A first,

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and then whatever the output of that is,

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you would then input that into B

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and you'd get the output of that.

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And that makes sense,

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because you can view
transformation matrices

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really as functions, functions

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that map points on the coordinate plane.

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So, in this situation,

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what would be the transformation matrix

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that is a composition of these two?

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Pause this video and think about that.

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All right, well, what would happen is,

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we would first transform any
point using these two vectors,

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the zero, five, the two, negative one,

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because that's the first
transformation we do.

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And then we would apply
this transformation

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to whatever the resulting vector is.

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Now, that seems pretty involved

00:02:13.630 --> 00:02:15.990
and we don't wanna write
it in terms of A or B's,

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we just wanna write it in terms
of a transformation matrix.

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So one way to think about it is,

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we can transform each of these vectors

00:02:24.570 --> 00:02:27.120
that you have in matrix A.

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Because remember, that
says, what do you turn

00:02:29.260 --> 00:02:32.140
the vectors one, zero and zero, one into?

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So if we transform zero,
five using the matrix B,

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and if we transform two,
negative one using the matrix B

00:02:41.220 --> 00:02:43.350
and we put them in their
respective columns,

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we should have the composition of this.

00:02:46.030 --> 00:02:48.720
So let me write it this way,
create a little bit of space.

00:02:48.720 --> 00:02:53.720
So let's say that the
composition B of A is equal to,

00:02:53.753 --> 00:02:57.060
I'll write a big two-by-two
matrix right over here.

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The first thing we can do is
apply transformation matrix B

00:03:01.170 --> 00:03:03.860
to the purple column right over here.

00:03:03.860 --> 00:03:05.640
And what is that going to tell us?

00:03:05.640 --> 00:03:09.470
Well, that's gonna be zero
times negative three, one.

00:03:09.470 --> 00:03:10.700
So let me write it that way.

00:03:10.700 --> 00:03:15.700
It's going to be zero
times negative three, one

00:03:17.440 --> 00:03:22.440
plus five times zero, four, zero, four.

00:03:23.320 --> 00:03:24.670
And this is going to give us

00:03:24.670 --> 00:03:27.320
a two by one vector right over here.

00:03:27.320 --> 00:03:28.680
So that you can view

00:03:28.680 --> 00:03:32.710
as filling up the first
column of this transformation,

00:03:32.710 --> 00:03:34.670
this composition, I guess you could say.

00:03:34.670 --> 00:03:37.560
And then let's think
about this second vector

00:03:37.560 --> 00:03:39.250
right over here, two, negative one.

00:03:39.250 --> 00:03:42.510
If you transform that using
B, what are you going to get?

00:03:42.510 --> 00:03:45.320
You're going to get two
times negative three, one.

00:03:45.320 --> 00:03:50.320
So right here, two times
negative three, one,

00:03:50.490 --> 00:03:53.390
plus negative one, or maybe
I just write it this way,

00:03:53.390 --> 00:03:58.390
minus one times zero, four.

00:04:00.030 --> 00:04:03.290
And this doesn't look
like a matrix just yet,

00:04:03.290 --> 00:04:05.870
but if you work through it,
it will become a matrix.

00:04:05.870 --> 00:04:08.260
For example, if I multiply,

00:04:08.260 --> 00:04:10.630
well, zero times all of
this is going to be zero,

00:04:10.630 --> 00:04:13.730
and then five times zero is going to be,

00:04:13.730 --> 00:04:15.080
let me just write it this way.

00:04:15.080 --> 00:04:17.470
This would turn into
five times zero is zero,

00:04:17.470 --> 00:04:20.320
five times four is 20.

00:04:20.320 --> 00:04:22.880
And then this matrix right over here

00:04:22.880 --> 00:04:26.060
is going to be two times
negative three, one,

00:04:26.060 --> 00:04:30.170
is going to be negative six, two,

00:04:30.170 --> 00:04:35.170
and then we have minus zero, four.

00:04:35.470 --> 00:04:37.090
And now if we wanted to write this clearly

00:04:37.090 --> 00:04:39.980
as a two-by-two matrix,
this would be equal to,

00:04:39.980 --> 00:04:42.140
and we get a little bit
of a drum roll here,

00:04:42.140 --> 00:04:45.040
the first column is zero, 20.

00:04:45.040 --> 00:04:47.910
And then the second column
is going to be, let's see,

00:04:47.910 --> 00:04:51.240
negative six minus zero
is still negative six

00:04:51.240 --> 00:04:54.720
and two minus four is negative two.

00:04:54.720 --> 00:04:55.720
And we're done.

00:04:55.720 --> 00:04:58.980
We have just created a
new transformation matrix.

00:04:58.980 --> 00:05:01.970
It's based on the composition B of A.

00:05:01.970 --> 00:05:05.630
So if you apply transformation
A first to any vector

00:05:05.630 --> 00:05:09.870
and then apply transformation
B to whatever you get there,

00:05:09.870 --> 00:05:13.140
that is equivalent to
just applying this one

00:05:13.140 --> 00:05:16.233
two-by-two transformation matrix, B of A.


Using matrices to transform the plane: Mapping a vector

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:04.040
- [Instructor] Let's say that
we have the vector three, two.

00:00:04.040 --> 00:00:07.190
We know that we can express
this as a weighted sum

00:00:07.190 --> 00:00:09.530
of the unit vectors in two dimensions

00:00:09.530 --> 00:00:12.040
or we can view it as a linear combination.

00:00:12.040 --> 00:00:14.600
And you could view this as three times

00:00:14.600 --> 00:00:16.650
the unit vector in the X direction

00:00:16.650 --> 00:00:21.390
which is one zero plus
two times the unit vector

00:00:21.390 --> 00:00:24.350
in the Y direction, which is zero, one.

00:00:24.350 --> 00:00:27.370
And we can graph three,
two by saying okay,

00:00:27.370 --> 00:00:30.210
we have three unit vectors
in the X direction.

00:00:30.210 --> 00:00:33.100
This would be one right over there.

00:00:33.100 --> 00:00:37.250
That would be two and
then that would be three.

00:00:37.250 --> 00:00:40.610
And then we have plus two unit
vectors in the Y direction.

00:00:40.610 --> 00:00:45.610
So one and then two, and then
we know where our vector is

00:00:46.790 --> 00:00:47.740
or what it would look like.

00:00:47.740 --> 00:00:51.070
The vector three, two
would look like this.

00:00:53.700 --> 00:00:58.160
Now let's apply a
transformation to this vector.

00:00:58.160 --> 00:01:01.670
And so let's say we have
the transformation matrix.

00:01:01.670 --> 00:01:02.820
I'll write it this way.

00:01:03.750 --> 00:01:07.040
Two, one, two, three.

00:01:07.040 --> 00:01:08.620
Now we've thought about this before.

00:01:08.620 --> 00:01:11.540
One way of thinking about
a transformation matrix is

00:01:11.540 --> 00:01:15.110
it gives you the image
of the unit vectors.

00:01:15.110 --> 00:01:17.650
And so instead of being
this linear combination

00:01:17.650 --> 00:01:19.090
of the unit vectors,

00:01:19.090 --> 00:01:21.010
it's going to be this linear combination

00:01:21.010 --> 00:01:22.990
of the images of the unit vectors

00:01:22.990 --> 00:01:24.670
when we take the transformation.

00:01:24.670 --> 00:01:25.900
What do I mean?

00:01:25.900 --> 00:01:28.460
Well, instead of having three one, zeros,

00:01:28.460 --> 00:01:32.010
we are now going to have three two, ones.

00:01:32.010 --> 00:01:33.870
Instead of having two zero, ones,

00:01:33.870 --> 00:01:37.210
we're now going to have two two, threes.

00:01:37.210 --> 00:01:39.490
So I could write it this way.

00:01:39.490 --> 00:01:43.960
Let me write it this way, the
image of our original vector.

00:01:43.960 --> 00:01:46.720
I'll put a prime here to say
we're talking about its image

00:01:46.720 --> 00:01:50.000
is going to be three times
instead of one, zero,

00:01:50.000 --> 00:01:53.840
it's going to be times two, one vectors.

00:01:53.840 --> 00:01:56.880
That's the image of the
one, zero unit vector

00:01:56.880 --> 00:01:58.950
under this transformation.

00:01:58.950 --> 00:02:03.390
And then we're gonna say plus
two instead of zero, one,

00:02:03.390 --> 00:02:04.860
we're gonna look at the image

00:02:04.860 --> 00:02:07.460
under the transformation
of the zero, one vector,

00:02:07.460 --> 00:02:09.580
which the transformation matrix gives us

00:02:09.580 --> 00:02:11.483
and that is the two, three, vector.

00:02:12.450 --> 00:02:16.130
Two, three, and we can graph this.

00:02:16.130 --> 00:02:19.350
If we have three two,
ones and two two, threes,

00:02:19.350 --> 00:02:23.910
what I could do is overlay
this extra grid to help us.

00:02:23.910 --> 00:02:27.630
So this is two, one, that's one to one,

00:02:27.630 --> 00:02:30.480
that is two two, ones going here.

00:02:30.480 --> 00:02:33.620
And then we have three
two, ones right over here.

00:02:33.620 --> 00:02:35.600
So there's three two, ones.

00:02:35.600 --> 00:02:37.790
Let me do this in this color.

00:02:37.790 --> 00:02:41.930
This part right over here
is going to be this vector.

00:02:41.930 --> 00:02:46.710
The three two, ones is
going to look like that.

00:02:46.710 --> 00:02:50.200
And then to that, we add two two, threes.

00:02:50.200 --> 00:02:51.520
So this is going to...

00:02:51.520 --> 00:02:52.910
Let's see, two and then three

00:02:52.910 --> 00:02:55.730
so this is going to be one two, three,

00:02:55.730 --> 00:02:58.280
and then we have two two, threes.

00:02:58.280 --> 00:03:01.800
So we end up right over there.

00:03:01.800 --> 00:03:04.110
And so let me actually
get rid of this grid

00:03:04.110 --> 00:03:07.510
so we can see things a
little bit more clearly.

00:03:07.510 --> 00:03:09.480
And so we have here in purple,

00:03:09.480 --> 00:03:11.720
we have our original three, two vector

00:03:11.720 --> 00:03:14.840
and now the image is going
to be three two, ones

00:03:14.840 --> 00:03:19.500
plus two two, threes so the
image of our three, two vector

00:03:19.500 --> 00:03:23.130
under this transformation
is going to be this vector

00:03:23.130 --> 00:03:25.150
that I'm drawing right here.

00:03:25.150 --> 00:03:27.420
And it looks when I eyeball it,

00:03:27.420 --> 00:03:31.120
it looks like it is the 10, nine vector

00:03:31.120 --> 00:03:33.940
and we can verify that by
doing the math right over here.

00:03:33.940 --> 00:03:34.773
So let's do that.

00:03:34.773 --> 00:03:39.490
This is going to be equal
to three times two is six,

00:03:39.490 --> 00:03:41.580
three times one is three.

00:03:41.580 --> 00:03:45.700
And we're going to add that
to two times two is four,

00:03:45.700 --> 00:03:48.320
two times three is six.

00:03:48.320 --> 00:03:51.110
And indeed you add the
corresponding entry,

00:03:51.110 --> 00:03:55.490
six plus four is 10 and
three plus six is nine.

00:03:55.490 --> 00:03:56.640
And we're done.

00:03:56.640 --> 00:03:58.010
The important takeaway here

00:03:58.010 --> 00:03:59.750
is that any vector can be represented

00:03:59.750 --> 00:04:02.760
as a linear combination
of the unit vectors.

00:04:02.760 --> 00:04:04.730
Now when we take the transformation,

00:04:04.730 --> 00:04:06.650
it's now going to be a linear combination

00:04:06.650 --> 00:04:08.010
not of the unit vectors,

00:04:08.010 --> 00:04:10.750
but of the images of the unit factors.

00:04:10.750 --> 00:04:12.040
And we saw that visually

00:04:12.040 --> 00:04:14.223
and we verified that mathematically.


Working with matrices as transformations of the plane

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:01.360
- [Instructor] In a previous video,

00:00:01.360 --> 00:00:04.330
I talked about how a
two-by-two matrix can be used

00:00:04.330 --> 00:00:08.290
to define a transformation for
the entire coordinate plane.

00:00:08.290 --> 00:00:09.500
What we're going to do in this video,

00:00:09.500 --> 00:00:10.860
is experiment with that little bit

00:00:10.860 --> 00:00:13.450
and see if we can think
about how to engineer

00:00:13.450 --> 00:00:16.580
two-by-two matrices to do
some of the transformations

00:00:16.580 --> 00:00:17.900
that you might be familiar with,

00:00:17.900 --> 00:00:22.100
like rotations, or
dilations, or reflections.

00:00:22.100 --> 00:00:26.360
So this is a website run
by the University of Texas,

00:00:26.360 --> 00:00:28.900
web.ma.utexas.edu.

00:00:28.900 --> 00:00:30.730
And you have the URL here,
I encourage you to go there

00:00:30.730 --> 00:00:32.490
and play around with it yourself.

00:00:32.490 --> 00:00:37.480
And what I have here is,
I have our two vectors,

00:00:37.480 --> 00:00:40.570
which any point on our
coordinate access can be defined

00:00:40.570 --> 00:00:44.120
by some combination of these two vectors.

00:00:44.120 --> 00:00:46.570
This in red here is the vector one, zero.

00:00:46.570 --> 00:00:50.150
It goes one in the x-direction,
zero in the y-direction,

00:00:50.150 --> 00:00:52.780
and you can see that is this
first column right over here

00:00:52.780 --> 00:00:54.270
in this identity matrix.

00:00:54.270 --> 00:00:56.270
And this blue vector right over here,

00:00:56.270 --> 00:00:58.040
this is the vector zero, one,

00:00:58.040 --> 00:01:00.500
which is the second column
in this identity matrix.

00:01:00.500 --> 00:01:02.350
It goes zero in the x-direction,

00:01:02.350 --> 00:01:05.760
and then one in the y-direction.

00:01:05.760 --> 00:01:09.940
Now, the way to engineer a
transformation is to say, well,

00:01:09.940 --> 00:01:12.690
what would that transformation
do to these two vectors,

00:01:12.690 --> 00:01:15.490
and then change the numbers accordingly.

00:01:15.490 --> 00:01:18.480
So, for example, let's say that we wanted

00:01:18.480 --> 00:01:23.210
to have a reflection about the x-axis.

00:01:23.210 --> 00:01:25.730
So if you did a reflection
about the x-axis,

00:01:25.730 --> 00:01:27.950
this red vector would not change.

00:01:27.950 --> 00:01:29.690
It would stay one, zero.

00:01:29.690 --> 00:01:31.520
But what would happen to this blue vector?

00:01:31.520 --> 00:01:33.260
Instead of being zero, one,

00:01:33.260 --> 00:01:35.610
it would be zero, negative one.

00:01:35.610 --> 00:01:38.040
So in the transformation matrix,

00:01:38.040 --> 00:01:39.810
if I go from the identity matrix here,

00:01:39.810 --> 00:01:41.840
but instead of zero, one, I now,

00:01:41.840 --> 00:01:43.390
it will no longer be the identity matrix

00:01:43.390 --> 00:01:44.930
if I put a negative one here.

00:01:44.930 --> 00:01:46.290
And when I press Enter,

00:01:46.290 --> 00:01:50.730
this should flip this blue
vector over the x-axis

00:01:50.730 --> 00:01:52.570
and essentially flip
everything else with it.

00:01:52.570 --> 00:01:53.403
So let's try that out.

00:01:53.403 --> 00:01:55.720
I'm gonna press Enter,
and there you have it.

00:01:55.720 --> 00:02:00.720
That cute little golden
retriever is now flipped over.

00:02:00.920 --> 00:02:02.880
So that met our intuition.

00:02:02.880 --> 00:02:04.670
So let's go back back to
what we were doing before.

00:02:04.670 --> 00:02:05.640
So that's a reflection,

00:02:05.640 --> 00:02:07.190
and you could think
about what would you do

00:02:07.190 --> 00:02:10.400
if you wanted to flip the
other way across the y-axis.

00:02:10.400 --> 00:02:12.880
Now, what about a dilation?

00:02:12.880 --> 00:02:17.080
What if we wanted to shrink
everything by a factor of two?

00:02:17.080 --> 00:02:19.610
How do you think we would
modify this matrix to do that?

00:02:19.610 --> 00:02:21.510
Pause this video and think about that?

00:02:22.870 --> 00:02:25.450
Well, if we want to scale everything down,

00:02:25.450 --> 00:02:27.230
what we would want is,
each of these vectors,

00:02:27.230 --> 00:02:28.920
and especially if it's by a factor of two,

00:02:28.920 --> 00:02:31.460
we'd want each of these
vectors to be half as long.

00:02:31.460 --> 00:02:33.440
So instead of one, zero and zero, one,

00:02:33.440 --> 00:02:38.440
we would do 0.5, zero and 0.5.

00:02:38.950 --> 00:02:40.360
Let me press Enter and see what happens.

00:02:40.360 --> 00:02:41.280
There you go.

00:02:41.280 --> 00:02:42.690
It indeed worked.

00:02:42.690 --> 00:02:44.920
And really this should
have showed this red vector

00:02:44.920 --> 00:02:47.110
gets smaller and this
blue vector gets smaller.

00:02:47.110 --> 00:02:48.950
But hopefully you get the idea.

00:02:48.950 --> 00:02:50.430
And so let me go.

00:02:50.430 --> 00:02:52.600
Or maybe this one that always show

00:02:52.600 --> 00:02:54.960
what we could kind of
call it, unit vectors.

00:02:54.960 --> 00:02:56.710
But let's go back to the original.

00:02:56.710 --> 00:02:58.280
And now let's think about our rotation.

00:02:58.280 --> 00:02:59.610
This is an interesting one.

00:02:59.610 --> 00:03:01.500
Pause this video and think
about how you would rotate it

00:03:01.500 --> 00:03:05.363
if you wanted to rotate this
clockwise by 90 degrees.

00:03:07.040 --> 00:03:07.873
All right.

00:03:07.873 --> 00:03:09.400
If you rotate clockwise by 90 degrees

00:03:09.400 --> 00:03:12.163
this red vector is no longer one, zero.

00:03:14.032 --> 00:03:16.550
It would become zero, negative one.

00:03:16.550 --> 00:03:17.550
So let me write that down.

00:03:17.550 --> 00:03:20.350
Zero, negative one.

00:03:20.350 --> 00:03:22.050
And the blue vector would then go

00:03:22.050 --> 00:03:24.870
to where this red vector is
and it would become one, zero.

00:03:24.870 --> 00:03:29.540
So let's see if we did it the right way.

00:03:29.540 --> 00:03:30.950
I'm gonna click Enter.

00:03:30.950 --> 00:03:31.900
And there you go.

00:03:31.900 --> 00:03:34.170
We got our 90 degree rotation.

00:03:34.170 --> 00:03:36.730
And so I just gave you some examples

00:03:36.730 --> 00:03:39.420
of how you can do a pure rotation,

00:03:39.420 --> 00:03:41.600
a pure dilation, or a pure reflection.

00:03:41.600 --> 00:03:44.410
But you can imagine you can
also do combinations of them,

00:03:44.410 --> 00:03:47.030
by manipulating this matrix accordingly.

00:03:47.030 --> 00:03:48.320
And I encourage you to play around.

00:03:48.320 --> 00:03:51.100
You can do some exotic
transformations if you want.

00:03:51.100 --> 00:03:53.220
Let's see what happens
if I make this a one.

00:03:53.220 --> 00:03:54.053
Press Enter.

00:03:54.053 --> 00:03:55.110
Oh, that's interesting.

00:03:55.110 --> 00:03:58.400
What happens if I then make this a two?

00:03:58.400 --> 00:03:59.510
Oh, that's interesting.

00:03:59.510 --> 00:04:00.830
So notice, you can do all sorts

00:04:00.830 --> 00:04:02.750
of really interesting
linear transformation.

00:04:02.750 --> 00:04:04.760
And just as a reminder,
linear transformation

00:04:04.760 --> 00:04:07.300
is one where the origin
always maps to itself,

00:04:07.300 --> 00:04:10.110
and any lines are mapped to other lines.

00:04:10.110 --> 00:04:11.470
Not necessarily the same line,

00:04:11.470 --> 00:04:14.120
but whatever it gets mapped
to will still be aligned.


Matrices as transformations of the plane

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:01.950
- [Instructor] In this
video, we're going to explore

00:00:01.950 --> 00:00:05.520
how a two by two matrix can be interpreted

00:00:05.520 --> 00:00:10.520
as representing a transformation
on the coordinate plane.

00:00:10.750 --> 00:00:13.410
So let's just start with some examples

00:00:13.410 --> 00:00:15.870
or some conceptual ideas.

00:00:15.870 --> 00:00:19.010
So the first conceptual idea is that

00:00:19.010 --> 00:00:21.680
any point on our coordinate plane here

00:00:21.680 --> 00:00:26.680
and this of course is our X
axis and this is our Y axis,

00:00:27.150 --> 00:00:31.610
can be represented by a
combination of two vectors.

00:00:31.610 --> 00:00:34.340
You could have this
vector right over here.

00:00:34.340 --> 00:00:38.080
That goes exactly one unit
in the horizontal direction.

00:00:38.080 --> 00:00:43.080
We can represent that as a
vector like this, one, zero,

00:00:43.490 --> 00:00:45.300
when you write a vector
vertically like this

00:00:45.300 --> 00:00:47.730
the convention is that the top number here

00:00:47.730 --> 00:00:49.440
is what we're doing in the X direction.

00:00:49.440 --> 00:00:51.200
And then the bottom number,

00:00:51.200 --> 00:00:53.350
the zero is what we're doing
in the vertical direction

00:00:53.350 --> 00:00:56.650
or the Y direction, so this
is the one, zero vector.

00:00:56.650 --> 00:01:00.953
And then this right over
here, what would we call that?

00:01:01.870 --> 00:01:05.110
Well that we would call
the zero, one vector

00:01:05.110 --> 00:01:08.030
because it doesn't go at
all in the X direction.

00:01:08.030 --> 00:01:11.460
And it only goes up
one in the Y direction.

00:01:11.460 --> 00:01:14.740
Now just to feel good that any
point on the coordinate plane

00:01:14.740 --> 00:01:18.560
can be represented as a
weighted sum of these.

00:01:18.560 --> 00:01:21.170
Let's just pick a point at random.

00:01:21.170 --> 00:01:23.960
Let's say this point, all right over here,

00:01:23.960 --> 00:01:26.160
let's call it point A.

00:01:26.160 --> 00:01:28.670
And you could represent that as a vector

00:01:28.670 --> 00:01:30.120
that looks something like this,

00:01:30.120 --> 00:01:31.890
I'll do it as a dotted line,

00:01:31.890 --> 00:01:33.300
but this could be represented

00:01:33.300 --> 00:01:36.150
as a positional vector like that.

00:01:36.150 --> 00:01:38.100
And of course, if we're
thinking about it in coordinates

00:01:38.100 --> 00:01:41.710
we would just say this is at
the coordinate three comma one.

00:01:41.710 --> 00:01:44.460
The X coordinate is a three,
the Y coordinate is one.

00:01:44.460 --> 00:01:47.240
But if we wanted to express
it in terms of a vector,

00:01:47.240 --> 00:01:51.850
we could write it out as three, one,

00:01:51.850 --> 00:01:55.040
the X direction we're moving
three from the origin,

00:01:55.040 --> 00:01:56.710
positive three and the Y direction

00:01:56.710 --> 00:01:58.190
we're moving one to get there.

00:01:58.190 --> 00:02:01.280
And you could see that
we can represent this

00:02:01.280 --> 00:02:03.820
as a weighted sum of these two vectors.

00:02:03.820 --> 00:02:08.820
We can write this as, this is
the same thing as three times

00:02:08.920 --> 00:02:13.920
our one, zero vector, one, zero

00:02:14.290 --> 00:02:19.290
plus one times our zero
one, vector zero one vector.

00:02:20.820 --> 00:02:22.530
And you can see it visually,

00:02:22.530 --> 00:02:27.210
this yellow vector that points
to point a right over there.

00:02:27.210 --> 00:02:32.210
You can have three of this
vector, one, two, and then three.

00:02:32.810 --> 00:02:35.963
And then one of the orange vector.

00:02:36.870 --> 00:02:41.870
Now I said that I would
explain how two by two matrices

00:02:41.930 --> 00:02:44.460
can represent a transformation.

00:02:44.460 --> 00:02:46.950
And the way that you
could think about it is

00:02:46.950 --> 00:02:51.940
if I have a two by two
matrix that looks like this,

00:02:51.940 --> 00:02:53.960
so let me just draw the matrix,

00:02:53.960 --> 00:02:56.990
where the first column is one, zero

00:02:56.990 --> 00:03:00.070
and then the second column is zero, one.

00:03:00.070 --> 00:03:03.850
This just tells you what to
do with these two vectors.

00:03:03.850 --> 00:03:06.170
I know this might be a little
bit confusing at first,

00:03:06.170 --> 00:03:08.390
but let's just walk through it together.

00:03:08.390 --> 00:03:09.640
So the way I've represented it,

00:03:09.640 --> 00:03:12.750
this first column says
what is the transformation

00:03:12.750 --> 00:03:15.220
you want to apply to
this one, zero vector,

00:03:15.220 --> 00:03:16.800
this first blue vector?

00:03:16.800 --> 00:03:18.640
Well, we're just keeping it one zero.

00:03:18.640 --> 00:03:20.560
So we're not changing it, it's the same.

00:03:20.560 --> 00:03:23.350
One, zero vector here
one, zero vector here.

00:03:23.350 --> 00:03:28.030
Likewise, zero, one vector
here, zero, one vector here.

00:03:28.030 --> 00:03:32.160
So this two by two matrix
actually represents

00:03:32.160 --> 00:03:34.800
what's sometimes known as
the identity transformation.

00:03:34.800 --> 00:03:38.220
It maps any point on the
coordinate plane back to itself.

00:03:38.220 --> 00:03:40.050
It doesn't change the points,

00:03:40.050 --> 00:03:42.210
but I'm showing this because
now I'm going to show

00:03:42.210 --> 00:03:43.940
a two by two matrix that represents

00:03:43.940 --> 00:03:46.370
a non identity transformation.

00:03:46.370 --> 00:03:48.943
For example, let me draw the matrix again.

00:03:50.050 --> 00:03:53.910
So let's say I have the matrix,

00:03:53.910 --> 00:03:56.850
instead of one, zero here I'm
going to write a two, one here

00:03:56.850 --> 00:04:00.613
instead of a zero, one here
let me write a one, two.

00:04:01.540 --> 00:04:04.510
So in this transformation,
what we're doing is

00:04:04.510 --> 00:04:08.300
we're turning this one, zero
vector into a two, one vector.

00:04:08.300 --> 00:04:10.680
You're going to see what I'm
talking about in a second.

00:04:10.680 --> 00:04:13.020
So what does a two, one vector look like?

00:04:13.020 --> 00:04:14.403
Let me do it in that color.

00:04:15.510 --> 00:04:18.450
Well, we go two in the X
direction one in the Y direction.

00:04:18.450 --> 00:04:20.700
So it's going to look like this,

00:04:20.700 --> 00:04:24.530
it's going to look like that.

00:04:24.530 --> 00:04:28.250
And then what does a one,
two vector look like?

00:04:28.250 --> 00:04:30.170
Well, it goes one in the X direction

00:04:30.170 --> 00:04:34.240
and then two in the Y direction,
so it looks like this.

00:04:34.240 --> 00:04:36.400
And the way that this
represents a transformation

00:04:36.400 --> 00:04:38.890
is that anything that was a weighted sum

00:04:38.890 --> 00:04:42.030
of the one, zero and the
zero, one vectors originally,

00:04:42.030 --> 00:04:44.510
you can now view as a
weighted sum of the two, one

00:04:44.510 --> 00:04:46.140
and the one, two vectors.

00:04:46.140 --> 00:04:50.040
And so we can now think
of another point A prime,

00:04:50.040 --> 00:04:52.530
that's not going to be
three of the ones zeros

00:04:52.530 --> 00:04:54.290
and one of the zero ones.

00:04:54.290 --> 00:04:57.270
We can think of it as,
let me write it over here,

00:04:57.270 --> 00:05:01.010
as three of the two, ones

00:05:03.340 --> 00:05:08.340
plus one of the one twos,
one of these one, two.

00:05:13.330 --> 00:05:15.180
So where would that put that now?

00:05:15.180 --> 00:05:17.400
What we're going to go,
go three of the two, ones.

00:05:17.400 --> 00:05:22.400
So this is one of them,
this is two of them

00:05:22.410 --> 00:05:25.520
and then this is three of them.

00:05:25.520 --> 00:05:28.810
And then I'm going to have
one of the one, two vectors,

00:05:28.810 --> 00:05:30.650
this orange vector right over here.

00:05:30.650 --> 00:05:33.850
And so I'm going to have one of those.

00:05:33.850 --> 00:05:37.300
And so that will take me to A prime.

00:05:37.300 --> 00:05:41.330
This is my new point
after the transformation,

00:05:41.330 --> 00:05:42.620
I go to A prime.

00:05:42.620 --> 00:05:46.190
So I've gone from this
point A to this A prime.

00:05:46.190 --> 00:05:51.190
And this two by two matrix is
telling us how to transform.

00:05:51.290 --> 00:05:53.990
Now we can do that with multiple points.

00:05:53.990 --> 00:05:56.810
Now, if I talk about a
point here at the origin

00:05:56.810 --> 00:05:59.040
that was originally at the origin point B,

00:05:59.040 --> 00:06:00.710
well that's zero of the orange vector,

00:06:00.710 --> 00:06:02.230
zero of the blue vectors.

00:06:02.230 --> 00:06:03.740
So even after the transformation,

00:06:03.740 --> 00:06:05.307
it's going to be zero
of the two, one vector

00:06:05.307 --> 00:06:06.890
and zero of the one, two vectors.

00:06:06.890 --> 00:06:08.400
So it's just going to stay in place.

00:06:08.400 --> 00:06:09.730
It's just going to map to itself.

00:06:09.730 --> 00:06:12.060
So B is equal to B prime,

00:06:12.060 --> 00:06:14.620
and we could also imagine another point.

00:06:14.620 --> 00:06:19.620
Let's say right over here,
let's call that point C.

00:06:20.050 --> 00:06:23.430
Well point C is originally
two of the blue vectors

00:06:23.430 --> 00:06:25.610
and none of the orange vectors.

00:06:25.610 --> 00:06:28.810
So after the mapping, it'll
be two of the two, one vectors

00:06:28.810 --> 00:06:31.140
and none of the one, two vectors.

00:06:31.140 --> 00:06:32.330
So two of the two ones.

00:06:32.330 --> 00:06:36.910
So if you go one, two, and
then none of the one, twos

00:06:36.910 --> 00:06:39.710
you're going to get C
prime right over there.

00:06:39.710 --> 00:06:43.260
And so notice, if originally
you had a triangle

00:06:43.260 --> 00:06:45.990
between A, B and C, let
me draw it like this.

00:06:45.990 --> 00:06:50.950
So originally you had
this triangle A, B, C

00:06:50.950 --> 00:06:54.180
what is it now gotten mapped,
what is it now mapped to?

00:06:54.180 --> 00:06:56.870
Well, it's now mapped to big triangle.

00:06:56.870 --> 00:07:00.470
I'll do my best to draw
it relatively straight

00:07:00.470 --> 00:07:01.590
so we land on target.

00:07:01.590 --> 00:07:03.250
So that's that side.

00:07:03.250 --> 00:07:08.250
And then we have this side
going from B prime to C prime

00:07:08.350 --> 00:07:13.170
and then we wanna connect that
side from C prime to A prime.

00:07:13.170 --> 00:07:14.003
Now you might be saying,

00:07:14.003 --> 00:07:16.850
so how do you know that the
lines map onto other lines?

00:07:16.850 --> 00:07:18.050
How do you know this transformation

00:07:18.050 --> 00:07:20.970
didn't all of a sudden make
this line squiggly or zigzag?

00:07:20.970 --> 00:07:22.570
And that's one of the
interesting properties

00:07:22.570 --> 00:07:24.560
of the type of transformation
we're talking about.

00:07:24.560 --> 00:07:29.330
A two by two matrix will
represent a linear transformation.

00:07:29.330 --> 00:07:31.560
And there's two ways to think
about it in this context,

00:07:31.560 --> 00:07:34.480
a linear transformation
will always map the origin

00:07:34.480 --> 00:07:38.850
onto itself and it will always
map a line onto another line,

00:07:38.850 --> 00:07:40.510
it won't turn that line into a curve,

00:07:40.510 --> 00:07:43.030
or it won't make it zigzag somehow.

00:07:43.030 --> 00:07:44.790
Now the last thing you
might be wondering is,

00:07:44.790 --> 00:07:46.210
Hey what about all these transformations

00:07:46.210 --> 00:07:49.130
we had from geometry, these
similarity transformations,

00:07:49.130 --> 00:07:52.230
things like rotations
reflections, dilations,

00:07:52.230 --> 00:07:53.950
can you do those with matrices?

00:07:53.950 --> 00:07:55.630
And the simple answer is yes,

00:07:55.630 --> 00:07:59.520
you can do them as long as
you keep the origin in place.

00:07:59.520 --> 00:08:02.400
And you can actually
using a two by two matrix

00:08:02.400 --> 00:08:05.490
come up with a whole series of
other linear transformations

00:08:05.490 --> 00:08:07.970
that are much more, let's call it exotic,

00:08:07.970 --> 00:08:10.523
than just the rotations,
reflections and dilations.


Using the tangent angle addition identity

WEBVTT
Kind: captions
Language: en

00:00:01.090 --> 00:00:03.580
- [Instructor] In this video,
we're going to try to compute

00:00:03.580 --> 00:00:08.580
what tangent of 13 pi over 12
is without using a calculator.

00:00:11.120 --> 00:00:13.260
But I will give you a few hints.

00:00:13.260 --> 00:00:16.900
First of all, you can rewrite
tangent of 13 pi over 12

00:00:16.900 --> 00:00:19.500
as tangent of,

00:00:19.500 --> 00:00:21.840
instead of 13 pi over
12, we can express that

00:00:21.840 --> 00:00:24.090
in terms of angles where we
might be able to figure out

00:00:24.090 --> 00:00:26.350
the tangents just based
on other things we know

00:00:26.350 --> 00:00:27.720
about the unit circle.

00:00:27.720 --> 00:00:31.817
13 pi over 12 is the same
thing as 15 pi over 12

00:00:33.900 --> 00:00:37.350
minus two pi over 12,

00:00:37.350 --> 00:00:41.680
which is the same thing as the tangent

00:00:41.680 --> 00:00:46.680
of five pi over four

00:00:47.320 --> 00:00:50.540
minus pi over six,

00:00:50.540 --> 00:00:54.660
or we could even view it as
plus negative pi over six.

00:00:54.660 --> 00:00:56.660
So that's my hint right over there.

00:00:56.660 --> 00:00:59.650
So pause this video and
see if you can keep going

00:00:59.650 --> 00:01:01.380
with this train of reasoning

00:01:01.380 --> 00:01:04.070
to evaluate what tangent
of 13 pi over 12 is

00:01:04.070 --> 00:01:05.733
without using a calculator.

00:01:07.390 --> 00:01:11.220
All right, now let's
keep on going together.

00:01:11.220 --> 00:01:13.270
Well, we already know what the tangent

00:01:13.270 --> 00:01:15.240
of the sum of two angles are.

00:01:15.240 --> 00:01:17.600
We've proven that in another video.

00:01:17.600 --> 00:01:19.390
We know that this is going to be equal

00:01:19.390 --> 00:01:24.390
to the tangent of the
first of these angles,

00:01:24.440 --> 00:01:27.740
five pi over four,

00:01:27.740 --> 00:01:30.120
plus the tangent of the second angle,

00:01:30.120 --> 00:01:34.830
tangent of negative pi over six,

00:01:34.830 --> 00:01:36.800
all of that is going to be over

00:01:38.300 --> 00:01:42.050
one minus tangent of the first angle,

00:01:42.050 --> 00:01:45.520
five pi over four,

00:01:45.520 --> 00:01:48.920
times the tangent of the second angle,

00:01:48.920 --> 00:01:51.370
negative pi over six.

00:01:51.370 --> 00:01:53.910
And so now we can break
out our unit circles

00:01:53.910 --> 00:01:56.490
to figure out what these things are.

00:01:56.490 --> 00:02:00.370
So I have pre-put some unit circles here.

00:02:00.370 --> 00:02:02.350
And so let's first think about

00:02:02.350 --> 00:02:04.700
what five pi over four looks like.

00:02:04.700 --> 00:02:05.780
Pi over four,

00:02:05.780 --> 00:02:07.990
you might already associate
it with 45 degrees.

00:02:07.990 --> 00:02:10.840
That's pi over four right over there.

00:02:10.840 --> 00:02:13.550
Two pi over four would get you here.

00:02:13.550 --> 00:02:15.590
Three pi over fours would get you there.

00:02:15.590 --> 00:02:17.520
Four pi over four, which
the same thing as pi,

00:02:17.520 --> 00:02:18.353
gets us over there.

00:02:18.353 --> 00:02:23.023
Five pi over four would
get us right about there.

00:02:23.930 --> 00:02:27.800
Now, you might already recognize
the tangent of an angle

00:02:27.800 --> 00:02:30.030
as the slope of the radius,

00:02:30.030 --> 00:02:32.420
and so you might already be able to intuit

00:02:32.420 --> 00:02:34.610
that the tangent here is going to be one,

00:02:34.610 --> 00:02:38.120
but we can also break out
our knowledge of triangles

00:02:38.120 --> 00:02:39.810
in the unit circle to figure this out

00:02:39.810 --> 00:02:41.530
if you didn't realize that.

00:02:41.530 --> 00:02:43.520
So what we need to do is
figure out the coordinates

00:02:43.520 --> 00:02:45.480
of that point right over there,

00:02:45.480 --> 00:02:46.880
and to help us do that,

00:02:46.880 --> 00:02:49.623
we can set up a little
bit of a right triangle,

00:02:50.660 --> 00:02:52.800
which you might immediately recognize

00:02:52.800 --> 00:02:55.550
is a 45-45-90 triangle.

00:02:55.550 --> 00:02:56.670
How do I know that?

00:02:56.670 --> 00:02:58.280
Well five pi over four,

00:02:58.280 --> 00:03:00.120
remember we go four pi
over four to get here,

00:03:00.120 --> 00:03:03.080
then we have one more pi
over four to go down here.

00:03:03.080 --> 00:03:05.120
So this angle right over
here is pi over four

00:03:05.120 --> 00:03:07.300
or you could view it as 45 degrees.

00:03:07.300 --> 00:03:09.520
And of course, if that's
45 degrees and that's 90,

00:03:09.520 --> 00:03:11.200
then this has to be 45 degrees

00:03:11.200 --> 00:03:14.300
'cause they all add up to 180 degrees.

00:03:14.300 --> 00:03:17.630
And we know a triangle like
this by the Pythagorean theorem,

00:03:17.630 --> 00:03:19.380
if our hypothesis is one,

00:03:19.380 --> 00:03:20.710
each of the other two sides

00:03:20.710 --> 00:03:23.440
is square root of two over
two times the hypotenuse.

00:03:23.440 --> 00:03:25.020
So this is square root of two over two

00:03:25.020 --> 00:03:27.740
and then this is square
root of two over two.

00:03:27.740 --> 00:03:29.630
Now, if we think about the coordinates,

00:03:29.630 --> 00:03:31.970
our X coordinate is square
root of two over two

00:03:31.970 --> 00:03:33.290
in the negative direction.

00:03:33.290 --> 00:03:37.260
So our X coordinate is negative
square root of two over two

00:03:37.260 --> 00:03:38.220
and our Y coordinate

00:03:38.220 --> 00:03:40.550
is square root of two over two going down,

00:03:40.550 --> 00:03:44.380
so that's also negative
square root of two over two.

00:03:44.380 --> 00:03:46.240
And the tangent is just

00:03:46.240 --> 00:03:50.360
the Y coordinate over
the X coordinate here.

00:03:50.360 --> 00:03:52.180
So the tangent is just going to be

00:03:52.180 --> 00:03:53.520
negative square root of two over two

00:03:53.520 --> 00:03:55.100
over negative square root of two over two,

00:03:55.100 --> 00:03:58.090
which is once again, one,
which was our intuition.

00:03:58.090 --> 00:04:03.090
So we can write the tangent
of five pi over four

00:04:03.590 --> 00:04:05.470
is equal to one.

00:04:05.470 --> 00:04:08.180
And then what about negative pi over six?

00:04:08.180 --> 00:04:09.340
Well, negative pi over six,

00:04:09.340 --> 00:04:12.130
you might recognize pi over
six as being a 30 degree angle.

00:04:12.130 --> 00:04:15.330
Pi is 180 degrees, so
divided by six is 30 degrees,

00:04:15.330 --> 00:04:17.120
and negative pi over six

00:04:17.120 --> 00:04:20.350
would be going 30 degrees
below the positive x-axis.

00:04:20.350 --> 00:04:22.640
So it would look just like that.

00:04:22.640 --> 00:04:25.970
And as we said, this angle
right over here is pi over six,

00:04:25.970 --> 00:04:29.290
which you can also view
as a 30 degree angle.

00:04:29.290 --> 00:04:32.420
If we were to drop a
perpendicular right over here,

00:04:32.420 --> 00:04:33.940
you might immediately recognize this

00:04:33.940 --> 00:04:37.970
as a 30-60-90 triangle.

00:04:37.970 --> 00:04:41.140
We know if the hypotenuse
is of length one,

00:04:41.140 --> 00:04:43.050
the side opposite the 30 degree side

00:04:43.050 --> 00:04:45.520
is 1/2 the hypotenuse,

00:04:45.520 --> 00:04:47.900
and then the longer non-hypotenuse side

00:04:47.900 --> 00:04:50.070
is going to be square root of
three times the shorter side,

00:04:50.070 --> 00:04:53.170
so square root of three over two.

00:04:53.170 --> 00:04:55.360
And so our coordinate's right over here,

00:04:55.360 --> 00:04:57.230
we are moving square
root of three over two

00:04:57.230 --> 00:04:58.580
in the positive X direction,

00:04:58.580 --> 00:05:00.400
square root of three over two,

00:05:00.400 --> 00:05:04.240
and then we're going negative
1/2 in the Y direction,

00:05:04.240 --> 00:05:07.780
so we put negative 1/2 right over there.

00:05:07.780 --> 00:05:12.780
And so now we know that the
tangent of negative pi over six

00:05:14.230 --> 00:05:16.800
is going to be equal to negative 1/2

00:05:18.120 --> 00:05:21.030
over square root of three over two,

00:05:21.030 --> 00:05:25.280
which is the same thing
as negative 1/2 times,

00:05:25.280 --> 00:05:26.610
let me write it this way, negative 1/2

00:05:26.610 --> 00:05:29.870
times two over the square root of three,

00:05:29.870 --> 00:05:32.170
which is equal to negative one

00:05:32.170 --> 00:05:34.340
over the square root of three.

00:05:34.340 --> 00:05:36.220
This right over here is one.

00:05:36.220 --> 00:05:37.250
We saw that there.

00:05:37.250 --> 00:05:39.420
This right over here is one.

00:05:39.420 --> 00:05:41.460
And then this right over here

00:05:41.460 --> 00:05:45.380
is negative one over the
square root of three.

00:05:45.380 --> 00:05:47.970
And then this is negative one
over the square root of three.

00:05:47.970 --> 00:05:51.420
And so I can rewrite this
entire expression as being equal

00:05:51.420 --> 00:05:56.420
to one minus one over the square to three,

00:05:56.990 --> 00:05:59.840
all of that over one,

00:05:59.840 --> 00:06:01.420
and then I have a negative
here and a negative here,

00:06:01.420 --> 00:06:03.030
so then that becomes, a
negative times a negative,

00:06:03.030 --> 00:06:07.330
so positive one plus one over
the square root of three.

00:06:07.330 --> 00:06:09.680
And if we multiply both the
numerator and the denominator

00:06:09.680 --> 00:06:12.450
by square root of three,

00:06:12.450 --> 00:06:14.000
what we're going to get in the numerator

00:06:14.000 --> 00:06:16.990
is square root of three minus one

00:06:16.990 --> 00:06:18.820
and then our denominator is going to be

00:06:18.820 --> 00:06:21.730
square root of three plus one.

00:06:21.730 --> 00:06:24.210
And we are done.

00:06:24.210 --> 00:06:28.353
That's tangent of 13 pi over
12 without using a calculator.


Interpreting solutions of trigonometric equations

WEBVTT
Kind: captions
Language: en

00:00:00.090 --> 00:00:02.040
- [Instructor] Alvaro presses the treadle

00:00:02.040 --> 00:00:04.720
of a spinning wheel with his foot.

00:00:04.720 --> 00:00:08.030
It moves a bar up and down
making the wheels spin.

00:00:08.030 --> 00:00:09.950
So just to be clear what a treadle is

00:00:09.950 --> 00:00:11.670
this is an old spinning wheel

00:00:11.670 --> 00:00:14.080
and this little pedal that is a treadle.

00:00:14.080 --> 00:00:17.400
And as this goes up and down,

00:00:17.400 --> 00:00:19.320
it's gonna pull on this bar,

00:00:19.320 --> 00:00:21.650
which is then going to spin this wheel

00:00:21.650 --> 00:00:24.993
which can then be used to
essentially power the machine.

00:00:26.020 --> 00:00:27.920
So it says the function B of t

00:00:27.920 --> 00:00:32.060
models the height in centimeters
of the top of the bar

00:00:32.060 --> 00:00:36.010
when Alvaro has pressed
the treadle for t seconds.

00:00:36.010 --> 00:00:37.900
So it's telling us the height of,

00:00:37.900 --> 00:00:40.160
I can barely see where
the top of the bar is

00:00:40.160 --> 00:00:41.230
someplace over here.

00:00:41.230 --> 00:00:42.720
And this isn't exactly

00:00:42.720 --> 00:00:45.510
what they're probably talking
about in this exercise here.

00:00:45.510 --> 00:00:47.680
But this is just to
give you a visualization

00:00:47.680 --> 00:00:49.910
of what a treadle is and what the bar is

00:00:49.910 --> 00:00:53.300
and then, what the spinning wheel is.

00:00:53.300 --> 00:00:56.330
Alvaro has pressed those
treadle for t seconds.

00:00:56.330 --> 00:00:58.460
So they give us B of t right over here

00:00:58.460 --> 00:01:00.753
and 90 minus 12 times sine of 5t.

00:01:01.670 --> 00:01:02.640
The first question is,

00:01:02.640 --> 00:01:05.280
what does the solution set to y is equal

00:01:05.280 --> 00:01:10.280
to 90 minus 12 times sine of
five times six, represent?

00:01:11.710 --> 00:01:14.083
Pause this video and see if
you can think about that.

00:01:14.950 --> 00:01:15.783
All right.

00:01:15.783 --> 00:01:17.830
So, it looks like right over here,

00:01:17.830 --> 00:01:20.470
so we have the 90, 90, 12, 12

00:01:20.470 --> 00:01:25.470
and we're subtracting 12 sine
of five times t, five times t.

00:01:25.610 --> 00:01:27.853
So this right over here is t.

00:01:29.090 --> 00:01:32.960
The solution set right over here tells us

00:01:32.960 --> 00:01:36.010
what is the height, because
that's what B of t is.

00:01:36.010 --> 00:01:38.070
So B of t is equal to y.

00:01:38.070 --> 00:01:40.930
What is the height when t is equal to six?

00:01:40.930 --> 00:01:43.530
And remember, t is in seconds.

00:01:43.530 --> 00:01:44.883
So, this is height,

00:01:46.510 --> 00:01:50.060
height of top of bar,

00:01:50.060 --> 00:01:55.060
top of bar at six seconds.

00:01:56.460 --> 00:01:59.300
All right, now we have
more questions here.

00:01:59.300 --> 00:02:01.493
The next question asks us,

00:02:02.370 --> 00:02:04.970
what does the solution set to 95

00:02:04.970 --> 00:02:09.130
equals 90 minus 12 sign of 5t represent?

00:02:09.130 --> 00:02:11.363
Pause the video and think about that.

00:02:13.100 --> 00:02:13.933
All right.

00:02:13.933 --> 00:02:18.153
So here, they're saying
that B of t is equal to 95.

00:02:19.480 --> 00:02:22.541
And so, the solution set,
you're really solving for t.

00:02:22.541 --> 00:02:25.950
So you're really solving
for all of the times

00:02:25.950 --> 00:02:30.940
when our height is going
to be 95 centimeters.

00:02:30.940 --> 00:02:35.940
So all times t

00:02:35.960 --> 00:02:40.960
when height of top of bar,

00:02:41.770 --> 00:02:44.770
of top of bar

00:02:45.720 --> 00:02:48.710
at 95 centimeters.

00:02:48.710 --> 00:02:50.490
And that's going to keep happening

00:02:50.490 --> 00:02:54.930
over and over and over again
as t goes forward in time.

00:02:54.930 --> 00:02:56.840
So you're going to have a very large,

00:02:56.840 --> 00:02:58.810
you're gonna have an infinite
solution set over here.

00:02:58.810 --> 00:03:01.020
You're gonna have an
infinite number of t's

00:03:01.020 --> 00:03:04.580
at which your solution at
which the top of the bar

00:03:04.580 --> 00:03:06.423
is at 95 centimeters.

00:03:07.270 --> 00:03:08.980
Now we have another question.

00:03:08.980 --> 00:03:12.070
This one is asking us,
what does the solution set

00:03:12.070 --> 00:03:16.170
to y is equal to 90 minus 12 sine

00:03:16.170 --> 00:03:19.310
of pi over two represent?

00:03:19.310 --> 00:03:21.383
So pause the video and think about that.

00:03:22.840 --> 00:03:24.640
All right, now this is pretty interesting.

00:03:24.640 --> 00:03:29.000
We can actually evaluate
what sine of pi over two is.

00:03:29.000 --> 00:03:33.030
So sign of pi over two
radians or sign of 90 degrees

00:03:33.030 --> 00:03:35.703
that is going to be equal to one.

00:03:36.840 --> 00:03:38.450
And so, that's the maximum value

00:03:38.450 --> 00:03:41.100
that this sign over here can take on.

00:03:41.100 --> 00:03:43.950
Now, we're going to
subtract 12 times that.

00:03:43.950 --> 00:03:46.600
So this is taking on a max.

00:03:46.600 --> 00:03:49.190
Then when you subtract 12 times that

00:03:49.190 --> 00:03:52.020
this is actually the minimum
value that you can take on.

00:03:52.020 --> 00:03:55.230
You're gonna have, you can't
get any lower than this.

00:03:55.230 --> 00:04:00.103
And so, this is going to be
the lowest, the lowest height

00:04:01.910 --> 00:04:06.910
for the top of the bar.

00:04:07.730 --> 00:04:08.563
And we're done.


Using specific values to test for inverses

WEBVTT
Kind: captions
Language: en

00:00:00.160 --> 00:00:01.450
- [Instructor] In this video,
we're gonna think about

00:00:01.450 --> 00:00:03.590
function inverses a little bit more,

00:00:03.590 --> 00:00:06.240
or whether functions are
inverses of each other,

00:00:06.240 --> 00:00:08.140
and specifically we're gonna think about

00:00:08.140 --> 00:00:10.440
can we tell that by essentially looking at

00:00:10.440 --> 00:00:14.030
a few inputs for the
functions and a few outputs?

00:00:14.030 --> 00:00:17.390
So for example, let's say we have f of x

00:00:17.390 --> 00:00:22.140
is equal to x squared plus three,

00:00:22.140 --> 00:00:27.140
and let's say that g of x
is equal to the square root,

00:00:29.410 --> 00:00:32.940
the principal root of x minus three.

00:00:32.940 --> 00:00:34.050
Pause this video and think about

00:00:34.050 --> 00:00:37.223
whether f and g are
inverses of each other.

00:00:38.870 --> 00:00:43.460
All right, now one approach
is to try out some values.

00:00:43.460 --> 00:00:47.410
So for example, let me make
a little table here for f,

00:00:47.410 --> 00:00:50.660
so this is x and then
this would be f of x.

00:00:50.660 --> 00:00:52.490
And then let me do the same thing for g.

00:00:52.490 --> 00:00:57.490
So we have x and then we have g of x.

00:00:58.020 --> 00:00:59.860
Now, first let's try a simple value.

00:00:59.860 --> 00:01:02.440
If we try out the value
one, what is f of one?

00:01:02.440 --> 00:01:04.010
Well, it's gonna be
one squared plus three.

00:01:04.010 --> 00:01:06.470
That's one plus three, that is four.

00:01:06.470 --> 00:01:11.470
So if g is an inverse of f,
then if I input four here,

00:01:12.360 --> 00:01:13.280
I should get one.

00:01:13.280 --> 00:01:15.110
Now, that wouldn't prove
that their inverses,

00:01:15.110 --> 00:01:16.660
but if it is an inverse,
we should at least

00:01:16.660 --> 00:01:17.493
be able to get that.

00:01:17.493 --> 00:01:19.290
So let's see if that's true.

00:01:19.290 --> 00:01:22.400
If we take four here,
four minus three is one.

00:01:22.400 --> 00:01:24.740
The principal root of that is one,

00:01:24.740 --> 00:01:26.090
so that's looking pretty good.

00:01:26.090 --> 00:01:27.900
Let's try one more value here.

00:01:27.900 --> 00:01:29.900
Let's try two.

00:01:29.900 --> 00:01:33.490
Two squared plus three is seven.

00:01:33.490 --> 00:01:36.033
Now let's try out seven here.

00:01:36.950 --> 00:01:39.330
Seven minus three is four.

00:01:39.330 --> 00:01:41.790
The principal root of that is two.

00:01:41.790 --> 00:01:44.650
So, so far it is looking pretty good.

00:01:44.650 --> 00:01:46.640
But then what happens if
we try a negative value?

00:01:46.640 --> 00:01:48.490
Pause the video and think about that.

00:01:49.430 --> 00:01:50.810
Let's do that.

00:01:50.810 --> 00:01:55.810
Let me put a negative two right over here.

00:01:57.340 --> 00:02:00.700
Now, if I have negative two
squared, that's positive four,

00:02:00.700 --> 00:02:03.920
plus three is seven, so I have seven here.

00:02:03.920 --> 00:02:07.430
But we already know that
when we input seven into g,

00:02:07.430 --> 00:02:09.820
we don't get negative two, we get two.

00:02:09.820 --> 00:02:12.580
In fact, there's no
way to get negative two

00:02:12.580 --> 00:02:15.100
out of this function right over here.

00:02:15.100 --> 00:02:16.860
So we have just found a case,

00:02:16.860 --> 00:02:19.340
and frankly any negative
number that you try to use

00:02:19.340 --> 00:02:21.450
would be a case where you could show

00:02:21.450 --> 00:02:24.570
that these are not inverses of each other.

00:02:24.570 --> 00:02:27.030
Not inverses.

00:02:27.030 --> 00:02:30.510
So you actually can use
specific points to determine

00:02:30.510 --> 00:02:33.110
that two functions like
this, especially functions

00:02:33.110 --> 00:02:36.240
that are defined over really
an infinite number of values,

00:02:36.240 --> 00:02:40.170
these are continuous functions,
that using specific points,

00:02:40.170 --> 00:02:43.270
you can show examples where
they are not inverses,

00:02:43.270 --> 00:02:45.260
but you actually can't use specific points

00:02:45.260 --> 00:02:46.880
to prove that they are inverses

00:02:46.880 --> 00:02:50.030
because there's an
infinite number of values

00:02:50.030 --> 00:02:51.360
that you could input into these functions,

00:02:51.360 --> 00:02:52.520
and there's no way that
you're going to be able

00:02:52.520 --> 00:02:54.070
to try out every value.

00:02:54.070 --> 00:02:58.490
For example, if I were
to tell you that h of x,

00:02:58.490 --> 00:03:01.680
really simple functions,
h of x is equal to four x,

00:03:01.680 --> 00:03:06.680
and let's say that j of x
is equal to x over four.

00:03:07.070 --> 00:03:09.350
We know that these are
inverse of each other.

00:03:09.350 --> 00:03:12.360
We'll prove it in other
ways in future videos,

00:03:12.360 --> 00:03:16.054
but you can't try every single input here

00:03:16.054 --> 00:03:19.560
and look at every single output,
and every single input here

00:03:19.560 --> 00:03:20.930
and every single output.

00:03:20.930 --> 00:03:23.620
So we need some other technique
other than just looking

00:03:23.620 --> 00:03:26.910
at specific values to
prove that two functions

00:03:26.910 --> 00:03:28.190
are inverses of each other.

00:03:28.190 --> 00:03:30.300
Although you can use
specific values to prove

00:03:30.300 --> 00:03:32.350
that they are not inverses of each other.


Verifying inverse functions from tables

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:01.800
- [Instructor] We're told
the following tables,

00:00:01.800 --> 00:00:04.270
give all of the input output pairs

00:00:04.270 --> 00:00:06.030
for the functions S and T.

00:00:06.030 --> 00:00:09.180
So we see this first table
here, we have some Xs

00:00:09.180 --> 00:00:12.020
and then they tell us what
the corresponding S of X is.

00:00:12.020 --> 00:00:14.460
And then in this table
that we have some Xs

00:00:14.460 --> 00:00:17.170
and they tell us the corresponding T of X.

00:00:17.170 --> 00:00:20.400
It says complete the table
for the composite function,

00:00:20.400 --> 00:00:25.310
S of T of X, we wanna fill
in these five entries here.

00:00:25.310 --> 00:00:28.700
And then they ask us our S and T inverses.

00:00:28.700 --> 00:00:31.430
So pause this video and see
if you can figure this out

00:00:31.430 --> 00:00:33.780
on your own before we
work through it together.

00:00:34.900 --> 00:00:37.500
All right, now let's work
through this together.

00:00:37.500 --> 00:00:38.780
So let's just remind ourselves

00:00:38.780 --> 00:00:41.510
what's going on with a
composite function like this.

00:00:41.510 --> 00:00:44.190
So you're going to take some X value

00:00:44.190 --> 00:00:46.230
and it looks like we're
first going to put it

00:00:46.230 --> 00:00:51.230
into the function T, that
is going to output T of X.

00:00:53.200 --> 00:00:55.630
And then we're gonna take
that output, take that T of X,

00:00:55.630 --> 00:00:59.020
and then it will be the input into S.

00:00:59.020 --> 00:01:01.600
So then we're gonna input that into S

00:01:01.600 --> 00:01:05.140
and then that would output
S of what we inputted

00:01:05.140 --> 00:01:08.170
which in this case is T of X.

00:01:08.170 --> 00:01:09.410
So let's go on that journey.

00:01:09.410 --> 00:01:11.820
So what we're gonna do is
first take these numbers,

00:01:11.820 --> 00:01:16.170
put them into the function,
T figure out what it outputs

00:01:16.170 --> 00:01:17.100
and then take that output

00:01:17.100 --> 00:01:19.480
and then put it into the function S,

00:01:19.480 --> 00:01:21.400
it's going to be a fun little ride.

00:01:21.400 --> 00:01:23.830
All right, so when X is equal to 12,

00:01:23.830 --> 00:01:26.830
we're gonna put it into
our function T first.

00:01:26.830 --> 00:01:29.660
So when X is an input into T

00:01:29.660 --> 00:01:32.710
the output is equal to negative one.

00:01:32.710 --> 00:01:35.850
So that's our T of X, and there
we're take this negative one

00:01:35.850 --> 00:01:39.460
and input it into S, so negative one here.

00:01:39.460 --> 00:01:44.010
And when you input that into
S you get as the output,

00:01:44.010 --> 00:01:48.320
S of negative one is 12,
so S of T of X is 12.

00:01:48.320 --> 00:01:52.070
So interestingly, this is 12.

00:01:52.070 --> 00:01:54.580
Now let's do the next one.

00:01:54.580 --> 00:01:59.580
So when we input 18 into
T, so the 18 is the input.

00:01:59.810 --> 00:02:03.000
The T of X, T of 18 is two.

00:02:03.000 --> 00:02:06.780
And then if we wanna do, if
we wanna input that into S,

00:02:06.780 --> 00:02:11.580
so this is gonna be the input
into S, the output is 18.

00:02:11.580 --> 00:02:16.180
Very interesting,
alright, let's keep going.

00:02:16.180 --> 00:02:21.180
So when we input 61 into
T, the output is eight.

00:02:21.890 --> 00:02:26.210
Then when we take eight
and we input it into S of X

00:02:26.210 --> 00:02:29.670
or S of eight I should
say, is going to be 61.

00:02:29.670 --> 00:02:33.060
All right, things are looking good so far

00:02:33.060 --> 00:02:35.370
and I'm running out of
colors, I'll do green.

00:02:35.370 --> 00:02:39.000
So when we take 70 and we input it into T,

00:02:39.000 --> 00:02:44.000
T of 7O is seven, when you
take seven and input it into S

00:02:45.740 --> 00:02:49.030
you get 70, all right.

00:02:49.030 --> 00:02:53.270
And then one last one I
will do in this blue color.

00:02:53.270 --> 00:02:58.270
When you take 100, input it
into T it outputs negative five

00:02:59.100 --> 00:03:04.100
you take negative five
input into S you get 100.

00:03:04.750 --> 00:03:08.500
So in every situation
that we have looked at

00:03:08.500 --> 00:03:11.420
right over here, in all
of these situations,

00:03:11.420 --> 00:03:16.420
we see that S of T of X is equal to X,

00:03:19.610 --> 00:03:22.320
which inclines us to believe
that they are inverses.

00:03:22.320 --> 00:03:24.360
Remember if these two are
inverses of each other,

00:03:24.360 --> 00:03:28.580
this would be true and also T of S of X

00:03:28.580 --> 00:03:30.760
is going to be equal to X.

00:03:30.760 --> 00:03:33.560
But we don't really know 100%

00:03:33.560 --> 00:03:36.750
unless we know that we have
looked at every combination

00:03:36.750 --> 00:03:39.440
in the domains for each of them.

00:03:39.440 --> 00:03:42.030
Now, when you look at these,
the two tables up here

00:03:42.030 --> 00:03:44.010
and I could have done this,
this is the one we looked at

00:03:44.010 --> 00:03:46.090
on our journey to get to
this 12 right over here.

00:03:46.090 --> 00:03:49.760
The following table gives
all of the input output pairs

00:03:49.760 --> 00:03:51.470
for the function S and T.

00:03:51.470 --> 00:03:55.890
So this right over here is
the domain for the function S

00:03:55.890 --> 00:04:00.320
and this right over here is
the domain for the function T.

00:04:00.320 --> 00:04:04.160
So because for every
member of the function S,

00:04:04.160 --> 00:04:06.470
every member of the
domain of the function S,

00:04:06.470 --> 00:04:09.440
the corresponding output right over there

00:04:09.440 --> 00:04:12.790
is the domain for the function T

00:04:12.790 --> 00:04:14.750
and it takes us back to where we began.

00:04:14.750 --> 00:04:16.750
And then the opposite is true as well,

00:04:16.750 --> 00:04:21.430
for every member of the
domain of T, what it outputs,

00:04:21.430 --> 00:04:24.470
that is the all of the
possible inputs for X

00:04:24.470 --> 00:04:26.890
and they all take us
back to where we began.

00:04:26.890 --> 00:04:30.043
So yes, the functions are inverses.


Reading inverse values from a table

WEBVTT
Kind: captions
Language: en

00:00:00.220 --> 00:00:01.500
- [Instructor] We're
told the following table

00:00:01.500 --> 00:00:05.110
shows a few inputs and outputs
of function G, all right.

00:00:05.110 --> 00:00:07.550
We have some possible inputs here for X

00:00:07.550 --> 00:00:10.530
and then the corresponding
outputs here, G of X.

00:00:10.530 --> 00:00:13.783
What is the value of G inverse of 54?

00:00:14.870 --> 00:00:16.800
So pause this video and see
if you can figure that out

00:00:16.800 --> 00:00:19.330
before we work through it together.

00:00:19.330 --> 00:00:20.163
All right.

00:00:20.163 --> 00:00:21.130
Let's just remind ourselves

00:00:21.130 --> 00:00:23.710
what an inverse function even does.

00:00:23.710 --> 00:00:26.910
So if you have some
value, X and you input it

00:00:26.910 --> 00:00:29.560
into some function, G,

00:00:29.560 --> 00:00:32.860
that function is going to output G of X.

00:00:32.860 --> 00:00:35.730
And an inverse function
takes us the other way.

00:00:35.730 --> 00:00:37.050
We could take this

00:00:37.050 --> 00:00:39.790
what was the output of G, G of X.

00:00:39.790 --> 00:00:43.140
We can input that into
an inverse function.

00:00:43.140 --> 00:00:44.750
The inverse function of G

00:00:44.750 --> 00:00:48.290
and that is actually going to give us X.

00:00:48.290 --> 00:00:50.610
It's going to get us back
to our original input

00:00:50.610 --> 00:00:51.930
right over here.

00:00:51.930 --> 00:00:55.860
So what we're focused on right
over here is G inverse of 54.

00:00:55.860 --> 00:00:57.584
So we can think about this part

00:00:57.584 --> 00:01:00.400
of this little chain that we set up.

00:01:00.400 --> 00:01:05.400
So what we're inputting into
this inverse function is 54.

00:01:05.540 --> 00:01:08.150
So what we want to say is, all right,

00:01:08.150 --> 00:01:11.800
when G of X is equal to 54, what is X?

00:01:11.800 --> 00:01:13.010
And we can see that right over here.

00:01:13.010 --> 00:01:17.530
When G of X is 54, the
corresponding input, original input,

00:01:17.530 --> 00:01:19.900
one way to think about it, is 62.

00:01:19.900 --> 00:01:21.940
So this will be equal to 62.

00:01:21.940 --> 00:01:24.450
Now some of you might have been
tempted to say, okay, look.

00:01:24.450 --> 00:01:27.420
It looks like I'm inputting
a 54 into a function.

00:01:27.420 --> 00:01:29.270
So I'll say, okay, X is the input.

00:01:29.270 --> 00:01:33.130
Let me just go to 54 right
over there as the input.

00:01:33.130 --> 00:01:38.130
But remember, this 54 isn't an
input into the inverse of G.

00:01:38.250 --> 00:01:40.550
This is an input into G of X.

00:01:40.550 --> 00:01:43.030
So if you wanted to evaluate this,

00:01:43.030 --> 00:01:46.753
if you wanted to evaluate G of 54,

00:01:47.660 --> 00:01:49.830
then you would look at the
54 up here and say, okay,

00:01:49.830 --> 00:01:52.140
that's going to be equal to 65.

00:01:52.140 --> 00:01:54.850
But we're looking at the inverse of G.

00:01:54.850 --> 00:01:59.700
So one way to think about it
is when 54 is the output in G,

00:02:00.540 --> 00:02:02.360
what is going to be the input?

00:02:02.360 --> 00:02:04.473
And we see that that is 62.


Meaningfully composing functions

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:02.040
- [Instructor] We're
told that Jaylen modeled

00:00:02.040 --> 00:00:05.040
the following relationships
about their bus ride.

00:00:05.040 --> 00:00:06.510
So there's three functions here.

00:00:06.510 --> 00:00:08.520
We have their inputs, and
we have their outputs.

00:00:08.520 --> 00:00:12.350
So function P, the input is
the time the bus arrives,

00:00:12.350 --> 00:00:14.000
given as lowercase b,

00:00:14.000 --> 00:00:16.460
and the output is the
probability that Jaylen

00:00:16.460 --> 00:00:18.210
gets to work on time.

00:00:18.210 --> 00:00:20.440
So P of b, all right.

00:00:20.440 --> 00:00:24.160
Function N, the input is
the time the bus arrives,

00:00:24.160 --> 00:00:25.330
given as k,

00:00:25.330 --> 00:00:28.350
and the output is the number
of people at the bus stop

00:00:28.350 --> 00:00:29.580
when the bus arrives.

00:00:29.580 --> 00:00:31.110
So it's the number of
people at the bus stop

00:00:31.110 --> 00:00:34.450
as a function of the time the bus arrives.

00:00:34.450 --> 00:00:37.430
And then T, centimeters of precipitation,

00:00:37.430 --> 00:00:41.910
the input is centimeters of
precipitation per hour, x.

00:00:41.910 --> 00:00:44.560
And this gives us the time the bus arrives

00:00:44.560 --> 00:00:46.570
as a function of that precipitation.

00:00:46.570 --> 00:00:47.860
Interesting.

00:00:47.860 --> 00:00:50.580
Now, they ask us which of the
following composed functions

00:00:50.580 --> 00:00:52.360
makes sense in this context?

00:00:52.360 --> 00:00:53.670
And we have to pick two.

00:00:53.670 --> 00:00:56.510
So pause this video and see
if you can have a go at it

00:00:56.510 --> 00:00:58.260
before we work through it together.

00:00:59.110 --> 00:01:00.360
All right, now let's look

00:01:00.360 --> 00:01:03.020
at this first composite function here.

00:01:03.020 --> 00:01:04.180
So let's see what's going on.

00:01:04.180 --> 00:01:07.570
One way to think about
this is we're taking x,

00:01:07.570 --> 00:01:11.310
we are inputting that
into our function, T,

00:01:11.310 --> 00:01:14.400
that will then output T of x,

00:01:14.400 --> 00:01:16.080
and then we're trying to input that

00:01:16.080 --> 00:01:19.030
into our function P,

00:01:19.030 --> 00:01:21.880
which would then output P of the input,

00:01:21.880 --> 00:01:24.250
which is T of x.

00:01:24.250 --> 00:01:25.690
Now, does this make sense?

00:01:25.690 --> 00:01:29.860
X is the centimeters of
precipitation per hour.

00:01:29.860 --> 00:01:34.140
T of x is a time the bus
arrives as a function of that.

00:01:34.140 --> 00:01:35.960
So we're gonna take the
time the bus arrives,

00:01:35.960 --> 00:01:38.980
is that a reasonable input into P?

00:01:38.980 --> 00:01:39.990
Well, let's see, function P,

00:01:39.990 --> 00:01:42.370
the input is the time the bus arrives.

00:01:42.370 --> 00:01:43.800
So this makes a lot of sense.

00:01:43.800 --> 00:01:46.230
We're taking this, which
is a time the bus arrives,

00:01:46.230 --> 00:01:48.100
and we're using that as an input into P,

00:01:48.100 --> 00:01:51.220
which is exactly what we
want to be the input into P.

00:01:51.220 --> 00:01:54.170
And so we have P as a function
of the time the bus arrives,

00:01:54.170 --> 00:01:55.450
which is then a function

00:01:55.450 --> 00:01:58.950
of the centimeters of
precipitation per hour.

00:01:58.950 --> 00:02:02.270
So I like this choice right over there.

00:02:02.270 --> 00:02:03.810
Now let's look at the next choice.

00:02:03.810 --> 00:02:07.780
P as a function of N of k.

00:02:07.780 --> 00:02:10.420
Let's see, so we're taking k,

00:02:10.420 --> 00:02:13.320
which is the time the bus arrives,

00:02:13.320 --> 00:02:17.320
we're inputting it into the function N,

00:02:17.320 --> 00:02:19.900
which then outputs N of k.

00:02:19.900 --> 00:02:21.430
Now the output here, N of k,

00:02:21.430 --> 00:02:23.530
is the number of people at the bus stop

00:02:23.530 --> 00:02:25.260
when the bus arrives.

00:02:25.260 --> 00:02:30.200
And then we're trying to put
that into our function P.

00:02:30.200 --> 00:02:32.940
Now, does that make sense
to take the number of people

00:02:32.940 --> 00:02:34.830
at the bus stop when the bus arrives

00:02:34.830 --> 00:02:37.960
and input it into a function
that expects as the input

00:02:37.960 --> 00:02:40.510
the time the bus arrives?

00:02:40.510 --> 00:02:42.570
That does not make sense.

00:02:42.570 --> 00:02:45.420
P wants as an input time the bus arrives,

00:02:45.420 --> 00:02:48.290
but we're giving it the number
of people at the bus stop.

00:02:48.290 --> 00:02:50.910
So I do not like this choice.

00:02:50.910 --> 00:02:54.190
Now let's look at choice C.

00:02:54.190 --> 00:02:56.500
So T of x, right over here,

00:02:56.500 --> 00:03:00.410
we already know that outputs
the time the bus arrives,

00:03:00.410 --> 00:03:04.000
which we know is a legitimate
input into function N.

00:03:04.000 --> 00:03:05.970
Function N takes the time the bus arrives

00:03:05.970 --> 00:03:08.180
and as a function, and
based on that input,

00:03:08.180 --> 00:03:10.190
it gives you the number
of people at the bus stop.

00:03:10.190 --> 00:03:11.350
So this makes sense.

00:03:11.350 --> 00:03:13.230
So I like this choice as well.

00:03:13.230 --> 00:03:15.290
So I'm guessing I won't like choice D,

00:03:15.290 --> 00:03:16.870
but let me validate that.

00:03:16.870 --> 00:03:18.690
So P of b.

00:03:18.690 --> 00:03:21.690
So this is going to
give us the probability

00:03:21.690 --> 00:03:23.490
that Jaylen gets to work on time,

00:03:23.490 --> 00:03:26.530
and then we're inputting
that into this function, T,

00:03:26.530 --> 00:03:29.410
that's expecting centimeters
of precipitation per hour.

00:03:29.410 --> 00:03:30.930
Well, that's not going to work out.

00:03:30.930 --> 00:03:32.460
You're trying to take a probability

00:03:32.460 --> 00:03:34.370
and you're inputting it into a function

00:03:34.370 --> 00:03:36.980
that expects centimeters
of precipitation per hour,

00:03:36.980 --> 00:03:38.870
so I would rule that one out as well.

00:03:38.870 --> 00:03:39.770
So I like A and C.


Finding the whole with a tape diagram

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:01.450
- [Instructor] We are told that Keisha

00:00:01.450 --> 00:00:03.920
can run 170 meters in one minute.

00:00:03.920 --> 00:00:07.060
This is 125% of the distance

00:00:07.060 --> 00:00:10.260
that she could run in one
minute three years ago.

00:00:10.260 --> 00:00:13.860
How far could Keisha run in
one minute three years ago?

00:00:13.860 --> 00:00:16.703
Pause this video, and see
if you can figure this out.

00:00:18.110 --> 00:00:19.650
All right, now let's do it together.

00:00:19.650 --> 00:00:22.450
And my brain wants to make
sure I know the difference

00:00:22.450 --> 00:00:27.450
between three years ago and today.

00:00:29.170 --> 00:00:34.170
So today, she can run
170 meters in one minute.

00:00:35.590 --> 00:00:36.540
And what we want to figure out

00:00:36.540 --> 00:00:40.790
is how much could she run in
one minute three years ago?

00:00:40.790 --> 00:00:45.790
Well, we know this 170 meters
is 125% of the distance

00:00:46.080 --> 00:00:47.690
that she could do three years ago.

00:00:47.690 --> 00:00:49.440
And the distance she
could do three years ago,

00:00:49.440 --> 00:00:51.840
of course, is 100% of the distance

00:00:51.840 --> 00:00:52.920
that she could do three years ago,

00:00:52.920 --> 00:00:55.750
because it's the exact same distance.

00:00:55.750 --> 00:00:58.290
But I like to think in terms of fractions.

00:00:58.290 --> 00:01:03.290
So 125%, I could rewrite
that as 125 over 100.

00:01:04.350 --> 00:01:07.860
If I divide both the numerator
and the denominator by 25,

00:01:07.860 --> 00:01:10.810
this is equivalent to five over four.

00:01:10.810 --> 00:01:12.740
So that 170 meters,

00:01:12.740 --> 00:01:15.433
that is five-fourths of what
she could do three years ago.

00:01:15.433 --> 00:01:17.080
And what you could do three years ago

00:01:17.080 --> 00:01:19.532
would be four-fourths of what
she could do three years ago,

00:01:19.532 --> 00:01:21.830
because that's 100%.

00:01:21.830 --> 00:01:25.100
And so to figure out if
five-fourths is 170 meters,

00:01:25.100 --> 00:01:26.760
what is four-fourths?

00:01:26.760 --> 00:01:30.463
Let us set up a tape
diagram right over here.

00:01:31.370 --> 00:01:34.640
And I'm going to try to hand
draw it as best as I can.

00:01:34.640 --> 00:01:37.550
I want to make five equal sections.

00:01:37.550 --> 00:01:38.700
And I know it's not exactly,

00:01:38.700 --> 00:01:40.840
but let's say for the sake
of argument for this video,

00:01:40.840 --> 00:01:42.400
this is five equal sections.

00:01:42.400 --> 00:01:44.720
And so if we imagine that
each of these are a fourth,

00:01:44.720 --> 00:01:46.120
this is five-fourths.

00:01:46.120 --> 00:01:48.340
And then this distance right over here

00:01:48.340 --> 00:01:50.720
is going to be 170 meters.

00:01:50.720 --> 00:01:52.220
That's what she could run today.

00:01:52.220 --> 00:01:54.830
And what we want to do is
figure out what is four-fourths?

00:01:54.830 --> 00:01:57.610
That's the distance that
she could run in one minute

00:01:57.610 --> 00:01:58.960
three years ago.

00:01:58.960 --> 00:02:01.090
So this is our question mark.

00:02:01.090 --> 00:02:01.923
Well to do that,

00:02:01.923 --> 00:02:05.340
we just have to figure out how
big is each of these fourths?

00:02:05.340 --> 00:02:08.070
And if five of them is 170 meters,

00:02:08.070 --> 00:02:12.550
well, I just have to divide five into 170.

00:02:12.550 --> 00:02:15.010
Five goes into 17 three times.

00:02:15.010 --> 00:02:17.180
Three times five is 15.

00:02:17.180 --> 00:02:21.710
Subtract, I get a two
here, bring down the zero.

00:02:21.710 --> 00:02:23.760
Five goes into 20 four times.

00:02:23.760 --> 00:02:27.640
Four times five is 20, and
it works out perfectly.

00:02:27.640 --> 00:02:32.220
So each of these
five-fourths are 34 meters.

00:02:32.220 --> 00:02:37.220
34 meters, 34 meters, 34 meters,
34 meters, and 34 meters.

00:02:38.400 --> 00:02:40.900
And so the distance that she
could run three years ago

00:02:40.900 --> 00:02:42.660
is going to be four of these fourths,

00:02:42.660 --> 00:02:45.020
or four of these 34 meters.

00:02:45.020 --> 00:02:50.020
So 34 times four, four times four is 16,

00:02:50.720 --> 00:02:54.350
three times four is 12, plus one is 13.

00:02:54.350 --> 00:02:56.640
So the mystery distance that
she could run in one minute

00:02:56.640 --> 00:03:01.090
three years ago is 136 meters.

00:03:01.090 --> 00:03:02.823
And we are done.


Finding percentages with a double number line

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:01.790
- [Instructor] We're
told that Omar's class

00:00:01.790 --> 00:00:03.580
has 28 students in it.

00:00:03.580 --> 00:00:06.530
21 of them take the bus to school.

00:00:06.530 --> 00:00:09.620
What percentage of the
students in Omar's class

00:00:09.620 --> 00:00:11.550
take the bus to school?

00:00:11.550 --> 00:00:14.403
Pause this video, and see
if you can figure that out.

00:00:15.750 --> 00:00:18.540
All right, well, I'm going
to try to visualize this

00:00:18.540 --> 00:00:22.830
with what we call a double number line.

00:00:22.830 --> 00:00:24.960
So let's say that is zero,

00:00:24.960 --> 00:00:28.540
and 28 is the total number of students

00:00:28.540 --> 00:00:30.840
in this classroom right over here.

00:00:30.840 --> 00:00:32.060
As a double number line,

00:00:32.060 --> 00:00:33.630
because I'm going to
make another number line

00:00:33.630 --> 00:00:34.920
right below it.

00:00:34.920 --> 00:00:38.570
But I'm going to write these
points in terms of percentages.

00:00:38.570 --> 00:00:41.090
So zero students would be 0%

00:00:41.090 --> 00:00:43.610
of the students in Omar's class,

00:00:43.610 --> 00:00:48.530
and 28 students would be 100%

00:00:48.530 --> 00:00:51.100
of the students in Omar's class.

00:00:51.100 --> 00:00:53.990
And so what we really need to do now is,

00:00:53.990 --> 00:00:56.910
we know that 21 of them take the bus.

00:00:56.910 --> 00:01:00.830
21 is going to be roughly around here.

00:01:00.830 --> 00:01:02.200
So that's 21.

00:01:02.200 --> 00:01:04.010
So we really just need to figure out,

00:01:04.010 --> 00:01:06.970
is what percentage is this going to be?

00:01:06.970 --> 00:01:11.310
One way to think about it is
what fraction is 21 of 28?

00:01:11.310 --> 00:01:13.683
Well, if I write 21 over 28,

00:01:14.860 --> 00:01:16.660
we know that we can
divide both the numerator

00:01:16.660 --> 00:01:18.720
and the denominator by seven,

00:01:18.720 --> 00:01:20.080
they're both divisible by seven,

00:01:20.080 --> 00:01:23.400
21 divided by seven is three,

00:01:23.400 --> 00:01:27.070
and 28 divided by seven is four.

00:01:27.070 --> 00:01:31.350
So 21 is three-fourths of 28.

00:01:31.350 --> 00:01:33.190
So let's divide this number line

00:01:33.190 --> 00:01:35.500
from zero to 28 into fourths.

00:01:35.500 --> 00:01:37.310
So that would be halfway,

00:01:37.310 --> 00:01:40.220
and now we have it divided into fourths,

00:01:40.220 --> 00:01:45.100
and we can see that 21 is one,
two, three of these fourths.

00:01:45.100 --> 00:01:48.010
Well, if 21 is three-fourths
of the way to 28,

00:01:48.010 --> 00:01:50.070
then whatever percentage this is here,

00:01:50.070 --> 00:01:52.630
that would be three-fourths
of the way to 100.

00:01:52.630 --> 00:01:55.740
So let's divide this into fourths as well.

00:01:55.740 --> 00:01:59.560
Now we know that one-fourth of 100 is 25%.

00:01:59.560 --> 00:02:02.220
Two-fourths of 100 is 50%.

00:02:02.220 --> 00:02:05.910
Three-fourths of 100 is 75%.

00:02:05.910 --> 00:02:07.860
So what percentage of the
students in Omar's class

00:02:07.860 --> 00:02:09.300
take the bus to school?

00:02:09.300 --> 00:02:11.213
75%.


Ways to rewrite a percentage

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:02.390
- [Instructor] We're asked
which of the following options

00:00:02.390 --> 00:00:07.050
have the same value as 2% of 90?

00:00:07.050 --> 00:00:09.130
Pause this video, and see
if you can figure it out.

00:00:09.130 --> 00:00:13.720
And as a reminder, they
say, pick two answers.

00:00:13.720 --> 00:00:16.320
All right, now let's work
through this together.

00:00:16.320 --> 00:00:18.110
So before I even look at the choices,

00:00:18.110 --> 00:00:20.160
I'm just going to try to
think of all of the ways

00:00:20.160 --> 00:00:23.810
that I can write 2% of 90.

00:00:23.810 --> 00:00:25.860
2%, this right over here,

00:00:25.860 --> 00:00:28.540
this literally means two per 100.

00:00:28.540 --> 00:00:32.530
So I could view it as the
same thing as two over 100,

00:00:32.530 --> 00:00:36.610
times 90, that's one
way to write 2% of 90.

00:00:36.610 --> 00:00:37.560
Two over a 100,

00:00:37.560 --> 00:00:41.400
I can divide both the numerator
and the denominator by two.

00:00:41.400 --> 00:00:43.700
And so two over a 100
is also the same thing

00:00:43.700 --> 00:00:45.320
as one over 50.

00:00:45.320 --> 00:00:50.320
And so it could also be
one over 50, times 90.

00:00:50.440 --> 00:00:52.880
Now let's see what other
choices they have here.

00:00:52.880 --> 00:00:54.910
Well, I could also write 2% as a decimal.

00:00:54.910 --> 00:00:58.300
Two-hundredths should
be written like this.

00:00:58.300 --> 00:01:00.900
Zero point, this is the tenths place.

00:01:00.900 --> 00:01:03.730
I have no tenths, but
I have two-hundredths.

00:01:03.730 --> 00:01:06.100
So I'm multiplying that by 90.

00:01:06.100 --> 00:01:07.570
All of these right over here

00:01:07.570 --> 00:01:11.630
are equivalent to the 2%, 2% of 90.

00:01:11.630 --> 00:01:13.470
All right, now let's look at the choices.

00:01:13.470 --> 00:01:16.260
This is 0.2 times 90.

00:01:16.260 --> 00:01:17.550
This is not 2%.

00:01:17.550 --> 00:01:20.790
This actually would be two-tenths, or 20%.

00:01:20.790 --> 00:01:22.793
So let me rule that one out.

00:01:23.650 --> 00:01:24.490
This one right over here.

00:01:24.490 --> 00:01:27.260
This is two-hundredths times 90.

00:01:27.260 --> 00:01:28.380
We already wrote that down.

00:01:28.380 --> 00:01:29.773
I like this choice.

00:01:30.950 --> 00:01:33.330
200 times 90.

00:01:33.330 --> 00:01:36.520
No, that would be far larger
than what we are looking for.

00:01:36.520 --> 00:01:38.470
We are looking for two-hundredths,

00:01:38.470 --> 00:01:40.930
not two hundreds times 90,

00:01:40.930 --> 00:01:43.210
so definitely rule that one out.

00:01:43.210 --> 00:01:45.020
Two times 90.

00:01:45.020 --> 00:01:49.640
Well, no, we're looking for
2% of 90, not two times 90.

00:01:49.640 --> 00:01:51.560
Once again, this is too large.

00:01:51.560 --> 00:01:54.640
And then we have two-hundredths times 90,

00:01:54.640 --> 00:01:56.430
which is the first thing that we wrote.

00:01:56.430 --> 00:01:59.033
And so I like this choice as well.