Adding 7 + 6

WEBVTT
Kind: captions
Language: en

00:00:00.074 --> 00:00:03.491
Voiceover:Let's think about
what seven plus six is,

00:00:03.491 --> 00:00:05.044
and I encourage you to pause this video

00:00:05.044 --> 00:00:06.488
and think about it on your own.

00:00:07.580 --> 00:00:09.534
I'm assuming you've given a go at it.

00:00:09.534 --> 00:00:10.795
Let's think about it.

00:00:10.795 --> 00:00:12.734
We could do this as seven objects

00:00:12.734 --> 00:00:14.555
plus six more objects,

00:00:14.581 --> 00:00:17.306
and then think about how
many total objects we had.

00:00:17.306 --> 00:00:20.439
For example, we could view
it as seven, say, tomatoes,

00:00:20.826 --> 00:00:24.845
so one tomato, two tomatoes,

00:00:25.276 --> 00:00:29.003
three tomatoes, four tomatoes,

00:00:30.049 --> 00:00:35.821
five, six, and seven,

00:00:35.960 --> 00:00:37.360
and then to that,

00:00:37.360 --> 00:00:39.132
we're going to add six more objects.

00:00:39.132 --> 00:00:40.337
Let's say they're blueberries,

00:00:40.337 --> 00:00:42.591
and we care about the
total amount of fruit.

00:00:42.591 --> 00:00:55.508
So, one, two, three, four, five, and six -

00:00:55.877 --> 00:00:57.535
six blueberries.

00:00:57.535 --> 00:00:59.376
Now, how many total pieces of fruit

00:00:59.376 --> 00:01:00.650
do I have in all now?

00:01:01.988 --> 00:01:04.136
I started with seven, so that's seven.

00:01:04.136 --> 00:01:06.168
If I keep counting, this is seven

00:01:06.168 --> 00:01:07.458
so this is going to be eight,

00:01:07.458 --> 00:01:13.344
nine, 10, 11, 12 and 13.

00:01:13.344 --> 00:01:17.661
I now have a total of 13 pieces of fruit -

00:01:17.661 --> 00:01:19.803
13 pieces of fruit.

00:01:19.803 --> 00:01:20.887
What are other ways

00:01:20.887 --> 00:01:22.435
that we could have thought about this?

00:01:22.604 --> 00:01:25.051
We could have also done
it on a number line.

00:01:26.451 --> 00:01:29.112
Let's draw ourselves a number line.

00:01:29.112 --> 00:01:32.753
Let me do this in a color

00:01:32.753 --> 00:01:34.706
that I haven't used yet, actually.

00:01:35.475 --> 00:01:37.574
Let's say I have a number line

00:01:40.450 --> 00:01:42.006
just like that.

00:01:42.006 --> 00:01:44.516
I can start the number line at seven.

00:01:44.840 --> 00:01:46.609
I could start at seven,

00:01:46.609 --> 00:01:48.219
and I'm going to add six more.

00:01:48.219 --> 00:01:50.260
I'm going to move six
more up the number line.

00:01:50.260 --> 00:01:56.141
This is one, two, three, four, five, six,

00:01:56.141 --> 00:01:57.222
and, of course, I could keep going.

00:01:57.222 --> 00:02:02.774
It's going to be eight,
nine, 10, 11, 12, 13.

00:02:02.774 --> 00:02:04.375
Of course, you could keep on going.

00:02:05.128 --> 00:02:07.289
Seven plus six - you could visualize this

00:02:07.642 --> 00:02:09.479
as starting at seven,

00:02:09.479 --> 00:02:12.502
and then making six jumps
up the number line -

00:02:12.502 --> 00:02:18.034
one, two, three, four, five, six.

00:02:18.034 --> 00:02:23.805
Either way, we get to 13.

00:02:24.374 --> 00:02:27.391
Another way to think about it is,

00:02:27.391 --> 00:02:29.856
look, we started with seven.

00:02:29.856 --> 00:02:32.964
We added three to get to 10,

00:02:32.964 --> 00:02:35.145
and so we have to add another three,

00:02:35.145 --> 00:02:38.382
which gets us to 13.

00:02:38.382 --> 00:02:39.615
That really goes to the heart

00:02:39.615 --> 00:02:42.545
of what the number 13 represents.

00:02:42.545 --> 00:02:47.928
The number 13 has a one as its left digit,

00:02:47.928 --> 00:02:50.112
This digit is in the tens place,

00:02:50.342 --> 00:02:53.653
so it literally represents one 10.

00:02:53.976 --> 00:02:59.493
So, it's one 10 plus
three, plus three ones.

00:02:59.493 --> 00:03:01.664
You see that right over here.

00:03:01.833 --> 00:03:03.614
When I added the pieces of fruit,

00:03:03.814 --> 00:03:07.565
this right over here is one group of 10.

00:03:07.918 --> 00:03:10.149
So, that's one group of 10.

00:03:11.318 --> 00:03:12.873
We had to add three to get

00:03:12.873 --> 00:03:14.432
to that one group of 10.

00:03:14.432 --> 00:03:17.094
We kind of filled that bucket,

00:03:17.094 --> 00:03:19.124
and then we had three more,

00:03:19.124 --> 00:03:21.256
so when you add seven plus six,

00:03:21.256 --> 00:03:24.440
you fill one whole group of 10

00:03:24.440 --> 00:03:27.450
and then you have three ones left over.

00:03:27.450 --> 00:03:30.648
This is the three ones right over here.

00:03:30.648 --> 00:03:32.535
Another way you could think about it is

00:03:32.535 --> 00:03:34.759
seven plus six is the same thing

00:03:35.574 --> 00:03:41.383
as 10 plus three, which, of course, is 13.

00:03:41.383 --> 00:03:44.106
This is the same thing - one 10 is 10,

00:03:44.875 --> 00:03:48.812
plus three, plus three ones -

00:03:48.812 --> 00:03:52.812
13 either way you look at it.


Classifying vectors and quantities example

WEBVTT
Kind: captions
Language: en

00:00:00.886 --> 00:00:05.205
Voiceover:Below is a drawing
of vectors U,V, and W.

00:00:05.205 --> 00:00:06.745
Sort the cards according

00:00:06.745 --> 00:00:08.726
to the quantity they represent.

00:00:08.726 --> 00:00:11.217
If a card doesn't represent
any of the quantities,

00:00:11.217 --> 00:00:12.890
it should be left unsorted.

00:00:12.890 --> 00:00:14.275
Let's see what they have over here.

00:00:14.275 --> 00:00:16.525
This, they've drawn vector V here in blue,

00:00:16.525 --> 00:00:19.238
vector W here in magenta,

00:00:19.238 --> 00:00:21.585
and then vector U here in purple.

00:00:21.585 --> 00:00:25.290
Let's see what they choices are.

00:00:25.290 --> 00:00:28.505
The rectangular coordinates 3,4.

00:00:28.505 --> 00:00:32.551
If we go 3 along the X axis
in the positive direction,

00:00:32.551 --> 00:00:36.213
and then we go up 4, 1,2,3,4,

00:00:36.213 --> 00:00:39.894
we get to the head of vector V.

00:00:39.894 --> 00:00:42.701
It's tail, it started at the origin.

00:00:42.701 --> 00:00:48.254
This could, right over
here, represent vector V.

00:00:48.254 --> 00:00:49.998
If we say hey look, the tail of our vector

00:00:49.998 --> 00:00:51.089
is going to be the origin,

00:00:51.089 --> 00:00:52.862
then this coordinate, right over here,

00:00:52.862 --> 00:00:54.241
represents the head of a vector.

00:00:54.241 --> 00:00:56.974
This could be a
representation for vector V.

00:00:56.974 --> 00:00:58.493
Another way to think about it is vector V,

00:00:58.493 --> 00:01:01.963
from it's tail, goes 3 in
the horizontal direction

00:01:01.963 --> 00:01:04.577
and 4 in the vertical direction.

00:01:04.577 --> 00:01:05.973
That's another way to think about it.

00:01:05.973 --> 00:01:07.518
I would say that's definitely

00:01:07.518 --> 00:01:09.023
a representation of vector V.

00:01:09.023 --> 00:01:11.406
The number 5.

00:01:11.406 --> 00:01:14.910
The number 5 won't represent a vector.

00:01:14.910 --> 00:01:16.525
This is just a number.

00:01:16.525 --> 00:01:17.750
This could be a magnitude,

00:01:17.750 --> 00:01:18.603
but to be a vector,

00:01:18.603 --> 00:01:20.666
we need a magnitude and a direction.

00:01:20.666 --> 00:01:23.679
Right over here we have
the magnitude of vector V.

00:01:23.679 --> 00:01:25.652
Let's think about whether 5

00:01:25.652 --> 00:01:27.307
could be the magnitude of vector V.

00:01:27.307 --> 00:01:29.466
If 5 is the magnitude,

00:01:29.466 --> 00:01:31.318
it would be the length of this vector.

00:01:31.318 --> 00:01:32.913
We could think about that

00:01:32.913 --> 00:01:34.405
from the Pythagorean theorem.

00:01:34.405 --> 00:01:37.365
This side, right over here, has length 3.

00:01:37.365 --> 00:01:39.859
That side over there has length 4.

00:01:39.859 --> 00:01:43.918
You might recognize this
as a 3,4,5 triangle.

00:01:43.918 --> 00:01:45.073
Another way to think about it

00:01:45.073 --> 00:01:46.662
if we use the Pythagorean theorem

00:01:46.662 --> 00:01:51.665
3 squared is 9 plus 4
squared plus 16 is 25,

00:01:51.665 --> 00:01:53.851
which is going to be
this magnitude squared.

00:01:53.851 --> 00:01:56.416
The magnitude of vector V is 5.

00:01:57.308 --> 00:01:59.044
We could throw that right over here.

00:01:59.044 --> 00:02:01.031
The magnitude of vector V is 5.

00:02:01.031 --> 00:02:05.918
The length of this, how
we depicted the vector.

00:02:05.918 --> 00:02:08.869
Let's look at vector U.

00:02:08.869 --> 00:02:13.810
Vector U is just a shifted
representation of vector V.

00:02:13.810 --> 00:02:15.401
Notice, we started its tail,

00:02:15.401 --> 00:02:21.160
3 to the right and 4 up.

00:02:21.160 --> 00:02:25.858
Vector U is really the
same thing as vector V.

00:02:25.858 --> 00:02:27.739
We don't care about where they start.

00:02:27.739 --> 00:02:30.510
We just care about, from
the tail to the head,

00:02:30.510 --> 00:02:33.093
how much do you move in
the horizontal direction,

00:02:33.093 --> 00:02:34.883
and how much do you move
in the vertical direction,

00:02:34.883 --> 00:02:37.207
and for both of them, you move positive 3

00:02:37.207 --> 00:02:40.022
in the horizontal and
positive 4 in the vertical.

00:02:40.022 --> 00:02:43.972
Vector U really is the
same thing as vector V

00:02:43.972 --> 00:02:45.815
if we're thinking in vector terms.

00:02:45.815 --> 00:02:52.026
Vector W, if we start at the tail

00:02:52.026 --> 00:02:54.023
to go to the head, we have to

00:02:54.023 --> 00:02:56.605
go negative 3 in the X direction

00:02:56.605 --> 00:03:00.014
and negative 4 in the Y direction.

00:03:00.014 --> 00:03:03.254
Vector W is the negative of vector V,

00:03:03.254 --> 00:03:04.865
or the negative of vector U,

00:03:04.865 --> 00:03:07.950
but it's not equal to either one of them.

00:03:07.950 --> 00:03:09.553
We're all done, we have set up.

00:03:09.553 --> 00:03:11.534
These are the quantities that

00:03:11.534 --> 00:03:14.495
are in the appropriate buckets.


Place value example with 25

WEBVTT
Kind: captions
Language: en

00:00:00.547 --> 00:00:01.921
Voiceover:What I want to
do in this video is think

00:00:01.921 --> 00:00:05.426
a little bit about what
numbers really represent.

00:00:05.426 --> 00:00:11.938
Let's say someone were to
write down the number 25.

00:00:11.938 --> 00:00:14.177
So what does this really mean?

00:00:14.177 --> 00:00:16.850
Well the five here, this
is in the one's place

00:00:16.850 --> 00:00:19.570
so it literally represents five ones.

00:00:19.570 --> 00:00:21.538
And this two right over here, this is in

00:00:21.538 --> 00:00:26.162
the 10's place, so let me write that down.

00:00:26.162 --> 00:00:29.088
That's the 10's place,
that's the one's place,

00:00:29.088 --> 00:00:31.314
so this represents two 10s.

00:00:31.314 --> 00:00:32.817
What do I mean by that?

00:00:32.817 --> 00:00:35.186
Well another way to write this is this two

00:00:35.186 --> 00:00:39.395
represents two 10s, just
let me write that down.

00:00:39.395 --> 00:00:50.734
This is equal to two 10s plus five ones.

00:00:51.087 --> 00:00:52.877
Now what are two 10s?

00:00:52.985 --> 00:00:55.622
Well that's literally if
you can imagine a 10 being

00:00:55.622 --> 00:00:59.431
a block of 10 things,
so you could imagine if

00:00:59.431 --> 00:01:00.983
this was a 10 right over here.

00:01:00.983 --> 00:01:04.794
Let me copy and paste this, so copy.

00:01:04.794 --> 00:01:07.038
So this is going to be the same thing,

00:01:07.038 --> 00:01:09.766
this is going to be
equal to, you have your

00:01:09.766 --> 00:01:15.574
two 10s, so one 10 and two tens,

00:01:15.574 --> 00:01:17.318
so you have your two 10s.

00:01:17.318 --> 00:01:19.799
So that's your two 10s right over there,

00:01:19.799 --> 00:01:24.054
that really represents
20 plus your five ones.

00:01:24.054 --> 00:01:28.070
Plus your five ones.

00:01:28.070 --> 00:01:29.942
Well if this is a one
right over here, and I'm

00:01:29.942 --> 00:01:31.974
just representing it
as kind of one vertical

00:01:31.974 --> 00:01:34.695
bar, five ones are going to look like,

00:01:34.695 --> 00:01:41.398
your going to have one,
two, I think you see

00:01:41.398 --> 00:01:55.095
where this is going,
three, four, and five.

00:01:55.095 --> 00:01:57.430
And so these are all different ways

00:01:57.430 --> 00:02:00.151
of representing the same quantity.

00:02:00.151 --> 00:02:03.558
25 is two 10s and five ones.

00:02:03.558 --> 00:02:05.878
This two is in the 10's place,

00:02:05.878 --> 00:02:07.462
the five is in the one's place.

00:02:07.462 --> 00:02:10.055
So two 10s plus five
ones, so you could then

00:02:10.055 --> 00:02:15.799
view it as two groups of 10 plus five ones

00:02:15.799 --> 00:02:16.630
right over here.

00:02:16.630 --> 00:02:18.683
And what's this going
to be the same thing as?

00:02:18.683 --> 00:02:22.950
Well two groups of 10, this
is the same thing as 20

00:02:22.950 --> 00:02:27.111
and this over here is equal to five.

00:02:27.111 --> 00:02:31.255
So once again, all ways to represent 25.

00:02:31.255 --> 00:02:35.846
You can view this as two
10s plus five ones or

00:02:35.846 --> 00:02:41.206
you can view this as 20,
let me write this down,

00:02:41.206 --> 00:02:46.519
25 you could view because
this two is two 10s

00:02:46.730 --> 00:02:50.730
that represents 20 plus the five ones.


28 Tall structures

WEBVTT
Kind: captions
Language: en

00:00:00.639 --> 00:00:01.966
- [Voiceover] The Empire State Building,

00:00:01.966 --> 00:00:05.083
the Sears Tower, the
Canadian National Tower,

00:00:05.083 --> 00:00:09.605
each of these structures
was the tallest in the world

00:00:09.605 --> 00:00:14.398
at the time they were built, or no errors.

00:00:14.398 --> 00:00:15.554
So, let's see.

00:00:15.554 --> 00:00:16.994
They list a bunch of structures here.

00:00:16.994 --> 00:00:19.450
The Empire State
Building, the Sears Tower,

00:00:19.450 --> 00:00:21.389
the Canadian National Tower.

00:00:21.389 --> 00:00:25.127
And then they talk about
each of these structures.

00:00:25.127 --> 00:00:26.706
So really they're talking about

00:00:26.706 --> 00:00:29.445
a singular subject right over here, each.

00:00:29.445 --> 00:00:31.256
If they said all of these structures,

00:00:31.256 --> 00:00:33.030
then we would talk about,

00:00:33.030 --> 00:00:33.969
we'd have a plural subject,

00:00:33.969 --> 00:00:36.081
but they're saying, "Hey,
take one of these at a time,"

00:00:36.081 --> 00:00:39.250
each of these structures
and so this makes sense

00:00:39.250 --> 00:00:41.963
that this is was 'cause
was would be the singular

00:00:41.963 --> 00:00:45.388
form of the verb, each
of these structures was,

00:00:45.388 --> 00:00:46.792
if this was all of these structures,

00:00:46.792 --> 00:00:48.333
then we'd put a were there.

00:00:48.333 --> 00:00:49.512
They say each of these structures

00:00:49.512 --> 00:00:52.673
was the tallest in the world

00:00:52.673 --> 00:00:56.254
at the time they were built.

00:00:56.254 --> 00:00:57.888
Now, this is interesting.

00:00:57.888 --> 00:01:01.588
This is a singular subject,
singular verb, that makes sense,

00:01:01.588 --> 00:01:03.677
but then we shouldn't be
referring to they now,

00:01:03.677 --> 00:01:05.596
we're talking about
each of the structures,

00:01:05.596 --> 00:01:08.414
so it was really at the time it was built,

00:01:08.414 --> 00:01:11.614
at the time it was built.

00:01:11.614 --> 00:01:12.884
Remember, we're talking about each

00:01:12.884 --> 00:01:16.249
of the structures taken individually,

00:01:16.249 --> 00:01:20.017
so each of these structures
was the tallest in the world

00:01:20.017 --> 00:01:22.655
at the time it was built.

00:01:22.655 --> 00:01:24.272
If you wanted to say they were built,

00:01:24.272 --> 00:01:26.482
you would have to say
all of these structures

00:01:26.482 --> 00:01:31.270
were the tallest in the world
at the time they were built,

00:01:31.270 --> 00:01:33.078
but we don't have the option of changing

00:01:33.078 --> 00:01:35.481
the each to the all right over here,

00:01:35.481 --> 00:01:38.314
so since we're talking
about a singular subject,

00:01:38.314 --> 00:01:41.437
we would keep the was
and instead of they were,

00:01:41.437 --> 00:01:43.516
it would be an it was,

00:01:43.516 --> 00:01:45.321
so let's just read the
whole thing over again.

00:01:45.321 --> 00:01:47.713
The Empire State
Building, the Sears Tower,

00:01:47.713 --> 00:01:50.108
the Canadian National Tower,

00:01:50.108 --> 00:01:55.108
each of these structures
was the tallest in the world

00:01:55.352 --> 00:01:58.415
at the time it was built.

00:01:58.415 --> 00:02:00.897
Yep, error in part D.


Matrix equations and systems

WEBVTT
Kind: captions
Language: en

00:00:00.581 --> 00:00:02.233
Voiceover:I have a system of 2 equations

00:00:02.233 --> 00:00:03.366
with 2 unknowns here.

00:00:03.366 --> 00:00:04.566
We've seen how to solve this,

00:00:04.566 --> 00:00:06.233
and there's multiple
techniques we've used,

00:00:06.233 --> 00:00:08.166
substitution, elimination,

00:00:08.166 --> 00:00:09.766
and we could do that right over here.

00:00:09.766 --> 00:00:11.235
In fact, you could just add the two.

00:00:11.235 --> 00:00:12.834
The left sides of the equations

00:00:12.834 --> 00:00:14.169
and the right sides of the equations,

00:00:14.169 --> 00:00:16.340
the s's would cancel out.

00:00:16.340 --> 00:00:17.366
Actually, let's just do it to show

00:00:17.366 --> 00:00:19.768
how that's relatively straightforward,

00:00:19.768 --> 00:00:21.501
for at least this example right over here.

00:00:21.501 --> 00:00:23.300
You add the left-hand sides.

00:00:23.300 --> 00:00:24.635
These cancel out.

00:00:24.635 --> 00:00:27.266
You're left with negative t.

00:00:27.340 --> 00:00:34.700
Negative t is equal to 7 plus
negative 6 is equal to 1,

00:00:34.700 --> 00:00:37.700
or you get the t is equal to negative 1.

00:00:37.700 --> 00:00:39.433
t is equal to negative 1.

00:00:39.433 --> 00:00:41.567
If t is equal to negative 1,

00:00:41.567 --> 00:00:45.633
this top equation, you
could use either one,

00:00:45.633 --> 00:00:48.600
would simplify to 2 times s.

00:00:49.366 --> 00:00:52.333
Negative 5 times negative 1 is plus 5.

00:00:52.701 --> 00:00:55.534
Plus 5 is equal to 7,

00:00:55.658 --> 00:00:57.330
is equal to ...

00:00:57.330 --> 00:00:59.635
let me use those same colors
that we have over there,

00:00:59.635 --> 00:01:05.366
is equal to 7, or, and we
could do this part in our head,

00:01:05.366 --> 00:01:08.766
2s must be equal to 2
and that s is equal to 1.

00:01:08.766 --> 00:01:11.100
2 times 1 plus 5 is 7,

00:01:11.100 --> 00:01:13.834
and so we have s is equal to 1.

00:01:13.834 --> 00:01:15.666
That was pretty straightforward.

00:01:15.777 --> 00:01:17.367
What we're going to do in this video

00:01:17.367 --> 00:01:20.301
is represent the same system,

00:01:20.301 --> 00:01:22.100
but we're going to
represent it esssentially

00:01:22.100 --> 00:01:23.834
as a matrix equation,

00:01:23.834 --> 00:01:27.679
and we're going to solve
it using inverse matrices.

00:01:28.077 --> 00:01:30.433
I'll give you a little bit of a warning.

00:01:30.433 --> 00:01:32.167
It's going to be more involved.

00:01:32.167 --> 00:01:33.967
It's going to take us more time to this,

00:01:33.967 --> 00:01:34.700
and you're probably going to say,

00:01:34.700 --> 00:01:36.900
"Well, why are we even going
through the trouble of it?"

00:01:36.900 --> 00:01:39.330
The value of what we're
going to do in this video

00:01:39.330 --> 00:01:41.379
is that it's very useful in computation

00:01:41.379 --> 00:01:44.435
where you might solve
almost the same system

00:01:44.435 --> 00:01:46.330
over and over and over again.

00:01:46.330 --> 00:01:48.330
Maybe the left-hand sides are the same,

00:01:48.330 --> 00:01:49.711
the right-hands keep changing,

00:01:49.711 --> 00:01:51.868
and this might be something
that you might see

00:01:51.868 --> 00:01:53.433
while writing a computer game

00:01:53.433 --> 00:01:58.373
or while working on some
type of a computer problem.

00:01:58.711 --> 00:01:59.833
And this is a general theme.

00:01:59.833 --> 00:02:02.700
A lot of the value of
matrices are they are ways

00:02:02.700 --> 00:02:04.167
to represent problems,

00:02:04.167 --> 00:02:05.500
mathematical problems,

00:02:05.500 --> 00:02:07.919
ways to represent data,

00:02:08.673 --> 00:02:11.981
and then we can use matrix
operations, matrix equations

00:02:11.981 --> 00:02:14.300
to essentially manipulate
them in appropriate ways

00:02:14.300 --> 00:02:16.900
if we're, for the most part,
writing computer programs

00:02:16.900 --> 00:02:18.933
or things like computer programs.

00:02:19.592 --> 00:02:21.930
Bear with me, you will
enjoy it eventually,

00:02:21.930 --> 00:02:23.119
what we're about to do,

00:02:23.119 --> 00:02:26.333
and one day, you will see that
it is actually quite useful.

00:02:26.700 --> 00:02:29.171
The first thing we need to
see or need to appreciate

00:02:29.171 --> 00:02:32.200
is that this can be represented
by a matrix equation.

00:02:32.635 --> 00:02:33.633
Now what I'm going to
do is I'm going to take

00:02:33.633 --> 00:02:35.433
the coefficients here.

00:02:36.003 --> 00:02:38.433
I'm going to take the coefficients here,

00:02:38.433 --> 00:02:42.334
so 2, negative 5,

00:02:42.673 --> 00:02:45.433
2, negative 5,

00:02:45.433 --> 00:02:47.188
negative 2,

00:02:47.188 --> 00:02:50.333
negative 2 and 4, and positive 4.

00:02:50.438 --> 00:02:54.340
All I've done is I've taken
the coefficients here,

00:02:54.340 --> 00:02:58.103
and I'm going to claim that
that times the column vector,

00:02:58.103 --> 00:03:00.333
column vector st,

00:03:00.673 --> 00:03:03.866
s and t,

00:03:04.234 --> 00:03:11.866
being equal to the column
vector 7, negative 6,

00:03:12.566 --> 00:03:15.200
7, negative 6,

00:03:16.501 --> 00:03:19.433
is the exact same thing as
what we have right over here.

00:03:19.433 --> 00:03:21.980
These are representing
the same constraints

00:03:21.980 --> 00:03:23.170
on the variable s and t.

00:03:23.170 --> 00:03:25.302
You say, "Wait, I don't quite get that."

00:03:25.302 --> 00:03:27.233
If you're saying you don't quite get that,

00:03:27.233 --> 00:03:28.733
multiply this out.

00:03:28.766 --> 00:03:31.252
Multiply this out and
think about which entries

00:03:31.252 --> 00:03:34.135
they need to be equal to
when you multiply it out.

00:03:34.368 --> 00:03:37.966
And you would see this entry,
this first row, first column,

00:03:37.966 --> 00:03:39.366
that's going to be this row.

00:03:39.366 --> 00:03:41.901
We're going to be dealing
with this row and that column,

00:03:41.901 --> 00:03:47.433
so if you think about it,
this tells us that 2 times s,

00:03:47.433 --> 00:03:53.970
2 times s plus negative 5 times t,

00:03:53.970 --> 00:03:58.033
so I could say minus 5 times t

00:03:58.033 --> 00:04:02.100
must be equal to the first entry up here,

00:04:02.100 --> 00:04:03.300
first row, first column,

00:04:03.300 --> 00:04:04.766
is equal to 7.

00:04:04.766 --> 00:04:06.567
All I did is I multiplied,

00:04:06.567 --> 00:04:08.566
I dealt with the first
row, first column and said,

00:04:08.566 --> 00:04:11.300
when I take essentially
the dot product of those,

00:04:11.300 --> 00:04:13.167
and if you don't know what a
dot product is, don't worry.

00:04:13.167 --> 00:04:14.767
We'll explain it at other places.

00:04:14.767 --> 00:04:16.033
It's essentially what I just did here,

00:04:16.033 --> 00:04:19.233
the first entry here
times the first entry,

00:04:19.233 --> 00:04:20.833
the second entry here
times the second entry,

00:04:20.833 --> 00:04:22.033
and we add them together,

00:04:22.033 --> 00:04:23.367
that that must be equal to 7,

00:04:23.367 --> 00:04:24.100
but when you do that,

00:04:24.100 --> 00:04:26.134
you essentially construct
this first equation.

00:04:26.366 --> 00:04:28.633
Now when you do it with the
second row and this column,

00:04:28.633 --> 00:04:30.167
you construct the second equation.

00:04:30.167 --> 00:04:32.700
You get negative 2 times s,

00:04:32.700 --> 00:04:36.100
negative 2 times s

00:04:36.100 --> 00:04:42.011
plus 4 times t,

00:04:42.011 --> 00:04:43.966
4 times t,

00:04:43.966 --> 00:04:50.720
is equal to negative 6.

00:04:50.720 --> 00:04:52.500
Hopefully you appreciate
that this contains

00:04:52.500 --> 00:04:54.400
the same information as that.

00:04:54.833 --> 00:04:56.600
And there is other ways
that I could have done it.

00:04:56.700 --> 00:04:58.633
For example, you could have,

00:04:58.633 --> 00:05:00.213
instead of writing it this way,

00:05:00.690 --> 00:05:02.933
this system is obviously the same thing,

00:05:02.992 --> 00:05:04.700
is obviously the same thing as ...

00:05:04.700 --> 00:05:06.700
and actually, let me
just copy and paste it,

00:05:06.700 --> 00:05:09.100
is the same thing as ...

00:05:09.100 --> 00:05:11.233
so copy and paste,

00:05:11.233 --> 00:05:12.700
is the same thing as this system,

00:05:12.700 --> 00:05:14.400
where I'm really just swapping.

00:05:14.767 --> 00:05:17.666
So once again, copy and paste.

00:05:18.499 --> 00:05:19.989
Obviously, I've written
the second one first,

00:05:19.989 --> 00:05:21.566
and I've written the first one second,

00:05:21.566 --> 00:05:23.866
so this is obviously the same system.

00:05:24.033 --> 00:05:28.500
If I wanted to construct a
matrix equation with this system,

00:05:28.500 --> 00:05:30.633
I would just swap all of the rows.

00:05:30.633 --> 00:05:33.112
The first row here would be negative 2, 4.

00:05:33.627 --> 00:05:35.900
I would swap the rows
for the coefficients,

00:05:35.900 --> 00:05:38.633
but I would still keep the
s and ts in the same order,

00:05:38.633 --> 00:05:39.766
and you could do that.

00:05:39.766 --> 00:05:42.634
Try to represent this right over here

00:05:42.634 --> 00:05:44.366
as a matrix equation.

00:05:44.366 --> 00:05:46.234
You would have the matrix here would be

00:05:46.234 --> 00:05:48.633
negative 2, 4, 2, negative 5,

00:05:48.633 --> 00:05:51.200
and this would be negative 6, 7.

00:05:51.373 --> 00:05:53.433
But now that we have set this up,

00:05:53.433 --> 00:05:56.033
how do we actually solve
something like this?

00:05:56.033 --> 00:05:57.733
Why do we even do this?

00:05:58.038 --> 00:05:59.900
To think about this, let's
actually think about it

00:05:59.900 --> 00:06:02.601
in terms of literally a matrix equation.

00:06:02.833 --> 00:06:07.066
Let's say that A, the matrix A
is this thing right over here.

00:06:07.166 --> 00:06:09.400
This thing over here is the matrix A.

00:06:09.701 --> 00:06:11.167
Let's say that this right over here,

00:06:11.167 --> 00:06:13.901
this is the column vector x.

00:06:14.685 --> 00:06:17.868
I'll write it as a
vector x right over here.

00:06:17.967 --> 00:06:19.500
You have the column vector x,

00:06:19.500 --> 00:06:20.967
and then this right over here,

00:06:20.967 --> 00:06:22.433
you could say that this is equal,

00:06:22.433 --> 00:06:24.536
and let's call this the column vector b.

00:06:24.835 --> 00:06:28.233
This is equal to the column vector b.

00:06:28.233 --> 00:06:30.502
We're essentially saying that A,

00:06:30.502 --> 00:06:35.367
the matrix A times the
column vector x is equal to,

00:06:35.367 --> 00:06:37.678
is equal to the column vector b.

00:06:37.678 --> 00:06:39.635
Let me write that again right over here,

00:06:39.635 --> 00:06:41.104
just to emphasize it.

00:06:41.104 --> 00:06:45.700
The matrix A times the column vector x

00:06:45.700 --> 00:06:48.167
is going to be equal to,

00:06:48.167 --> 00:06:52.533
is equal to the column vector B.

00:06:52.634 --> 00:06:53.900
This is what they're talking about

00:06:53.900 --> 00:06:56.368
when they say a matrix equation.

00:06:56.368 --> 00:06:58.439
Actually, before we even
think about computation

00:06:58.439 --> 00:07:00.166
and computer graphics and all of that,

00:07:00.166 --> 00:07:03.166
you will see a lot of
things like this in physics,

00:07:03.166 --> 00:07:04.634
where they're speaking in general terms,

00:07:04.634 --> 00:07:06.034
or they might not even be specifying

00:07:06.034 --> 00:07:07.501
the dimensions of the matrix

00:07:07.501 --> 00:07:09.234
or the dimensions of this vector,

00:07:09.234 --> 00:07:11.566
but they're talking about
some general property,

00:07:11.566 --> 00:07:13.100
in, say, physics.

00:07:13.100 --> 00:07:17.633
You will see matrix vector
equations like this a lot

00:07:17.633 --> 00:07:20.466
as you get into higher
and higher sciences.

00:07:21.033 --> 00:07:22.248
But once again,

00:07:22.633 --> 00:07:25.366
let's just go back to our core issue

00:07:25.366 --> 00:07:27.400
of how do we actually solve this?

00:07:27.766 --> 00:07:29.100
One way you could think about it,

00:07:29.100 --> 00:07:33.166
we've already seen that
if a matrix is invertable,

00:07:33.166 --> 00:07:35.766
that means that there is a
matrix A inverse that exists,

00:07:35.766 --> 00:07:41.301
such that A inverse times
A is equal to the identity,

00:07:41.301 --> 00:07:44.002
is equal to the identity matrix.

00:07:44.233 --> 00:07:47.366
What if we multiplied the left-hand sides

00:07:47.366 --> 00:07:49.300
of both sides of this
equation by A inverse?

00:07:49.300 --> 00:07:51.073
Remember, order matters

00:07:51.073 --> 00:07:53.200
when we are multiplying matrices.

00:07:53.368 --> 00:07:56.766
We multiply the left-hand
sides of both sides

00:07:56.766 --> 00:07:58.233
of the equation by A inverse,

00:07:58.233 --> 00:08:07.367
which would get us A
inverse times A times x

00:08:07.367 --> 00:08:12.566
is equal to A inverse.

00:08:12.566 --> 00:08:14.301
Remember, I'm multiplying
the left-hand sides

00:08:14.301 --> 00:08:18.866
of both equations, A inverse,
times the column vector B.

00:08:19.166 --> 00:08:20.500
Now why is this interesting?

00:08:20.500 --> 00:08:23.368
Well, we just said that
the inverse times A,

00:08:23.368 --> 00:08:24.900
assuming that A is invertable,

00:08:24.900 --> 00:08:26.367
that this right over
here is going to be equal

00:08:26.367 --> 00:08:28.268
to the identity matrix.

00:08:28.566 --> 00:08:30.720
That's going to be the identity matrix

00:08:30.720 --> 00:08:34.700
times column vector x,

00:08:34.700 --> 00:08:38.666
and that's going to be
equal to this stuff.

00:08:39.435 --> 00:08:41.000
That's going to be equal to that.

00:08:41.834 --> 00:08:43.667
Let me just copy and paste that.

00:08:43.967 --> 00:08:45.467
That's going to be equal to that.

00:08:45.569 --> 00:08:47.100
Now why is that interesting?

00:08:47.100 --> 00:08:52.700
The identity matrix
times some other matrix,

00:08:52.700 --> 00:08:56.167
this column vector essentially
is a 2 by 1 matrix,

00:08:56.167 --> 00:08:58.567
that's just going to be
this column vector again.

00:08:58.567 --> 00:09:03.700
It's just going to simplify
to that our column vector

00:09:03.700 --> 00:09:05.700
is equal to,

00:09:05.700 --> 00:09:10.567
our column vector is equal
to the inverse matrix

00:09:10.567 --> 00:09:13.101
times our column vector,

00:09:13.101 --> 00:09:16.233
or our column vector x is
equal to the inverse matrix

00:09:16.233 --> 00:09:18.000
times the column vector b.

00:09:18.033 --> 00:09:21.103
Once again, emphasizing
why is this useful,

00:09:21.103 --> 00:09:23.100
yes, you do have to go through the trouble

00:09:23.100 --> 00:09:25.370
of calculating A inverse,

00:09:25.370 --> 00:09:26.633
but once you've done that,

00:09:26.633 --> 00:09:29.633
you could keep swapping in
different, different b's.

00:09:29.633 --> 00:09:32.435
This one is 7, negative 6, but
you could have other b's here,

00:09:32.435 --> 00:09:34.303
and if you're running a computer program,

00:09:34.303 --> 00:09:35.366
you want to do this over and over again,

00:09:35.366 --> 00:09:38.133
you just have to do multiple
matrix multiplications.

00:09:38.633 --> 00:09:40.168
I'll leave that thought here.

00:09:40.168 --> 00:09:42.166
I've realized that I'm
approaching 10 minutes,

00:09:42.166 --> 00:09:44.568
which I never like to
cross in these videos.

00:09:44.568 --> 00:09:48.901
In the next video, we'll actually
compute what A inverse is

00:09:48.901 --> 00:09:53.268
and calculate what the
solution vector x is.


Solving the matrix vector equation

WEBVTT
Kind: captions
Language: en

00:00:00.539 --> 00:00:02.294
that we could take a
system of two equations

00:00:02.294 --> 00:00:04.519
with two unknowns and represent it

00:00:04.519 --> 00:00:07.852
as a matrix equation where the matrix A's

00:00:07.852 --> 00:00:10.525
are the coefficients here
on the left-hand side.

00:00:10.525 --> 00:00:12.402
The column vector X has our two

00:00:12.402 --> 00:00:15.343
unknown variables, S and T.

00:00:15.343 --> 00:00:16.578
Then the column vector B is essentially

00:00:16.578 --> 00:00:19.262
representing the
right-hand side over here.

00:00:19.262 --> 00:00:20.379
What was interesting about it,

00:00:20.379 --> 00:00:22.119
then that would be the equation A,

00:00:22.119 --> 00:00:23.963
the matrix A times the column vector X

00:00:23.963 --> 00:00:26.338
being equal to the column vector B.

00:00:26.338 --> 00:00:28.633
What was interesting about that is we saw

00:00:28.633 --> 00:00:29.373
well, look, if A is invertible,

00:00:29.373 --> 00:00:33.064
we can multiply both the
left and the right-hand sides

00:00:33.064 --> 00:00:34.273
of the equation,

00:00:34.273 --> 00:00:36.143
and we have to multiply
them on the left-hand sides

00:00:36.143 --> 00:00:38.173
of their respective sides by A inverse

00:00:38.173 --> 00:00:39.611
because remember matrix,

00:00:39.611 --> 00:00:41.856
when matrix multiplication order matters,

00:00:41.856 --> 00:00:43.644
we're multiplying the left-hand side

00:00:43.644 --> 00:00:45.399
of both sides of the equation.

00:00:45.399 --> 00:00:47.930
If we do that then we
can get to essentially

00:00:47.930 --> 00:00:51.413
solving for the unknown column vector.

00:00:51.413 --> 00:00:52.282
If we know what column vector X is,

00:00:52.282 --> 00:00:54.713
then we know what S and T are.

00:00:54.713 --> 00:00:56.033
Then we've essentially solved

00:00:56.033 --> 00:00:58.112
this system of equations.

00:00:58.112 --> 00:00:59.646
Now let's actually do that.

00:00:59.646 --> 00:01:02.264
Let's actually figure
out what A inverse is

00:01:02.264 --> 00:01:04.266
and multiply that times
the column vector B

00:01:04.266 --> 00:01:06.713
to figure out what the column vector X is,

00:01:06.713 --> 00:01:08.852
and what S and T are.

00:01:08.852 --> 00:01:14.234
A inverse, A inverse is equal to

00:01:14.234 --> 00:01:16.580
one over the determinant of A,

00:01:16.580 --> 00:01:20.504
the determinant of A for a two-by-two here

00:01:20.504 --> 00:01:25.513
is going to be two times
four minus negative

00:01:25.513 --> 00:01:27.149
two times negative five.

00:01:27.149 --> 00:01:31.292
It's going to be eight minus positive 10,

00:01:31.292 --> 00:01:33.114
eight minus positive 10,

00:01:33.114 --> 00:01:34.764
which would be negative two.

00:01:34.764 --> 00:01:38.023
This would become negative
two right over here.

00:01:38.076 --> 00:01:40.847
Once again, two times four is eight minus

00:01:40.847 --> 00:01:43.433
negative two times negative five

00:01:43.433 --> 00:01:47.693
so minus positive 10 which
gets us negative two.

00:01:47.693 --> 00:01:48.850
You multiply one over the determinant

00:01:48.850 --> 00:01:53.604
times what is sometimes
called the adjoint of A

00:01:53.604 --> 00:01:56.638
which is essentially swapping the top left

00:01:56.638 --> 00:01:59.580
and bottom right or at least
for a two-by-two matrix.

00:01:59.580 --> 00:02:02.374
This would be a four.

00:02:02.374 --> 00:02:04.372
This would be a two.

00:02:04.372 --> 00:02:05.653
Notice I just swapped these,

00:02:05.653 --> 00:02:07.024
and making these two negative,

00:02:07.024 --> 00:02:09.176
the negative of what they already are.

00:02:09.176 --> 00:02:10.652
This is from a negative two this is

00:02:10.652 --> 00:02:12.234
going to become a positive two,

00:02:12.234 --> 00:02:13.360
and this right over
here is going to become

00:02:13.360 --> 00:02:14.970
a positive five.

00:02:14.970 --> 00:02:17.696
If all of this looks
completely unfamiliar to you,

00:02:17.696 --> 00:02:20.263
you might want to review the tutorial

00:02:20.263 --> 00:02:21.426
on inverting matrices

00:02:21.426 --> 00:02:23.973
because that's all I'm doing here.

00:02:23.973 --> 00:02:27.284
So A inverse is going to be equal to,

00:02:27.284 --> 00:02:30.653
A inverse is going to be equal to,

00:02:30.653 --> 00:02:34.453
let's see, this is negative 1/2 times four

00:02:34.453 --> 00:02:35.506
is negative two.

00:02:35.506 --> 00:02:41.438
Negative 1/2, negative 1/2 times five

00:02:41.438 --> 00:02:47.042
is negative 2.5, negative 2.5.

00:02:47.042 --> 00:02:51.543
And negative 1/2 times
two is negative one.

00:02:51.543 --> 00:02:53.913
Negative 1/2 times two is negative one.

00:02:53.913 --> 00:02:55.719
So that's A inverse right over here.

00:02:55.719 --> 00:02:58.104
Now let's multiply A inverse times

00:02:58.104 --> 00:03:00.453
our column vector, seven, negative six.

00:03:00.453 --> 00:03:02.443
Let's do that.

00:03:02.443 --> 00:03:03.908
This is A inverse. I'll rewrite it.

00:03:03.908 --> 00:03:08.166
Negative two, negative 2.5, negative one,

00:03:08.166 --> 00:03:13.890
negative one times seven and negative six.

00:03:13.890 --> 00:03:16.696
Times, I'll just write
them all in white here now.

00:03:16.696 --> 00:03:18.413
Seven, negative six.

00:03:18.413 --> 00:03:22.594
We've had a lot of practice
multiplying matrices.

00:03:22.594 --> 00:03:25.128
So what is this going to be equal to?

00:03:25.128 --> 00:03:26.853
The first entry is
going to be negative two

00:03:26.853 --> 00:03:32.253
times seven which is negative 14 plus

00:03:32.253 --> 00:03:37.281
negative 2.5 times negative six.

00:03:37.281 --> 00:03:39.513
Let's see. That's going to be positive.

00:03:39.513 --> 00:03:42.484
That's going to be 12 plus another 3.

00:03:42.484 --> 00:03:44.399
That's going to be plus 15.

00:03:44.399 --> 00:03:46.436
Plus 15.

00:03:46.436 --> 00:03:48.620
Negative 2.5 times negative six

00:03:48.620 --> 00:03:50.864
is positive 15.

00:03:50.864 --> 00:03:52.876
Then we're going to have negative one

00:03:52.876 --> 00:03:56.419
times seven which is negative seven plus

00:03:56.419 --> 00:03:58.863
negative one times negative six.

00:03:58.863 --> 00:04:01.430
Well, that is positive six.

00:04:01.430 --> 00:04:05.263
So the product A inverse B

00:04:05.263 --> 00:04:07.545
which is the same things
as a column vector X

00:04:07.545 --> 00:04:08.732
is equal to,

00:04:08.732 --> 00:04:10.573
we deserve a little
bit of a drum roll now,

00:04:10.573 --> 00:04:14.449
the column vector one, negative one.

00:04:14.449 --> 00:04:17.647
We have just shown that this is equal to

00:04:17.647 --> 00:04:20.982
one, negative one or that X is equal to

00:04:20.982 --> 00:04:22.585
one, negative one,

00:04:22.585 --> 00:04:26.624
or we could even say
that the column vector,

00:04:26.624 --> 00:04:31.453
the column vector ST,

00:04:31.453 --> 00:04:35.505
column vector with the
entries S and T is equal to,

00:04:35.505 --> 00:04:42.039
is equal to one, negative one,

00:04:42.039 --> 00:04:45.329
is equal to one, negative one

00:04:45.329 --> 00:04:46.326
which is another way of saying

00:04:46.326 --> 00:04:47.983
that S is equal to one

00:04:47.983 --> 00:04:50.144
and T is equal to negative one.

00:04:50.144 --> 00:04:51.203
I know what you're saying.

00:04:51.203 --> 00:04:52.272
I said this in the last video

00:04:52.272 --> 00:04:53.541
and I'll say it again in this video.

00:04:53.541 --> 00:04:54.905
You're like, "Well, you
know, it was so much easier

00:04:54.905 --> 00:04:56.848
"to just solve this system directly

00:04:56.848 --> 00:04:59.860
"just with using elimination
or using substitution."

00:04:59.860 --> 00:05:04.643
I agree with you, but
this is a useful technique

00:05:04.643 --> 00:05:06.397
because when you are doing problems

00:05:06.397 --> 00:05:08.858
in computation there may be situations

00:05:08.858 --> 00:05:10.844
where you have the left-hand side

00:05:10.844 --> 00:05:13.533
of this system stays the same,

00:05:13.533 --> 00:05:15.095
but there's many, many,
many different values

00:05:15.095 --> 00:05:17.007
for the right-hand side of the system.

00:05:17.007 --> 00:05:19.159
So it might be easier to just compute

00:05:19.159 --> 00:05:22.640
the inverse once and
just keep multiplying,

00:05:22.640 --> 00:05:24.999
keep multiplying this inverse times

00:05:24.999 --> 00:05:28.618
the different what we have
on the right-hand side.

00:05:28.618 --> 00:05:30.865
You probably are familiar with some types,

00:05:30.865 --> 00:05:32.602
you have graphics processors,

00:05:32.602 --> 00:05:34.333
and graphics cards on computers

00:05:34.333 --> 00:05:36.513
and they talk about
special graphic processors.

00:05:36.513 --> 00:05:38.130
What these are really all about

00:05:38.130 --> 00:05:40.608
are the hardware that is special-purposed

00:05:40.608 --> 00:05:44.213
for really fast matrix multiplication

00:05:44.213 --> 00:05:46.286
because when you're
doing graphics processing

00:05:46.286 --> 00:05:47.597
when you're thinking about modeling things

00:05:47.597 --> 00:05:48.612
in three dimensions,

00:05:48.612 --> 00:05:50.120
and you're doing all
these transformations,

00:05:50.120 --> 00:05:51.574
you're really just doing a lot of

00:05:51.574 --> 00:05:54.018
matrix multiplications
really, really, really fast

00:05:54.018 --> 00:05:56.426
in real time so that to
the user playing the game

00:05:56.426 --> 00:05:58.213
or whatever they're doing,

00:05:58.213 --> 00:05:59.569
it feels like they're in some type

00:05:59.569 --> 00:06:02.599
of a 3D, real-time reality.

00:06:02.599 --> 00:06:04.494
Anyway, I just want to point that out.

00:06:04.494 --> 00:06:08.712
This wouldn't be, if I
saw this just randomly

00:06:08.712 --> 00:06:12.030
my instincts would be to
solve this with elimination,

00:06:12.030 --> 00:06:15.663
but this ability to think of this

00:06:15.663 --> 00:06:20.375
as a matrix equation is a
very, very useful concept,

00:06:20.375 --> 00:06:21.973
one actually not just in computation,

00:06:21.973 --> 00:06:25.293
but also as you go into
higher level sciences

00:06:25.293 --> 00:06:27.611
especially physics, you will see a lot of

00:06:27.611 --> 00:06:30.559
matrix vector equations like this

00:06:30.559 --> 00:06:32.089
that kind of speak in generalities.

00:06:32.089 --> 00:06:33.304
It's really important to think about

00:06:33.304 --> 00:06:35.241
what these actually represent


Is matrix multiplication commutative

WEBVTT
Kind: captions
Language: en

00:00:00.434 --> 00:00:01.901
Voiceover:We know that the multiplication

00:00:01.901 --> 00:00:04.233
of scalar quantities is commutative.

00:00:04.233 --> 00:00:11.566
For example, 5 times 7 is
the same thing as 7 times 5,

00:00:11.566 --> 00:00:13.366
and that's obviously just
a particular example.

00:00:13.366 --> 00:00:14.900
I could give many, many more.

00:00:14.900 --> 00:00:22.233
3 times negative 11 is the same
thing as negative 11 times 3

00:00:22.233 --> 00:00:24.434
and the whole point of commutativity ...

00:00:24.434 --> 00:00:25.366
I could never say it ...

00:00:25.366 --> 00:00:27.033
is that it doesn't matter what order

00:00:27.033 --> 00:00:28.366
that I'm multiplying in.

00:00:28.366 --> 00:00:31.767
This is the same thing
as negative 11 times 3.

00:00:31.767 --> 00:00:33.766
Or if we wanted to speak in general terms,

00:00:33.766 --> 00:00:39.233
if I have the scalar a and I
multiply it times the scalar b,

00:00:39.233 --> 00:00:40.366
that's going to be the same thing

00:00:40.366 --> 00:00:45.801
as multiplying the scalar
b times the scalar a.

00:00:45.966 --> 00:00:48.300
Now what I want to do in
this video is think about

00:00:48.300 --> 00:00:51.367
whether this property of commutativity,

00:00:51.367 --> 00:00:52.767
whether the commutative property

00:00:52.767 --> 00:00:54.500
of multiplication of scalars,

00:00:54.500 --> 00:00:56.233
whether there is a similar property

00:00:56.233 --> 00:00:58.900
for the multiplication of matrices,

00:00:58.900 --> 00:01:01.567
whether it's the case that
if I had two matrices,

00:01:01.567 --> 00:01:08.300
let's say matrix capital
A and matrix capital B,

00:01:08.300 --> 00:01:10.901
whether it's always the
case that that product,

00:01:10.901 --> 00:01:13.500
the resulting matrix here is the same

00:01:13.500 --> 00:01:18.500
as the product of matrix B and matrix A,

00:01:18.500 --> 00:01:20.300
just swapping the order.

00:01:20.300 --> 00:01:23.367
I encourage you ... so
is this always true?

00:01:23.367 --> 00:01:24.433
It might be sometimes true,

00:01:24.433 --> 00:01:27.900
but in order for us to say
that matrix multiplication

00:01:27.900 --> 00:01:29.700
is commutative, that it
doesn't matter what order

00:01:29.700 --> 00:01:31.905
we are multiplying it,

00:01:31.905 --> 00:01:36.700
we have to figure out is
this always going to be true?

00:01:36.700 --> 00:01:38.367
I encourage you to pause this video

00:01:38.367 --> 00:01:40.467
and think about that for a little bit.

00:01:40.946 --> 00:01:42.694
Let's just think through a few things.

00:01:42.694 --> 00:01:45.034
First of all, let's just
think about matrices

00:01:45.034 --> 00:01:47.100
of different dimensions.

00:01:47.100 --> 00:01:49.500
Let's say I have a matrix here.

00:01:49.500 --> 00:01:54.567
Let's say that matrix
A is a, I don't know,

00:01:54.567 --> 00:01:58.734
let's say it is a 5 by 2 matrix,

00:01:58.834 --> 00:02:00.867
5 by 2 matrix,

00:02:00.900 --> 00:02:08.666
and matrix B is a 2 by 3 matrix.

00:02:08.900 --> 00:02:14.400
The product AB is going
to have what dimensions?

00:02:14.433 --> 00:02:18.433
If I multiply these two, you're
going to get a third matrix.

00:02:18.433 --> 00:02:21.392
Let's just call that C for now.

00:02:21.392 --> 00:02:22.700
You're going to get a third matrix C.

00:02:22.700 --> 00:02:25.484
What are going to be the dimensions of C?

00:02:25.822 --> 00:02:30.900
We know, first of all, that
this product is defined

00:02:30.900 --> 00:02:33.114
under our convention of
matrix multiplication

00:02:33.114 --> 00:02:35.433
because the number of columns that A has

00:02:35.433 --> 00:02:38.234
is the same as the number of rows B has,

00:02:38.234 --> 00:02:40.633
and the resulting rows and column

00:02:40.633 --> 00:02:43.833
are going to be the rows
of A and the columns of B.

00:02:43.833 --> 00:02:48.066
So C is going to be a 5 by 3 matrix,

00:02:48.567 --> 00:02:50.400
a 5 by 3 matrix.

00:02:50.700 --> 00:02:52.500
Now what about the other way around?

00:02:52.500 --> 00:02:54.100
What would B times A be?

00:02:54.100 --> 00:02:56.366
Once again, I encourage
you to pause the video.

00:02:57.688 --> 00:02:59.884
If you were to take B,

00:03:00.500 --> 00:03:03.067
let me copy and paste that,

00:03:03.700 --> 00:03:05.767
and multiply that times A,

00:03:05.767 --> 00:03:08.800
so I'm really just switching
the order of the multiplication

00:03:09.167 --> 00:03:12.467
so copy and paste.

00:03:13.241 --> 00:03:15.767
If we take that product right over there,

00:03:15.767 --> 00:03:17.734
what is that going to be equal to?

00:03:17.900 --> 00:03:19.667
What is this?

00:03:19.700 --> 00:03:23.269
What is this right over
here going to be equal to?

00:03:25.233 --> 00:03:29.100
The first question is, is matrix
multiplication even defined

00:03:29.100 --> 00:03:30.767
for these two matrices?

00:03:30.767 --> 00:03:35.366
When you look at the number
of columns that B has

00:03:35.366 --> 00:03:38.169
and the number of rows that A has,

00:03:38.169 --> 00:03:40.168
you see that it actually is not defined,

00:03:40.168 --> 00:03:42.167
that we have a different
number of columns for B

00:03:42.167 --> 00:03:44.434
and a different number of rows for A.

00:03:44.434 --> 00:03:46.866
Here, the product is not defined,

00:03:47.234 --> 00:03:49.400
is not defined,

00:03:49.900 --> 00:03:52.366
so this immediately is a pretty big clue

00:03:52.366 --> 00:03:54.102
that this isn't always going to be true.

00:03:54.102 --> 00:03:57.701
Here, AB, the product AB is defined,

00:03:57.701 --> 00:04:00.166
and you'll end up with a 5 by 3 matrix.

00:04:00.166 --> 00:04:03.833
The product here, BA, isn't even defined.

00:04:03.833 --> 00:04:05.766
This is already ...

00:04:05.766 --> 00:04:09.633
We're already seeing that
this is not the case,

00:04:09.633 --> 00:04:12.833
that order matters when
you are multiplying,

00:04:12.833 --> 00:04:15.167
when you are multiplying matrices.

00:04:15.167 --> 00:04:16.700
To make things a little bit more concrete,

00:04:16.700 --> 00:04:18.500
let's actually look at a matrix.

00:04:18.500 --> 00:04:19.700
You might be saying, oh,
maybe this doesn't work

00:04:19.700 --> 00:04:20.834
only when it's not defined,

00:04:20.834 --> 00:04:23.168
but hey, maybe it works
if we're always to do

00:04:23.168 --> 00:04:26.102
square matrices or matrices
where both products

00:04:26.102 --> 00:04:27.701
are always defined in some way,

00:04:27.701 --> 00:04:30.500
or maybe some other case.

00:04:30.500 --> 00:04:31.900
Let's look at a case where we're dealing

00:04:31.900 --> 00:04:36.666
with 2 by 2 matrices and
see whether order matters.

00:04:36.700 --> 00:04:38.734
Let's say I have the matrix.

00:04:39.300 --> 00:04:47.066
Let's say I have the matrix
1, 2, negative 3, negative 4,

00:04:47.433 --> 00:04:50.766
and I want to multiply that by the matrix,

00:04:50.766 --> 00:04:56.100
by the matrix negative
2, 0, 0, negative 3.

00:04:56.100 --> 00:04:57.767
What's that product going to be?

00:04:57.767 --> 00:04:59.500
Once again, I encourage
you to pause the video

00:04:59.500 --> 00:05:00.938
and think about that.

00:05:02.965 --> 00:05:06.370
Let's think it through, and
we've done this many times now.

00:05:07.322 --> 00:05:09.369
This first entry here is going to be,

00:05:09.369 --> 00:05:12.567
we're essentially going to look
at this row and this column,

00:05:12.567 --> 00:05:14.901
so it's 1 times negative
2, which is negative 2,

00:05:14.901 --> 00:05:16.033
plus 2 times 0.

00:05:16.033 --> 00:05:18.233
This is going to be negative 2.

00:05:18.233 --> 00:05:20.503
Now, for this entry,

00:05:20.503 --> 00:05:21.366
for this entry over here,

00:05:21.366 --> 00:05:24.700
we'll look at this row and this column,

00:05:24.700 --> 00:05:27.566
1 times 0, which is 0,
plus 2 times negative 3,

00:05:27.566 --> 00:05:29.710
which is negative 6.

00:05:29.710 --> 00:05:35.167
Then for this entry, we
would look at this row

00:05:35.167 --> 00:05:37.680
and this column.

00:05:37.680 --> 00:05:40.233
Negative 3 times negative 2 is positive 6

00:05:40.233 --> 00:05:42.933
plus negative 4 times 0,
which is just positive 6.

00:05:43.202 --> 00:05:44.935
We're going to have positive 6.

00:05:45.033 --> 00:05:47.101
Then finally, for this entry,

00:05:47.101 --> 00:05:49.247
it's going to be the second
row times the second column.

00:05:49.247 --> 00:05:51.440
Negative 3 times 0 is 0.

00:05:51.440 --> 00:05:54.868
Negative 4 times negative
3 is positive 12,

00:05:55.366 --> 00:05:56.396
so fair enough.

00:05:56.396 --> 00:05:58.134
Now what if we did it
the other way around?

00:05:58.234 --> 00:06:06.368
What if we were to multiply
negative 2, 0, 0, negative 3

00:06:06.368 --> 00:06:13.666
times 1, 2, negative 3, negative 4?

00:06:13.674 --> 00:06:15.501
What's this going to be equal to?

00:06:15.501 --> 00:06:18.100
As always, it's a good
idea to try to pause it

00:06:18.100 --> 00:06:19.800
and work through it on your own.

00:06:20.479 --> 00:06:21.900
Let's think about this.

00:06:21.900 --> 00:06:29.505
Negative 2 times 1 is negative
2, plus 0 times negative 3,

00:06:29.505 --> 00:06:31.835
so that's going to be negative 2.

00:06:31.835 --> 00:06:34.335
So far, it's looking pretty good.

00:06:34.453 --> 00:06:39.233
Then if you have negative
2 times 2 is negative 4,

00:06:39.233 --> 00:06:42.766
plus 0 times negative 4 is negative 4.

00:06:42.766 --> 00:06:44.833
We already see that these two things

00:06:44.833 --> 00:06:46.833
aren't going to be equal,
but let's just finish it,

00:06:46.833 --> 00:06:49.602
just so that we have a
feeling of completion.

00:06:50.100 --> 00:06:51.367
This entry right over here is going to be

00:06:51.367 --> 00:06:52.966
the second row, first column,

00:06:52.966 --> 00:06:56.573
0 times 1 plus negative 3
times negative 3 is positive 9.

00:06:56.573 --> 00:06:58.233
Once again, it doesn't match up.

00:06:58.233 --> 00:07:01.500
Then finally, 0 times 2 is 0

00:07:01.500 --> 00:07:06.301
plus negative 3 times
negative 4 is positive 12.

00:07:06.301 --> 00:07:07.566
That one actually did match up,

00:07:07.566 --> 00:07:10.833
but clearly, these two products
are not the same thing.

00:07:10.833 --> 00:07:13.530
The order with which even those defined,

00:07:14.669 --> 00:07:15.642
it doesn't matter whether you take

00:07:15.642 --> 00:07:17.366
the yellow one times the purple one

00:07:17.366 --> 00:07:18.770
or the purple one times the yellow one.

00:07:18.770 --> 00:07:21.433
Both of those result in a defined product,

00:07:21.433 --> 00:07:23.700
but we see it's not the same product.

00:07:23.700 --> 00:07:26.043
Once again, another case showing

00:07:26.043 --> 00:07:30.918
that multiplication of
matrices is not commutative.


Associative property of matrix multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.338 --> 00:00:01.724
Voiceover:What I want to do in this video,

00:00:01.724 --> 00:00:05.053
is show that matrix
multiplication is associative.

00:00:05.053 --> 00:00:07.230
At least I'll show it for 2 by 2 matrices.

00:00:07.230 --> 00:00:09.888
And what I do in this
video you can extend it to

00:00:09.888 --> 00:00:12.634
really any dimension of matrices for which

00:00:12.634 --> 00:00:16.171
of the matrix multiplication
is actually defined.

00:00:16.171 --> 00:00:19.006
So let's look at 3
matrices, so let's say this

00:00:19.006 --> 00:00:26.032
first matrix is A, B, C,
and D and this second matrix

00:00:26.032 --> 00:00:34.846
is E, F, G, H and then
finally this third matrix

00:00:34.846 --> 00:00:41.538
is I, J, and that's not the
imaginary unit I, just letter I,

00:00:41.538 --> 00:00:43.840
and this isn't E, this
is just the letter E.

00:00:43.840 --> 00:00:49.384
J, K and L, and I want
to look at 2 scenarios.

00:00:49.384 --> 00:00:52.030
So let me actually just
copy and paste this,

00:00:52.030 --> 00:00:56.915
So copy and paste.

00:00:56.915 --> 00:01:00.379
So I want to look at this
scenario where first,

00:01:00.379 --> 00:01:03.214
I multiply the orange
and the yellow matrix.

00:01:03.214 --> 00:01:05.251
And I multiply that
times the purple matrix

00:01:05.251 --> 00:01:08.322
And then another scenario
where first I multiply

00:01:08.322 --> 00:01:10.718
the yellow and the
purple and multiply that

00:01:10.718 --> 00:01:13.451
times the orange and if
these two products based on

00:01:13.451 --> 00:01:17.232
how I, which ones I do
first come out the same then

00:01:17.232 --> 00:01:20.282
I've just shown that at least
for three 2 by 2 matrices,

00:01:20.282 --> 00:01:24.109
that matrix multiplication is associative.

00:01:24.109 --> 00:01:25.968
If we already seen that
it's not commutative,

00:01:25.968 --> 00:01:27.660
let's see whether it's associative.

00:01:27.660 --> 00:01:30.103
And actually I'll give
you the punchline, it is.

00:01:30.103 --> 00:01:31.534
But let's work through
it and I encourage you

00:01:31.534 --> 00:01:32.874
to actually pause the video yourself ,

00:01:32.874 --> 00:01:34.769
and try to work through
it with these letters

00:01:34.769 --> 00:01:38.236
and then see if you got
the result that I just said

00:01:38.236 --> 00:01:39.890
that you should be getting.

00:01:39.890 --> 00:01:42.845
Alright, so let's
multiply these first two.

00:01:42.845 --> 00:01:46.499
So this product, I'm gonna
make it a little bit big.

00:01:46.499 --> 00:01:58.160
So it's going to be AE + BG, then AF + BH,

00:01:58.160 --> 00:02:05.037
and then it's going to be CE + DG,

00:02:05.037 --> 00:02:11.002
and then finally it's gonna be CF + DH.

00:02:11.002 --> 00:02:14.346
And we're going to multiply
it times the matrix

00:02:14.346 --> 00:02:22.062
the matrix, I, J, K, and L
and what does this give us?

00:02:22.062 --> 00:02:25.552
And I'll just give us some
space to do this with.

00:02:25.552 --> 00:02:28.883
So this will give us, let
me give myself an ample

00:02:28.883 --> 00:02:31.455
amount of space, so it's
going to be this stuff,

00:02:31.455 --> 00:02:35.997
times I, so we could write this as I,

00:02:35.997 --> 00:02:38.410
actually let me just distribute the I.

00:02:38.410 --> 00:02:51.198
IAE + IBG + this stuff
times K, + KAF + KBH.

00:02:51.198 --> 00:02:53.280
I'm already gonna run out of space here,

00:02:53.280 --> 00:02:57.914
so let me clear this,
let me just keep going.

00:02:57.914 --> 00:03:00.419
And then we're gonna
multiply this, essentially,

00:03:00.419 --> 00:03:02.558
we're going to consider
this row and this column.

00:03:02.558 --> 00:03:05.558
First row, second column
and so you are going

00:03:05.558 --> 00:03:22.990
to have JAE + JBG + LAF + LBH so,

00:03:22.990 --> 00:03:25.627
these matrices are bigger
than I expected they would be.

00:03:25.627 --> 00:03:28.055
And then you're going to
have over here you're going

00:03:28.055 --> 00:03:30.783
to have this times this
plus this times this.

00:03:30.783 --> 00:03:40.995
So ICE + IDG + KCF + KDH and then finally,

00:03:40.995 --> 00:03:43.265
this times this plus this plus this,

00:03:43.265 --> 00:03:46.203
or this times that plus this times that.

00:03:46.203 --> 00:03:58.718
So JCE + JDG + LCF + LDH, alright.

00:03:58.718 --> 00:04:01.252
Now see if we can power
through this one over here.

00:04:01.252 --> 00:04:04.270
So what is this product going to be?

00:04:04.270 --> 00:04:07.359
This product if I multiply
by these two first.

00:04:07.359 --> 00:04:18.052
It's going to be EI +
FK + EJ, no not plus,

00:04:18.052 --> 00:04:26.382
this is the next entry, EJ + FL,

00:04:26.382 --> 00:04:40.683
Then we have GI + HK and
then finally we have GJ + HL.

00:04:40.683 --> 00:04:44.067
Then it's all going to
be multiplied times ABCD.

00:04:44.067 --> 00:04:50.519
A, B, C, and D, and I'm going
to need some real estate

00:04:50.519 --> 00:04:54.906
to do this, so let me do it
down here, I'll do it in green.

00:04:54.906 --> 00:04:58.239
So let me do a little arrow to
show, or actually I can just

00:04:58.239 --> 00:05:01.101
scroll over a little bit, that
actually might work out better.

00:05:01.101 --> 00:05:05.530
So what am I going to
get, so A times this,

00:05:05.530 --> 00:05:19.438
plus B times this, so
AEI + AFK + BGI + BHK,

00:05:19.438 --> 00:05:22.220
then you're going to have
this row and this column,

00:05:22.220 --> 00:05:39.298
So it's AEJ + AFL + BGJ +
BHL, close the brackets,

00:05:39.298 --> 00:05:42.672
now you're going to have C
times this plus D times this.

00:05:42.672 --> 00:05:53.495
So CEI + CFK + DGI + DHK and then finally,

00:05:53.495 --> 00:05:57.354
home stretch, C times
that plus D times this.

00:05:57.354 --> 00:06:06.596
So CEJ + CFL, and then
you're going to have,

00:06:06.596 --> 00:06:18.106
+ DGJ + DHL, now are these
two things equivalent?

00:06:18.106 --> 00:06:20.197
Well let's look at entry by entry.

00:06:20.197 --> 00:06:25.161
So, IAE, this is equivalent to AEI.

00:06:25.161 --> 00:06:28.247
Because we know scalar
multiplication is commutative,

00:06:28.247 --> 00:06:35.205
Now IBJ or IBG, you see it
there and you see it there,

00:06:35.205 --> 00:06:42.813
KAF, you see it there and
this is the same thing as AFK.

00:06:42.813 --> 00:06:47.573
And then KBH, this is
the same thing as BHK.

00:06:47.573 --> 00:06:49.926
And you can go entry by entry, actually,

00:06:49.926 --> 00:06:52.779
let's just do that, I'll do
that really fast, so let's do,

00:06:52.779 --> 00:06:56.732
so ICE is the same thing as CEI.

00:06:56.732 --> 00:06:59.497
IDG is the same thing as DGI.

00:06:59.497 --> 00:07:03.189
KCF is the same thing as CFK,

00:07:03.189 --> 00:07:05.788
KDH is the same thing as DHK,

00:07:05.788 --> 00:07:08.015
and we go to the second columns,

00:07:08.015 --> 00:07:15.711
JAE, AEJ, JBG is the same thing as BGJ

00:07:15.711 --> 00:07:19.722
LAF is the same thing as AFL,

00:07:19.722 --> 00:07:23.016
And LBH is the same thing as BHL

00:07:23.016 --> 00:07:28.005
and then finally JCE is
the same thing as CEJ

00:07:28.005 --> 00:07:32.087
JDG is the same thing as DGJ,

00:07:32.087 --> 00:07:37.327
LEF, LEF, or is that LCF?

00:07:37.327 --> 00:07:41.159
Let's see, LDH is this right over here,

00:07:41.159 --> 00:07:45.020
And so this one must be an
LCF is the same thing as CFL.

00:07:45.020 --> 00:07:47.881
Is this one right over here
an LCF, let me make sure,

00:07:47.881 --> 00:07:50.890
Cause that would throw
a major monkey wrench

00:07:50.890 --> 00:07:54.029
into the whole operation, so
this entry right over here,

00:07:54.029 --> 00:07:57.598
is going to be, we get
that from multiplying

00:07:57.598 --> 00:08:00.614
the second row times the second column

00:08:00.614 --> 00:08:07.712
and we're going to get, we get JCE + JDG

00:08:07.712 --> 00:08:11.794
and then we have LCF,
yeah that's LCF + LDH,

00:08:11.794 --> 00:08:14.478
and so [you will] see
that these two quantities

00:08:14.478 --> 00:08:16.586
are the same it doesn't
matter if I multiply

00:08:16.586 --> 00:08:19.259
the first two first and
then multiply by the third.

00:08:19.259 --> 00:08:21.206
Or multiply the second and
third and then multiply

00:08:21.206 --> 00:08:23.174
by the first, now once again,

00:08:23.174 --> 00:08:25.518
this is the associative
property, I'm keeping them

00:08:25.518 --> 00:08:27.257
essentially in the same order.

00:08:27.257 --> 00:08:31.796
Order matters, but as we
see, we can associate these.

00:08:31.796 --> 00:08:33.133
We can do the first two first

00:08:33.133 --> 23:59:59.999
or we can do the second two first.


Equivalent matrix expressions

WEBVTT
Kind: captions
Language: en

00:00:00.252 --> 00:00:02.405
Voiceover:In order to
get into Battle School

00:00:02.405 --> 00:00:04.823
cadets have to pass a
rigorous entrance exam

00:00:04.823 --> 00:00:06.964
which includes mathematics.

00:00:06.964 --> 00:00:10.222
Help Commander Graff grade the next wave

00:00:10.222 --> 00:00:11.456
of students' tests.

00:00:11.456 --> 00:00:12.994
The last step of a problem

00:00:12.994 --> 00:00:15.739
in the matrix multiplication section is

00:00:15.739 --> 00:00:18.329
the matrix A times B times C

00:00:18.329 --> 00:00:20.769
where A, B, and C are square matrices.

00:00:20.769 --> 00:00:23.377
Which of the following
candidates have answers

00:00:23.377 --> 00:00:25.473
that are equivalent to this expression?

00:00:25.473 --> 00:00:29.784
Select all who are right
for any A, B, and C.

00:00:29.784 --> 00:00:31.495
Select all that apply.

00:00:31.495 --> 00:00:34.041
I encourage you to pause this
video and think about it.

00:00:34.041 --> 00:00:37.017
Which of these expressions
for any square matrices

00:00:37.017 --> 00:00:38.207
A, B, and C

00:00:38.207 --> 00:00:41.491
are equivalent to this right over here.

00:00:42.475 --> 00:00:44.115
I'm assuming you've given a go at it.

00:00:44.115 --> 00:00:45.196
Let's think through each of them.

00:00:45.196 --> 00:00:46.644
This one is B, A, C

00:00:46.644 --> 00:00:48.759
so if they have changed the order

00:00:48.759 --> 00:00:51.369
and we've already seen
that matrix multiplication

00:00:51.369 --> 00:00:53.789
is not commutative in general

00:00:53.789 --> 00:00:57.224
and so this will not be
true for any square set

00:00:57.224 --> 00:00:59.332
of matrices A, B, and C,

00:00:59.332 --> 00:01:01.240
so this is not going to be true.

00:01:01.240 --> 00:01:03.920
Matrix multiplication is not commutative.

00:01:04.843 --> 00:01:09.367
Here you have Bernard, who
says A times, C times B.

00:01:09.367 --> 00:01:11.436
We already know that that's going to be

00:01:11.436 --> 00:01:14.874
the equivalent to A, C, B

00:01:14.874 --> 00:01:16.943
which once again they've swapped the order

00:01:16.943 --> 00:01:18.290
between the B and the C,

00:01:18.290 --> 00:01:20.676
matrix multiplication is not commutative.

00:01:20.676 --> 00:01:22.233
You can't just swap order

00:01:22.233 --> 00:01:23.916
and expect to get the same product

00:01:23.916 --> 00:01:26.958
for any square matrices A, B, and C

00:01:26.958 --> 00:01:28.506
so we could rule that one out.

00:01:28.506 --> 00:01:30.503
A times, B, C,

00:01:30.503 --> 00:01:32.366
so we've already seen
matrix multiplication

00:01:32.366 --> 00:01:33.544
is associative,

00:01:33.544 --> 00:01:38.737
so this is the same thing
as A times B, times C

00:01:38.737 --> 00:01:42.614
which of course is the
same thing as A, B, C.

00:01:42.614 --> 00:01:45.306
What Caren has right
over here, that is right,

00:01:45.306 --> 00:01:47.987
that is equivalent for any
square matrices A, B, and C

00:01:47.987 --> 00:01:51.327
that is equivalent to A, B, C.

00:01:52.342 --> 00:01:54.667
Now Ducheval, let's see,

00:01:54.667 --> 00:01:57.252
now this looks like a
bit of a crazy expression

00:01:57.252 --> 00:01:59.375
but let's think it through a little bit.

00:02:00.698 --> 00:02:03.282
First of all matrix multiplication,

00:02:03.282 --> 00:02:05.126
as long as you keep the order right,

00:02:05.126 --> 00:02:07.322
the distributive property does hold.

00:02:07.322 --> 00:02:10.927
This first part right over
here is equivalent to ...

00:02:10.927 --> 00:02:12.599
Let me write this down,
this one's interesting.

00:02:12.599 --> 00:02:23.749
We have A times B, C
plus A, minus A squared.

00:02:23.749 --> 00:02:27.097
You can actually distribute this A

00:02:27.097 --> 00:02:28.932
and I encourage you to
prove it for yourself

00:02:28.932 --> 00:02:32.009
maybe using some two by two
matrices for simplicity.

00:02:32.009 --> 00:02:34.656
This is going to be equal to,

00:02:34.656 --> 00:02:38.588
this part over here is going to be A, B, C

00:02:38.588 --> 00:02:44.048
plus A A, A times A which
we could write as A squared

00:02:44.048 --> 00:02:47.793
and then we're going
to subtract A squared.

00:02:47.793 --> 00:02:50.213
These two things are going to cancel out,

00:02:50.213 --> 00:02:52.183
they're going to end up
being the zero matrix

00:02:52.183 --> 00:02:53.667
and if you take the zero,

00:02:53.667 --> 00:02:56.178
so these are going to be the
zero matrix right over here.

00:02:56.178 --> 00:02:58.456
If you take the zero matrix
and add it to A, B, C

00:02:58.456 --> 00:03:04.331
you're just going to end up with A, B, C.

00:03:04.331 --> 00:03:06.124
This one was a little bit tricky,

00:03:06.124 --> 00:03:08.145
this one actually is equivalent.

00:03:08.145 --> 00:03:12.338
This one is right and this one is right.

00:03:12.338 --> 00:03:15.506
Here this is A times B, plus C

00:03:15.506 --> 00:03:19.312
so this is kind of not-y right over here.

00:03:19.312 --> 00:03:21.257
They're not even multiplying B and C,

00:03:21.257 --> 00:03:24.720
so this one's definitely
not going to be true

00:03:24.720 --> 00:03:28.265
for all square matrices A, B, and C.


Matrix transformation on triangle

WEBVTT
Kind: captions
Language: en

00:00:00.328 --> 00:00:01.592
Voiceover:We've already used

00:00:01.592 --> 00:00:04.008
a transformation matrix
to transform one point,

00:00:04.008 --> 00:00:05.688
what I want to do in this video,

00:00:05.688 --> 00:00:07.712
is transform a series of points.

00:00:07.712 --> 00:00:09.243
So, I have these position vectors,

00:00:09.243 --> 00:00:11.176
p1, p2, and p3,

00:00:11.176 --> 00:00:13.608
and I've plotted them right over here.

00:00:13.608 --> 00:00:15.624
And you can imagine them to even be

00:00:15.624 --> 00:00:18.128
vertices of a triangle

00:00:18.128 --> 00:00:19.528
that looks something like this

00:00:19.528 --> 00:00:21.680
(that's one side, that's another side,

00:00:21.680 --> 00:00:23.936
and that's another side, right like that)

00:00:23.936 --> 00:00:25.632
and what I'm curious about is

00:00:25.632 --> 00:00:30.192
what happens if I transform
these three points?

00:00:30.192 --> 00:00:31.448
And like the last video,

00:00:31.448 --> 00:00:33.464
I could apply this transformation matrix

00:00:33.464 --> 00:00:35.160
separately to each of them

00:00:35.160 --> 00:00:36.904
to see what they transform into,

00:00:36.904 --> 00:00:39.696
or instead I could take
this transformation

00:00:39.696 --> 00:00:44.448
matrix, and multiply it times a

00:00:44.448 --> 00:00:48.528
matrix composed of these position vectors,

00:00:48.528 --> 00:00:50.184
so let me do that...

00:00:50.184 --> 00:00:54.472
Let me take my transformation matrix,

00:00:54.472 --> 00:00:57.432
let me copy and paste that.

00:00:57.432 --> 00:01:01.496
So, copy and paste.

00:01:01.496 --> 00:01:04.344
So, I'm going to take
my transformation matrix

00:01:04.344 --> 00:01:07.136
and I am going to multiply that

00:01:07.136 --> 00:01:10.944
by a matrix that has all three of these

00:01:10.944 --> 00:01:13.552
position vectors in it,

00:01:13.552 --> 00:01:15.344
where each of the columns of this matrix

00:01:15.344 --> 00:01:17.408
is going to be one of
these position vectors.

00:01:17.408 --> 00:01:21.200
So, the first one is 2, 1.

00:01:21.200 --> 00:01:26.040
Then we have -2, 0

00:01:26.040 --> 00:01:30.472
Then we have 0, 2.

00:01:30.472 --> 00:01:31.920
So, one way you could think about it

00:01:31.920 --> 00:01:35.560
is we're taking our transformation matrix

00:01:35.560 --> 00:01:37.376
and we're multiplying it

00:01:37.376 --> 00:01:43.584
by a matrix that is composed of

00:01:43.584 --> 00:01:48.496
the first column is position vector 1,

00:01:48.496 --> 00:01:51.720
the second column is position vector 2,

00:01:51.720 --> 00:01:58.224
and the third column is position vector 3.

00:01:58.224 --> 00:02:02.064
Now, what is this going to give us?

00:02:02.064 --> 00:02:06.744
Well, this is a 2x2 matrix.

00:02:06.744 --> 00:02:12.856
We're multiplying it by a 2x3 matrix,

00:02:12.856 --> 00:02:16.200
so matrix multiplication
is defined over here

00:02:16.200 --> 00:02:18.048
because the number of columns here

00:02:18.048 --> 00:02:20.256
is the same as the number of rows here.

00:02:20.256 --> 00:02:25.512
And it's going to result in a 2x3 matrix.

00:02:25.512 --> 00:02:31.416
So, it's going to result in a 2x3 matrix.

00:02:31.416 --> 00:02:34.728
So, 2 rows and 3 columns,

00:02:34.728 --> 00:02:36.904
which we could imagine represents

00:02:36.904 --> 00:02:41.584
three new position vectors.

00:02:41.584 --> 00:02:43.624
So, what's this going to be?

00:02:43.624 --> 00:02:45.256
Let's go step by step.

00:02:45.256 --> 00:02:46.800
So, this first entry,

00:02:46.800 --> 00:02:48.184
first row, first column,

00:02:48.184 --> 00:02:49.776
is this row times this column,

00:02:49.776 --> 00:02:52.144
so 2x2 which is 4,

00:02:52.144 --> 00:02:55.272
plus 1 plus 1, so it's 4 plus 1

00:02:55.272 --> 00:02:57.970
so this is going to be 5.

00:02:57.970 --> 00:03:00.979
Let me do it in a different color.

00:03:00.979 --> 00:03:04.954
-1 times 2 is -2.

00:03:05.994 --> 00:03:11.826
plus 2 times 1 plus 2,

00:03:11.826 --> 00:03:15.554
that's -2 plus 2 which is 0.

00:03:15.554 --> 00:03:17.402
So, we already see it transformed

00:03:17.402 --> 00:03:20.114
2,1 to 5,0.

00:03:20.114 --> 00:03:23.042
1, 2, 3, 4, 5.

00:03:23.042 --> 00:03:27.122
So if we consider this p1,

00:03:27.122 --> 00:03:30.978
we consider this p1 prime,

00:03:30.978 --> 00:03:33.618
p1 after our transformation.

00:03:33.618 --> 00:03:36.122
Now, let's go to p2.

00:03:36.122 --> 00:03:39.210
2 times -2 is -4,

00:03:39.210 --> 00:03:43.074
plus 1 times 0, so it's -4 plus 0,

00:03:43.074 --> 00:03:45.970
which is just -4.

00:03:45.970 --> 00:03:49.514
And then -1 times 2 is positive 2,

00:03:49.514 --> 00:03:52.786
plus 2 times 0, which
is just going to be 0.

00:03:52.786 --> 00:03:55.298
So it's going to be positive 2 plus 0,

00:03:55.298 --> 00:03:57.490
which is just 2.

00:03:57.490 --> 00:04:03.994
So, -4, 2. Negative 1, 2, 3, 4 comma 2.

00:04:03.994 --> 00:04:06.682
Right over here.

00:04:06.682 --> 00:04:08.626
So this is p2

00:04:08.626 --> 00:04:10.626
This was p2 right over here.

00:04:10.626 --> 00:04:12.882
This is p2 prime.

00:04:12.882 --> 00:04:17.731
This is position vector p2 prime.

00:04:17.731 --> 00:04:19.786
Or the position that position vector p2

00:04:19.786 --> 00:04:22.138
prime would specify.

00:04:22.138 --> 00:04:25.218
And then finally, let's look at p3.

00:04:25.218 --> 00:04:27.986
So we have 2 times 0,

00:04:27.986 --> 00:04:29.986
which is 0, plus 1 times 2,

00:04:29.986 --> 00:04:32.730
so that's zero plus 2 or just 2,

00:04:32.730 --> 00:04:35.746
and then we have -1 times 0, which is 0.

00:04:35.746 --> 00:04:38.730
plus 2 times 2 is four.

00:04:38.730 --> 00:04:41.962
So, we get the point 2, 4.

00:04:41.962 --> 00:04:44.730
So, 2 comma 1, 2, 3, 4...

00:04:44.730 --> 00:04:46.794
we go right over here.

00:04:46.794 --> 00:04:49.882
So, this is p3 right over here.

00:04:49.882 --> 00:04:54.676
This is p3 prime.

00:04:54.676 --> 00:04:56.378
And so something interesting has happened.

00:04:56.378 --> 00:04:58.338
We now have the vertices you could imagine

00:04:58.338 --> 00:05:00.762
of a new triangle.

00:05:00.762 --> 00:05:03.826
Of a new triangle that
looks something like this.

00:05:03.826 --> 00:05:08.634
That looks something like this.

00:05:08.634 --> 00:05:10.882
So, what you can imagine is...

00:05:10.882 --> 00:05:12.490
Actually, let me draw it...

00:05:12.490 --> 00:05:15.762
Let me draw our new one
with this blue color.

00:05:15.762 --> 00:05:17.970
so we can see it a little better.

00:05:17.970 --> 00:05:22.666
So we went from that smaller triangle,

00:05:22.666 --> 00:05:25.010
we went from that smaller triangle

00:05:25.010 --> 00:05:27.738
to the larger one, this is the smaller one

00:05:27.738 --> 00:05:29.418
right over here.

00:05:29.418 --> 00:05:31.538
That's our smaller triangle

00:05:31.538 --> 00:05:33.474
to the larger one,

00:05:33.474 --> 00:05:35.266
or another way you could think of it,

00:05:35.266 --> 00:05:37.346
this entire triangle was transformed,

00:05:37.346 --> 00:05:39.530
and right now we only
transformed the vertices,

00:05:39.530 --> 00:05:40.962
but it actually turns out,

00:05:40.962 --> 00:05:42.202
and I'm not proving this video,

00:05:42.202 --> 00:05:43.490
that if you transformed-

00:05:43.490 --> 00:05:45.434
if you took any of these
points on this triangle,

00:05:45.434 --> 00:05:47.018
it would have transformed to a

00:05:47.018 --> 00:05:51.146
corresponding point on
this larger triangle.

00:05:51.146 --> 00:05:53.026
And what's neat about this is hopefully

00:05:53.026 --> 00:05:54.354
you're starting to appreciate

00:05:54.354 --> 00:05:57.714
the power of a transformation matrix.

00:05:57.714 --> 00:05:59.738
And hopefully you're
starting to appreciate

00:05:59.738 --> 00:06:02.186
why this is useful as you
start to think about things

00:06:02.186 --> 00:06:05.298
like computer games and animation,

00:06:05.298 --> 00:06:07.418
because what transformation matrixes

00:06:07.418 --> 00:06:08.506
allow us to do,

00:06:08.506 --> 00:06:10.410
and this is what these computer programs

00:06:10.410 --> 00:06:11.594
allow us to do -

00:06:11.594 --> 00:06:13.682
view things from different perspectives,

00:06:13.682 --> 00:06:15.826
what they're actually
doing under the covers,

00:06:15.826 --> 00:06:18.026
is they're using transformation matrixes,

00:06:18.026 --> 00:06:20.442
and they're multiplying
them times coordinates,

00:06:20.442 --> 00:06:22.602
in order to get new coordinates based on

00:06:22.602 --> 00:06:25.146
the position or the
perspective of the player,

00:06:25.146 --> 00:06:27.930
or the position or the
perspective of the camera,

00:06:27.930 --> 00:06:32.490
or the virtual camera in
a computer graphics world.

00:06:32.490 --> 00:06:35.002
So two, I guess, several neat things here,

00:06:35.002 --> 00:06:37.202
is we haven't just
transformed a point now,

00:06:37.202 --> 00:06:38.690
we've transformed three points,

00:06:38.690 --> 00:06:40.322
which could represent the
vertices of a triangle,

00:06:40.322 --> 00:06:43.170
and you see it as this kind of expansion

00:06:43.170 --> 00:06:48.706
and rotation that seems to have happened

00:06:48.706 --> 00:06:51.586
when we used this transformation matrix.

00:06:51.586 --> 00:06:53.322
If we use a different
transformation matrix,

00:06:53.322 --> 00:06:55.442
we would have a different transformation

00:06:55.442 --> 00:06:57.410
and not only did we do it,

00:06:57.410 --> 00:06:59.858
but we saw that we could
do it with multiple

00:06:59.858 --> 00:07:02.066
position vectors as the same time.

00:07:02.066 --> 00:07:03.938
I could have done it independently

00:07:03.938 --> 00:07:05.218
and gotten the same result,

00:07:05.218 --> 00:07:06.378
but this is hopefully

00:07:06.378 --> 00:07:08.202
starting to show you the power

00:07:08.202 --> 00:07:10.130
of matrixes, and why
it also could be useful

00:07:10.130 --> 00:07:12.266
in things like computer
graphics and animation

00:07:12.266 --> 00:07:14.213
and things like that.


Transformation matrix for position vector

WEBVTT
Kind: captions
Language: en

00:00:00.430 --> 00:00:04.103
Voiceover:Let's say that
we've got a position vector, P

00:00:04.103 --> 00:00:07.305
and it is equal to or
represented as a column vector,

00:00:07.305 --> 00:00:09.616
right over here, 2, 1.

00:00:09.616 --> 00:00:12.712
If we wanted to plot this,
and that is what I'll do.

00:00:12.712 --> 00:00:14.269
Let's plot it.

00:00:15.545 --> 00:00:17.202
If we were to plot it,

00:00:17.202 --> 00:00:19.102
that's my Y axis,

00:00:19.855 --> 00:00:23.743
this right over here is my X axis.

00:00:25.973 --> 00:00:27.252
If we assume that the first entry

00:00:27.252 --> 00:00:32.759
here is X coordinate, 1, 2, 1.

00:00:32.759 --> 00:00:35.557
Our position vector is
going to be right over here.

00:00:35.557 --> 00:00:37.572
We could represent it.

00:00:37.572 --> 00:00:38.451
We could represent it in

00:00:38.451 --> 00:00:40.903
this kind of vector symbol, like that.

00:00:40.903 --> 00:00:42.537
Where the tail is at the origin

00:00:42.537 --> 00:00:44.912
and the tip, the head, is at that point

00:00:44.912 --> 00:00:48.472
or we could say it's really
representing this position,

00:00:48.472 --> 00:00:50.353
right over here on the coordinate plan.

00:00:51.153 --> 00:00:52.568
What I wanted to do in this video

00:00:52.568 --> 00:00:57.175
is apply a transformation
to this position vector.

00:00:57.175 --> 00:00:58.344
The way I'm going to do it

00:00:58.344 --> 00:01:01.003
is I'm going to multiply our vector, P

00:01:01.003 --> 00:01:04.249
by a matrix and then the resulting product

00:01:04.249 --> 00:01:07.225
is going to give me
another position vector.

00:01:07.225 --> 00:01:09.015
So, what do I mean by that?

00:01:09.015 --> 00:01:13.354
I could have a transformation
matrix, capital T

00:01:13.354 --> 00:01:23.673
and let's just say it is
equal to 2, 1, -1, and 2.

00:01:24.365 --> 00:01:27.092
What happens if I multiply TxP?

00:01:27.092 --> 00:01:28.865
Let's do that right over here.

00:01:28.865 --> 00:01:35.217
TxP, so first let's just make sure that

00:01:35.217 --> 00:01:38.772
this is actually a valid operation

00:01:38.772 --> 00:01:40.734
that matrix multiplication

00:01:40.734 --> 00:01:43.531
or matrix vector
multiplication is defined here.

00:01:43.531 --> 00:01:47.072
Let's just look at what T and P look like.

00:01:47.087 --> 00:01:48.077
I'm gonna copy and paste.

00:01:49.984 --> 00:01:54.393
That is T and this is P right over here.

00:01:58.976 --> 00:02:03.364
Can we multiply a 2x2 matrix

00:02:03.364 --> 00:02:06.128
times a column vector like this,

00:02:06.128 --> 00:02:09.106
which is essentially 2x1 matrix.

00:02:09.106 --> 00:02:11.217
Well sure, as as we know

00:02:11.217 --> 00:02:13.230
matrix multiplication is only defined,

00:02:13.230 --> 00:02:14.776
or at least conventional
matrix multiplication

00:02:14.776 --> 00:02:19.739
is only defined if the first matrix number

00:02:19.739 --> 00:02:24.325
of columns is equal to the number

00:02:24.325 --> 00:02:27.357
of rows in the second
matrix, right over here.

00:02:27.357 --> 00:02:29.399
We see there, both of those are 2.

00:02:29.399 --> 00:02:34.987
This is going to result in a 2x1 matrix.

00:02:34.987 --> 00:02:39.478
This is going to result in a 2x1 matrix.

00:02:39.478 --> 00:02:43.743
What's interesting about that is

00:02:43.743 --> 00:02:45.444
that this is another column vector.

00:02:45.444 --> 00:02:46.841
This is another position vector.

00:02:46.841 --> 00:02:48.862
You took this vector P,

00:02:48.862 --> 00:02:50.870
multiplied it by this
transformation vector

00:02:50.870 --> 00:02:52.381
and we're gonna get another 2x1,

00:02:52.381 --> 00:02:54.498
which we could think of
as a position vector,

00:02:54.498 --> 00:02:55.765
which we could plot.

00:02:55.765 --> 00:02:57.179
What's essentially happened is

00:02:57.179 --> 00:02:59.787
this transformation vector has moved,

00:02:59.787 --> 00:03:02.531
has essentially given this point.

00:03:02.531 --> 00:03:04.412
It has given us a new point.

00:03:04.412 --> 00:03:06.032
Let's think about what that is.

00:03:07.032 --> 00:03:08.318
This first entry here,

00:03:08.318 --> 00:03:09.618
we're essentially thinking about,

00:03:09.618 --> 00:03:12.575
we're going to deal with the first row

00:03:12.575 --> 00:03:15.355
and the first and the only column, really.

00:03:15.355 --> 00:03:17.453
Let me do, use a color
that I haven't used yet.

00:03:18.606 --> 00:03:21.877
We're gonna think about
this row and this column.

00:03:21.877 --> 00:03:23.965
It's gonna be 2x2, which is 4,

00:03:23.965 --> 00:03:26.334
plus 1x1, which is 1

00:03:26.334 --> 00:03:29.887
or this is just going to be equal to 5.

00:03:29.887 --> 00:03:32.271
For the second entry, right over here.

00:03:32.271 --> 00:03:35.521
It's going for the
second row, first column.

00:03:35.521 --> 00:03:37.608
It's gonna deal with the second row here

00:03:37.608 --> 00:03:40.024
and the one and only
column, right over here.

00:03:40.024 --> 00:03:46.541
-1x2 is -2, plus 2x1, which is 2.

00:03:46.541 --> 00:03:52.761
-2+2 is just equal to 0.

00:03:52.761 --> 00:03:55.884
Now, we have the position 5, 0,

00:03:55.884 --> 00:03:57.567
which is right over here.

00:03:57.567 --> 00:04:03.989
1,2,3,4,5

00:04:03.989 --> 00:04:05.987
So, we started with this point,

00:04:05.987 --> 00:04:08.111
this position vector, I
guess you could say, P

00:04:08.111 --> 00:04:11.350
As transformed it to this
other position vector

00:04:11.350 --> 00:04:16.786
and I guess we could call this P prime.

00:04:16.786 --> 00:04:18.568
If you wanted to draw them as vectors,

00:04:18.568 --> 00:04:21.582
this right over, kinda
tradition vector form,

00:04:21.582 --> 00:04:24.479
that right over there is P prime

00:04:24.479 --> 00:04:27.304
and this right over here is P.

00:04:27.304 --> 00:04:31.040
This right over here is P.

00:04:31.040 --> 00:04:33.382
This is P actually, let me
make sure there's a prime.

00:04:33.382 --> 00:04:34.881
This is P prime

00:04:34.881 --> 00:04:37.022
and the way we got from P to P prime

00:04:37.022 --> 00:04:41.315
is using this transformation matrix.


Transforming a quadrilateral

WEBVTT
Kind: captions
Language: en

00:00:00.541 --> 00:00:04.240
If the transformation matrix T is equal to

00:00:04.240 --> 00:00:08.423
three zero zero three, choose
which sketch can represent

00:00:08.423 --> 00:00:12.273
this transformation when applied
to the red quadrilateral.

00:00:12.288 --> 00:00:13.556
This is fascinating!

00:00:14.172 --> 00:00:17.445
They don't give us any coordinates,
save for the vertices of

00:00:17.445 --> 00:00:19.877
the quadrilateral, which are
really the most useful points

00:00:19.877 --> 00:00:23.620
to use when thinking about
potential transformations.

00:00:23.620 --> 00:00:26.176
So let's just make up some
just to see what would happen

00:00:26.176 --> 00:00:28.738
to the particular coordinates
that we're looking at.

00:00:28.738 --> 00:00:30.036
And I think that will give us enough

00:00:30.036 --> 00:00:32.189
information to think about this.

00:00:32.189 --> 00:00:34.714
I encourage you to do
it on your own first.

00:00:34.714 --> 00:00:35.951
Pause the video.

00:00:35.951 --> 00:00:40.753
Come up with some coordinates
for this red quadrilateral.

00:00:40.753 --> 00:00:43.246
Then, see what transformation
you get and which of these

00:00:43.246 --> 00:00:46.273
seem to be closest to
the one that you got.

00:00:47.240 --> 00:00:49.200
I'm assuming you've had a go at it.

00:00:49.200 --> 00:00:51.722
Let's just say for the sake of argument

00:00:51.722 --> 00:00:53.834
that this point right over here.

00:00:53.834 --> 00:00:56.401
That right there, you could
say that's the position vector.

00:00:56.417 --> 00:00:58.828
I'll represent it as a column vector.

00:00:58.843 --> 00:01:01.471
Let's say this is a square.

00:01:01.471 --> 00:01:03.732
So this is one comma one.

00:01:03.732 --> 00:01:09.582
I'll just write that as
a column vector one one.

00:01:09.582 --> 00:01:16.395
And let's say this one then
would be one negative one.

00:01:16.395 --> 00:01:18.606
One negative one.

00:01:19.191 --> 00:01:23.834
Then, this one over here
could be represented

00:01:23.834 --> 00:01:27.267
by the position vector
negative one negative one.

00:01:27.267 --> 00:01:28.762
Negative one negative one.

00:01:28.762 --> 00:01:31.978
Negative one negative one.

00:01:31.994 --> 00:01:35.891
Finally, this point
right over here could be

00:01:35.891 --> 00:01:40.259
represented by the position
vector negative one one.

00:01:40.275 --> 00:01:42.261
Negative one one.

00:01:42.261 --> 00:01:44.719
So let's see what the
transformation matrix would do

00:01:44.719 --> 00:01:46.802
when it transforms these four points.

00:01:46.818 --> 00:01:47.989
The way I'm gonna think about it,

00:01:47.989 --> 00:01:51.001
let me just take our
transformation matrix.

00:01:51.002 --> 00:01:55.853
So, three zero zero three.

00:01:55.868 --> 00:01:59.134
And I'm gonna multiply it
by a two by four matrix

00:01:59.150 --> 00:02:02.668
that represents all of
these position vectors.

00:02:02.684 --> 00:02:08.765
I'm gonna multiply it by,
we have this point one one.

00:02:08.765 --> 00:02:10.229
That's our first point.

00:02:10.245 --> 00:02:12.414
We have the point one negative one.

00:02:12.414 --> 00:02:14.601
One negative one.

00:02:14.601 --> 00:02:18.023
We have this point which
is negative one one.

00:02:18.023 --> 00:02:19.872
Negative one one.

00:02:19.887 --> 00:02:23.193
And then we have this point
which negative one negative one.

00:02:23.193 --> 00:02:25.974
Negative one negative one.

00:02:25.974 --> 00:02:27.866
These were convenient points to pick since

00:02:27.866 --> 00:02:29.166
they didn't give us the points because

00:02:29.181 --> 00:02:32.202
it'll make the math
fairly straightforward.

00:02:32.447 --> 00:02:33.785
What is this going to be equal to?

00:02:33.785 --> 00:02:35.466
We have a two by two.

00:02:35.466 --> 00:02:37.044
Two by two.

00:02:37.060 --> 00:02:40.193
Times a two by four.

00:02:40.193 --> 00:02:42.314
Matrix multiplication is defined here

00:02:42.314 --> 00:02:44.521
cause we have the same number of columns

00:02:44.521 --> 00:02:47.258
as we have rows, right over here.

00:02:47.273 --> 00:02:51.448
This is going to result
in a two by four matrix.

00:02:51.448 --> 00:02:53.660
This is going to give us
another two by four matrix,

00:02:53.660 --> 00:02:55.644
which makes sense, cause
we're going to need four

00:02:55.644 --> 00:02:59.516
column vectors here for our
four new transformed points.

00:02:59.516 --> 00:03:01.403
Let's figure out what it is.

00:03:02.157 --> 00:03:04.796
The first column vector, right over here.

00:03:04.796 --> 00:03:07.509
We can think about this
row and this column.

00:03:07.985 --> 00:03:10.436
Or actually, the first
position right over here

00:03:10.436 --> 00:03:12.627
the first row, first entry
is this row, this column.

00:03:12.643 --> 00:03:14.103
The second one's going to be

00:03:14.103 --> 00:03:15.986
the second row and the first column.

00:03:16.223 --> 00:03:20.779
So let's see, three times
one plus zero times one.

00:03:20.779 --> 00:03:25.309
Well that's three plus zero
so this is going to be three.

00:03:25.309 --> 00:03:29.645
Then over here, zero times
one plus three times one

00:03:29.645 --> 00:03:32.470
is going to give us three as well.

00:03:32.470 --> 00:03:35.300
I think you see already a pattern here.

00:03:35.300 --> 00:03:38.917
To get the the x-coordinate for each

00:03:38.933 --> 00:03:42.177
of these vectors, we're
involving this row.

00:03:42.192 --> 00:03:46.708
And this row, we really just multiplied

00:03:46.708 --> 00:03:48.732
the three times the x-coordinate here

00:03:48.732 --> 00:03:50.453
and then we don't involve the y-coordinate

00:03:50.453 --> 00:03:52.220
because we're multiplying it times zero.

00:03:52.220 --> 00:03:53.989
We'll see that over and over again.

00:03:53.990 --> 00:03:57.621
So over here you have three times one.

00:03:57.621 --> 00:04:01.352
Three times one plus
zero times negative one.

00:04:01.352 --> 00:04:04.250
So it's just really three
times one, which is three.

00:04:04.250 --> 00:04:06.649
Then over here you see
this for the y-coordinate

00:04:06.649 --> 00:04:09.042
each time we're only
involving the y-coordinate

00:04:09.042 --> 00:04:12.131
of the point before transformation.

00:04:12.131 --> 00:04:14.554
You see zero times one is zero.

00:04:14.554 --> 00:04:17.934
So we're essentially not
thinking about the x-coordinate.

00:04:17.934 --> 00:04:19.197
And then it's just three times

00:04:19.197 --> 00:04:22.234
negative one, which is negative three.

00:04:22.234 --> 00:04:24.441
You see what its doing, it's just scaling

00:04:24.441 --> 00:04:27.133
each of these up by a factor of three.

00:04:27.149 --> 00:04:31.914
Three times negative one is negative three

00:04:31.914 --> 00:04:35.080
plus zero times one,
so it's negative three.

00:04:35.080 --> 00:04:37.044
Then, it's going to be
zero times negative one

00:04:37.044 --> 00:04:39.543
is zero plus three times one is three

00:04:39.712 --> 00:04:44.182
and then finally, three
times negative one.

00:04:44.505 --> 00:04:45.682
Three times negative one plus

00:04:45.682 --> 00:04:49.187
zero times negative one is negative three.

00:04:49.203 --> 00:04:50.997
And zero times negative one plus

00:04:50.997 --> 00:04:54.156
three times negative one
is negative three again.

00:04:54.156 --> 00:04:55.988
So what's going to happen to it?

00:04:55.988 --> 00:04:57.735
Well, each of these
coordinates essentially

00:04:57.735 --> 00:05:02.738
get pushed out by a factor of three.

00:05:02.738 --> 00:05:05.578
Actually this one seems to be the closest

00:05:05.578 --> 00:05:06.814
to the one we're thinking about.

00:05:06.814 --> 00:05:07.742
How do I know that?

00:05:07.742 --> 00:05:08.987
Well look at this.

00:05:08.987 --> 00:05:13.679
This is the point one
comma one right over there.

00:05:13.679 --> 00:05:15.339
One comma one.

00:05:15.355 --> 00:05:17.747
It gets mapped to the
point three comma three.

00:05:17.762 --> 00:05:20.250
So, one two three.

00:05:20.265 --> 00:05:22.011
One two three.

00:05:22.011 --> 00:05:24.201
This is three comma
three right over there.

00:05:24.201 --> 00:05:27.543
This got mapped to that and
we see it with each of them.

00:05:27.543 --> 00:05:29.245
One negative one.

00:05:29.245 --> 00:05:33.305
One negative one gets mapped
to three negative three.

00:05:33.305 --> 00:05:36.946
Then, we have negative one one.

00:05:36.946 --> 00:05:38.807
Negative one one.

00:05:38.822 --> 00:05:43.990
Negative one one is getting
mapped to negative three three.

00:05:43.990 --> 00:05:48.100
Finally, of course,
negative one negative one.

00:05:48.100 --> 00:05:49.615
Negative one negative one gets mapped

00:05:49.630 --> 00:05:52.605
to negative three negative three.

00:05:52.605 --> 00:05:54.778
It's definitely the second.

00:05:54.778 --> 00:05:58.166
The second diagram is the one that

00:05:58.166 --> 00:06:03.865
represents the transformation matrix T

00:06:03.865 --> 00:06:07.435
being applied to the red quadrilateral.