WEBVTT

1
00:00:05.706 --> 00:00:09.860
Game Basic Episode
Basic Math that Upholds the Game 4 (Matrix, Inverse matrix)

2
00:00:09.860 --> 00:00:12.078
GCC Academy

3
00:00:29.444 --> 00:00:30.894
Hello, everyone

4
00:00:30.894 --> 00:00:32.700
This is Lee Deukwoo of Game Math

5
00:00:32.700 --> 00:00:35.350
This time, I prepared a lecture about

6
00:00:35.350 --> 00:00:38.747
matrix and inverse matrix, which serve an important role

7
00:00:38.747 --> 00:00:42.180
in the computer graphics process that we will use

8
00:00:42.180 --> 00:00:45.580
Matrix is divided into 4 types

9
00:00:45.580 --> 00:00:47.930
First, we'll learn about what matrix is

10
00:00:47.930 --> 00:00:49.979
and we'll learn about the method

11
00:00:49.979 --> 00:00:52.229
of how it is calculated

12
00:00:52.229 --> 00:00:54.355
Second, using the matrix

13
00:00:54.355 --> 00:00:57.455
we'll learn about how to calculate and apply

14
00:00:57.455 --> 00:01:00.200
the linear transformation we learned before

15
00:01:00.200 --> 00:01:02.400
Third, when the linear transformation process

16
00:01:02.400 --> 00:01:05.339
is connected to matrix

17
00:01:05.339 --> 00:01:07.980
we'll learn about how it can be visualized

18
00:01:07.980 --> 00:01:11.373
Once you learn about this principle

19
00:01:11.373 --> 00:01:13.623
you'll be able to perform the wanted linear transformation

20
00:01:13.623 --> 00:01:17.379
by designing it with a matrix

21
00:01:17.379 --> 00:01:20.739
Finally, we'll learn about the product of matrix

22
00:01:20.739 --> 00:01:22.489
Through the multiplication of matrices

23
00:01:22.489 --> 00:01:26.109
we'll learn about why they allow us to calculate quickly

24
00:01:26.109 --> 00:01:28.059
We'll learn about

25
00:01:28.059 --> 00:01:30.521
the basic mechanism

26
00:01:30.521 --> 00:01:32.471
For inverse matrix

27
00:01:32.471 --> 00:01:34.961
I'll explain in two parts

28
00:01:34.961 --> 00:01:38.059
First, I'll explain about the concept of inverse matrix

29
00:01:38.059 --> 00:01:40.809
And second, I'll talk about

30
00:01:40.809 --> 00:01:41.900
the calculation method of inverse matrix

31
00:01:42.149 --> 00:01:45.792
Matrix

32
00:01:46.211 --> 00:01:49.221
We'll first learn about the matrix

33
00:01:49.221 --> 00:01:52.671
To define a matrix in simple terms

34
00:01:52.671 --> 00:01:54.900
it's simply a rectangular frame

35
00:01:54.900 --> 00:01:58.900
with numbers arranged in rows and columns

36
00:01:58.900 --> 00:02:01.091
It's very simple, right?

37
00:02:01.091 --> 00:02:04.741
Rather than the matrix itself having some meaning

38
00:02:04.741 --> 00:02:09.091
the numbers are arranged inside the rectangular frame

39
00:02:09.091 --> 00:02:12.702
And it's a tool that helps us

40
00:02:12.702 --> 00:02:15.259
calculate something conveniently

41
00:02:15.259 --> 00:02:19.159
I'll explain about this again later but

42
00:02:19.159 --> 00:02:22.139
the linear transformation that we mentioned earlier

43
00:02:22.139 --> 00:02:26.092
can be performed in a very easy, fast, and simple way

44
00:02:26.092 --> 00:02:28.899
through this calculating tool

45
00:02:28.899 --> 00:02:31.709
So when we assign the matrix

46
00:02:31.709 --> 00:02:35.491
we can create them in two concepts

47
00:02:35.491 --> 00:02:37.541
First is transformation

48
00:02:37.541 --> 00:02:41.179
We can design the matrix in the concept of a function

49
00:02:41.179 --> 00:02:46.613
Second, in the perspective of a vector within a space

50
00:02:46.613 --> 00:02:50.820
we can use a matrix to perform the transformation we want

51
00:02:50.820 --> 00:02:54.620
For the transformation in 2D vector space

52
00:02:54.620 --> 00:02:59.930
the matrix is expressed with the size 2x2

53
00:02:59.930 --> 00:03:05.286
Next, for the element of 2D vector space, which is 2D vector

54
00:03:05.286 --> 00:03:11.434
it's expressed with a matrix with 2 rows and 1 column

55
00:03:11.434 --> 00:03:14.084
So these matrices

56
00:03:14.084 --> 00:03:18.108
can be used as a calculating tool

57
00:03:18.108 --> 00:03:20.508
to transform a vector

58
00:03:20.508 --> 00:03:22.899
into the form we want

59
00:03:22.899 --> 00:03:25.724
First, I'll explain about the basic calculation method

60
00:03:25.724 --> 00:03:27.259
of a matrix

61
00:03:27.259 --> 00:03:30.309
Matrices have the adding operation

62
00:03:30.309 --> 00:03:34.609
This can only applied to

63
00:03:34.609 --> 00:03:37.380
matrices with the same size, rows, and columns

64
00:03:37.380 --> 00:03:39.820
You just need to add the numbers in each location

65
00:03:39.820 --> 00:03:41.259
Very easy, right?

66
00:03:41.259 --> 00:03:45.009
So take the number corresponding to the location of "a"

67
00:03:45.009 --> 00:03:49.140
add the two numbers together, assign it to that location

68
00:03:49.140 --> 00:03:51.820
then you can perform the adding operation

69
00:03:51.820 --> 00:03:53.920
Next, matrix and scalar

70
00:03:53.920 --> 00:03:58.129
When a scalar value "k" is assigned to a matrix

71
00:03:58.129 --> 00:04:00.648
you just need to perform a multiplication operation

72
00:04:00.648 --> 00:04:02.460
by multiplying "k" to all elements

73
00:04:02.460 --> 00:04:06.836
It's the same result as

74
00:04:06.836 --> 00:04:09.420
multiplying scalar to a vector

75
00:04:09.420 --> 00:04:12.220
Next, there's something called the transpose operation of a matrix

76
00:04:12.220 --> 00:04:15.820
This is processed using superscript T

77
00:04:15.820 --> 00:04:19.633
Transposing means switching

78
00:04:19.633 --> 00:04:21.594
We're switching the rows and columns

79
00:04:21.594 --> 00:04:24.980
Thus, rows become columns, and columns become rows

80
00:04:24.980 --> 00:04:30.230
When a matrix with 3 rows and 2 columns gets transposed

81
00:04:30.230 --> 00:04:34.630
it changes into a matrix with 2 rows and 3 columns

82
00:04:34.630 --> 00:04:38.404
As you can see here, it flips to the other side

83
00:04:38.404 --> 00:04:41.859
with "a" and "f" as the center point

84
00:04:41.859 --> 00:04:44.709
Using "a" and "f" as the pivot

85
00:04:44.709 --> 00:04:48.578
it flips to the other side

86
00:04:48.578 --> 00:04:51.660
That's the transformation it undergoes

87
00:04:51.660 --> 00:04:54.660
For square matrix, it looks the same even when you flip it

88
00:04:54.660 --> 00:04:56.110
because the rows and columns are the same

89
00:04:56.110 --> 00:04:59.019
Matrices with identical rows and columns are called square matrices

90
00:04:59.019 --> 00:05:03.619
In that case, keep "a" and "d" as-is

91
00:05:03.619 --> 00:05:09.398
and for the rest of the elements, "b" and "c"

92
00:05:09.398 --> 00:05:12.002
flip them to the other side

93
00:05:12.002 --> 00:05:14.779
then you have performed the transpose operation

94
00:05:14.779 --> 00:05:18.500
That's called the transpose operation

95
00:05:18.500 --> 00:05:22.140
The next important operation is the multiplication of matrices

96
00:05:22.140 --> 00:05:24.090
This is a bit complicated

97
00:05:24.090 --> 00:05:28.497
When there are two matrices like this

98
00:05:28.497 --> 00:05:31.297
the one on the left is the row

99
00:05:31.297 --> 00:05:33.978
We use its rows

100
00:05:33.978 --> 00:05:36.522
For the one on the right, we use its columns

101
00:05:36.522 --> 00:05:41.847
So criss-cross the rows and columns, and multiply

102
00:05:41.847 --> 00:05:44.370
Multiply and add each element

103
00:05:44.370 --> 00:05:48.550
Add the product of "a", "e" and the product of "b", "g"

104
00:05:48.550 --> 00:05:51.019
Then you will end up with "ae" plus "bg"

105
00:05:51.019 --> 00:05:54.852
For the second element, multiply "a" and "f"

106
00:05:54.852 --> 00:05:57.980
and add it to the product of "b" and "h"

107
00:05:57.980 --> 00:06:00.730
For the third one, row 2 column 1

108
00:06:00.730 --> 00:06:05.043
Multiply "c" and "e" and add it to the product of "d" and "g"

109
00:06:05.043 --> 00:06:09.761
For the fourth, final one, row 2 column 2

110
00:06:09.761 --> 00:06:15.402
Multiply "c" and "f", and add the product of "d" and "h"

111
00:06:15.402 --> 00:06:18.519
Just like that, cross the rows and columns

112
00:06:18.519 --> 00:06:20.660
and add the elements after multiplying them

113
00:06:20.660 --> 00:06:23.810
The multiplication operation is complicated

114
00:06:23.810 --> 00:06:27.619
so first-timers will have to practice many times

115
00:06:27.619 --> 00:06:30.775
Through these kinds of operation system

116
00:06:30.775 --> 00:06:34.786
we can calculate mechanically

117
00:06:34.786 --> 00:06:38.633
It's good to know about the characteristics of operations beforehand

118
00:06:38.633 --> 00:06:42.233
The characteristic of the multiplication operation of matrices

119
00:06:42.233 --> 00:06:46.551
is that it doesn't satisfy the commutative property

120
00:06:46.551 --> 00:06:50.876
That is, when matrices "a" and "b" are multiplied in reverse order

121
00:06:50.876 --> 00:06:54.292
we can't guarantee that the value will be the same

122
00:06:54.292 --> 00:06:55.530
We'll take a look

123
00:06:55.530 --> 00:06:59.330
When "a" and "b" are defined as matrices, like the following

124
00:06:59.330 --> 00:07:03.757
the first element will be ae+bg

125
00:07:03.757 --> 00:07:07.357
But if we switch these two

126
00:07:07.357 --> 00:07:10.879
it becomes ea+fc

127
00:07:10.879 --> 00:07:15.756
Since the commutative property is valid for ea

128
00:07:15.756 --> 00:07:18.700
the first is the same, but the second is bg

129
00:07:18.700 --> 00:07:20.303
and we get cf here

130
00:07:20.303 --> 00:07:22.380
So we can see that the value is different

131
00:07:22.380 --> 00:07:26.246
For the value to be the same even with the switched order

132
00:07:26.246 --> 00:07:30.081
a particular condition would have to be established

133
00:07:30.081 --> 00:07:32.081
Anyways, for normal cases

134
00:07:32.081 --> 00:07:34.859
the commutative property is not satisfied

135
00:07:34.859 --> 00:07:37.072
as you should know

136
00:07:37.072 --> 00:07:40.422
However, switching the order

137
00:07:40.422 --> 00:07:44.430
switching the multiplication order when calculating

138
00:07:44.430 --> 00:07:49.080
And by order, I mean the direction is the same

139
00:07:49.080 --> 00:07:51.656
Commutative property is when the direction itself

140
00:07:51.656 --> 00:07:55.006
the multiplying direction itself is flipped

141
00:07:55.006 --> 00:07:58.106
For associative property, the direction is the same

142
00:07:58.106 --> 00:08:00.256
but it's about whether multiplication is in the front or the back

143
00:08:00.256 --> 00:08:02.374
It's about whether we're multiplying the front first or the back first

144
00:08:02.374 --> 00:08:03.871
That's the difference

145
00:08:03.871 --> 00:08:06.471
So as long as we maintain the same direction

146
00:08:06.471 --> 00:08:08.565
whether we multiply the front first or the back first

147
00:08:08.565 --> 00:08:11.500
the result will be the same

148
00:08:11.500 --> 00:08:14.823
I will explain about these properties of a matrix later

149
00:08:14.823 --> 00:08:18.288
but this gives you a very important, big advantage

150
00:08:18.288 --> 00:08:21.540
The property of a matrix satisfies the associative property

151
00:08:21.540 --> 00:08:25.140
Since it satisfies like this

152
00:08:25.140 --> 00:08:27.714
whether we multiply first or later

153
00:08:27.714 --> 00:08:31.420
you can understand that the results are the same

154
00:08:31.420 --> 00:08:34.120
The third thing we need to know

155
00:08:34.120 --> 00:08:40.431
is the transpose of the products of matrices

156
00:08:40.431 --> 00:08:46.802
When multiplied products are transposed

157
00:08:46.802 --> 00:08:52.452
it's the same thing as multiplying

158
00:08:52.452 --> 00:08:54.594
the transposed matrices

159
00:08:54.594 --> 00:09:00.294
What that means, is that

160
00:09:00.294 --> 00:09:03.881
when we multiply the matrices "A" and "B"

161
00:09:03.881 --> 00:09:10.500
the transpose matrix of this would be ehfg

162
00:09:10.500 --> 00:09:14.460
And the transpose of this would be adbc

163
00:09:14.460 --> 00:09:17.510
But multiplying first

164
00:09:17.510 --> 00:09:20.659
has the same results as

165
00:09:20.659 --> 00:09:24.260
transposing, and then multiplying

166
00:09:24.260 --> 00:09:26.380
Then we end up with this kind of formula

167
00:09:26.380 --> 00:09:30.979
We're going to use it later, so it would be good to know

168
00:09:30.979 --> 00:09:34.968
Next, although it is complicated, it also satisfies the distributive property

169
00:09:34.968 --> 00:09:36.979
which uses the multiplication and addition of matrices

170
00:09:36.979 --> 00:09:39.980
The calculation is complicated, so I didn't write it down but

171
00:09:39.980 --> 00:09:42.260
just know that it is satisfied

172
00:09:42.260 --> 00:09:44.272
Actually, it's good to know but

173
00:09:44.272 --> 00:09:45.822
among what we're going to learn right now

174
00:09:45.822 --> 00:09:47.472
distributive property won't be used

175
00:09:47.472 --> 00:09:50.492
That's why I didn't write it down

176
00:09:50.492 --> 00:09:52.892
We now learned about the characteristics

177
00:09:52.892 --> 00:09:54.901
of basic operations

178
00:09:54.901 --> 00:09:57.301
Then why do we use matrices?

179
00:09:57.301 --> 00:10:00.260
I've briefly mentioned earlier, but

180
00:10:00.260 --> 00:10:03.760
in linear transformation, maintaining the structure of the vector space as-is

181
00:10:03.760 --> 00:10:06.878
while also having linearity

182
00:10:06.878 --> 00:10:10.528
and transforming into a new space, this process itself

183
00:10:10.528 --> 00:10:13.978
can be corresponded to a matrix, and that's the merit

184
00:10:13.978 --> 00:10:16.659
Therefore, rather than calculating each linear transformation

185
00:10:16.659 --> 00:10:19.859
simply using matrices

186
00:10:19.859 --> 00:10:22.783
to organize them and conduct multiplication

187
00:10:22.783 --> 00:10:25.473
we quickly can perform the wanted linear transformation

188
00:10:25.473 --> 00:10:27.780
Because of that merit, we use matrices

189
00:10:27.780 --> 00:10:29.785
We'll learn about how matrices

190
00:10:29.785 --> 00:10:32.299
correspond to linear transformation

191
00:10:32.299 --> 00:10:35.349
Earlier, we said that vector functions, like the following

192
00:10:35.349 --> 00:10:40.726
satisfy linearity

193
00:10:40.726 --> 00:10:45.623
When we combine two independent elements, x and y

194
00:10:45.623 --> 00:10:51.129
to make a vector, and change that in some form

195
00:10:51.129 --> 00:10:55.318
and when we make the first and second elements

196
00:10:55.318 --> 00:10:59.986
with this combination of linear equation, with no impurities

197
00:10:59.986 --> 00:11:02.640
then in the vector space R2

198
00:11:02.640 --> 00:11:05.820
it's a linear transformation that has correspondence to

199
00:11:05.820 --> 00:11:07.291
R2

200
00:11:07.291 --> 00:11:10.641
Same thing with this, when I define

201
00:11:10.641 --> 00:11:13.923
an arbitrary matrix as abcd

202
00:11:13.923 --> 00:11:17.269
and an arbitrary vector as x, y

203
00:11:17.269 --> 00:11:21.169
then based on the rule of multiplication of matrices

204
00:11:21.169 --> 00:11:24.069
the result of the matrix is like the following

205
00:11:24.069 --> 00:11:28.602
The first element is ax+by

206
00:11:28.602 --> 00:11:31.402
and the second element is cx+dy

207
00:11:31.402 --> 00:11:33.663
It is made into a vector

208
00:11:33.663 --> 00:11:36.203
If you look here

209
00:11:36.203 --> 00:11:39.653
the original x, y corresponds to this vector

210
00:11:39.653 --> 00:11:41.353
and then the result

211
00:11:41.353 --> 00:11:45.844
of using the vector function

212
00:11:45.844 --> 00:11:48.694
is the same as

213
00:11:48.694 --> 00:11:50.566
ax+by, cx+dy

214
00:11:50.566 --> 00:11:55.463
Then this matrix, abcd, ultimately

215
00:11:55.463 --> 00:12:02.503
corresponds x and y to a point in ax+by, cx+dy

216
00:12:02.503 --> 00:12:06.160
in a linear transformation

217
00:12:06.160 --> 00:12:11.643
Thus, it conducts the role of a function

218
00:12:11.643 --> 00:12:15.479
Then now we're ready to conduct linear transformation

219
00:12:15.479 --> 00:12:19.579
using these matrices

220
00:12:19.579 --> 00:12:21.756
Before getting into it

221
00:12:21.756 --> 00:12:24.556
there are two ways to express a matrix

222
00:12:24.556 --> 00:12:28.163
There's the column major matrix, and the row major matrix

223
00:12:28.163 --> 00:12:31.440
Usually, when you use computer graphics

224
00:12:31.440 --> 00:12:35.140
or make an independent game engine

225
00:12:35.140 --> 00:12:39.659
there are two ways to use the matrix

226
00:12:39.659 --> 00:12:42.909
The normal way we use is

227
00:12:42.909 --> 00:12:47.706
treating the vector as the column during calculation

228
00:12:47.706 --> 00:12:49.906
And we make the matrix into a square matrix

229
00:12:49.906 --> 00:12:51.405
Make the function into a square matrix

230
00:12:51.405 --> 00:12:56.405
The result is also a column vector, as we call it

231
00:12:56.405 --> 00:12:59.933
We can see that it is made into a column vector

232
00:12:59.933 --> 00:13:03.973
This notation method is called OPEN GL

233
00:13:03.973 --> 00:13:05.323
In graphic libraries

234
00:13:05.323 --> 00:13:07.273
or when we conduct matrices in normal math

235
00:13:07.273 --> 00:13:09.684
we often use this method

236
00:13:09.684 --> 00:13:12.560
Since this expresses the vector as column

237
00:13:12.560 --> 00:13:15.960
it's called a column major matrix

238
00:13:15.960 --> 00:13:19.344
However, there's also a way to express based on the row

239
00:13:19.344 --> 00:13:22.344
This method expresses the vector as a row

240
00:13:22.344 --> 00:13:24.640
And

241
00:13:24.640 --> 00:13:26.890
when we use DirectX or something

242
00:13:26.890 --> 00:13:31.100
to implement the game engine ourselves

243
00:13:31.100 --> 00:13:34.297
or when we take a look at the source of business engines

244
00:13:34.297 --> 00:13:38.347
some cases are written as row major matrix

245
00:13:38.347 --> 00:13:41.552
When we use the vector as a row

246
00:13:41.552 --> 00:13:45.402
we spread out x, y horizontally

247
00:13:45.402 --> 00:13:49.201
and its order comes to the front

248
00:13:49.201 --> 00:13:54.140
And when we look at abcd

249
00:13:54.140 --> 00:13:58.923
this one is arranged in horizontal order

250
00:13:58.923 --> 00:14:02.423
but here, abcd is arranged in a vertical order

251
00:14:02.423 --> 00:14:04.979
Then the direction would change now

252
00:14:04.979 --> 00:14:07.529
Finally, for the result of calculating this

253
00:14:07.529 --> 00:14:09.979
since we calculate the row and column

254
00:14:09.979 --> 00:14:12.876
ax+by is correct

255
00:14:12.876 --> 00:14:17.099
And then cx+dy, the values are the same

256
00:14:17.099 --> 00:14:18.852
However, this has a result of a column

257
00:14:18.852 --> 00:14:22.612
but this has a result of a row

258
00:14:22.612 --> 00:14:26.862
Then let's say

259
00:14:26.862 --> 00:14:29.120
we made a new vector on a vector

260
00:14:29.120 --> 00:14:34.604
When ax+by, cx+dy is set as v'

261
00:14:34.604 --> 00:14:36.829
and when x, y is set as v

262
00:14:36.829 --> 00:14:39.929
it gets this form of equation

263
00:14:39.929 --> 00:14:41.729
in column major matrix

264
00:14:41.729 --> 00:14:47.257
But when we change this into row major

265
00:14:47.257 --> 00:14:50.815
the result, v'

266
00:14:50.815 --> 00:14:54.940
v', expressed in column major, is now changed

267
00:14:54.940 --> 00:14:56.390
into row major

268
00:14:56.390 --> 00:15:01.571
Then changing the v' of column major into a row major

269
00:15:01.571 --> 00:15:03.886
ultimately means transposing

270
00:15:03.886 --> 00:15:06.736
Transposing this is taking Av

271
00:15:06.736 --> 00:15:09.136
the equation we used earlier to make this

272
00:15:09.136 --> 00:15:11.843
this equation of multiplying matrix A and v

273
00:15:11.843 --> 00:15:15.348
will be like transposed in the result

274
00:15:15.348 --> 00:15:18.748
What that becomes is

275
00:15:18.748 --> 00:15:22.063
transposing a product of matrices

276
00:15:22.063 --> 00:15:23.604
is the same as switching the two

277
00:15:23.604 --> 00:15:26.833
and multiplying the transpose of each

278
00:15:26.833 --> 00:15:29.283
Therefore, for this equation

279
00:15:29.283 --> 00:15:35.916
put the transposed v^T first

280
00:15:35.916 --> 00:15:41.316
and put the transposed matrix A in the back

281
00:15:41.316 --> 00:15:44.780
It's the same result of calculation

282
00:15:44.780 --> 00:15:48.980
When we use the vector as the column

283
00:15:48.980 --> 00:15:51.719
here's the difference between using it as the column

284
00:15:51.719 --> 00:15:53.288
and using it as the row

285
00:15:53.288 --> 00:15:57.290
The matrix that actually conducts the transformation

286
00:15:57.290 --> 00:15:59.290
has a simple difference

287
00:15:59.290 --> 00:16:00.609
in the transposition

288
00:16:00.609 --> 00:16:03.609
The only difference is the A and A^T

289
00:16:03.609 --> 00:16:08.940
Therefore, using row major versus column major

290
00:16:08.940 --> 00:16:10.740
doesn't make a big difference

291
00:16:10.740 --> 00:16:14.388
For the matrix that actually conducts transformation

292
00:16:14.388 --> 00:16:17.620
row major method is simply transposing

293
00:16:17.620 --> 00:16:20.413
the matrix used in column major

294
00:16:20.413 --> 00:16:21.613
Same goes for the other way

295
00:16:21.613 --> 00:16:23.459
If you want to use it as row major

296
00:16:23.459 --> 00:16:26.859
same thing, simply transpose it

297
00:16:26.859 --> 00:16:31.937
The one we'll talk about in this lesson

298
00:16:31.937 --> 00:16:33.787
is column major

299
00:16:33.787 --> 00:16:37.038
but if you are using other DirectX or referring to

300
00:16:37.038 --> 00:16:39.988
source codes of Unreal Engine, and you find a matrix

301
00:16:39.988 --> 00:16:43.374
they will be expressed as row major

302
00:16:43.374 --> 00:16:45.739
Since they're different, you might think that you have to

303
00:16:45.739 --> 00:16:49.735
learn it again from the start, but it will be solved by simply

304
00:16:49.735 --> 00:16:51.585
transposing it

305
00:16:51.585 --> 00:16:56.739
I summarized it here because it would be good to know

306
00:16:56.739 --> 00:16:59.539
To see how matrices that conduct these linear transformation

307
00:16:59.539 --> 00:17:02.500
get changed

308
00:17:02.500 --> 00:17:06.535
we'll draw an actual picture

309
00:17:06.535 --> 00:17:08.735
Before the transformation

310
00:17:08.735 --> 00:17:11.142
let's say there's a vector space

311
00:17:11.142 --> 00:17:14.459
The x, y here is

312
00:17:14.459 --> 00:17:18.609
Actually, the x and y

313
00:17:18.609 --> 00:17:23.859
are combined in a uniform, equal relationship

314
00:17:23.859 --> 00:17:26.309
with no interference

315
00:17:26.309 --> 00:17:31.483
So for the arbitrary point x, y

316
00:17:31.483 --> 00:17:36.009
if we think about the structure of how it is created

317
00:17:36.009 --> 00:17:39.540
e1 and e2 of a standard basis vector

318
00:17:39.540 --> 00:17:45.590
were each multiplied by x and y, then added together

319
00:17:45.590 --> 00:17:48.335
Through a combined equation like this

320
00:17:48.335 --> 00:17:53.200
the vector was created in the vector space

321
00:17:53.200 --> 00:17:56.000
Then some space was changed

322
00:17:56.000 --> 00:17:59.000
Actually, a space contains infinite elements

323
00:17:59.000 --> 00:18:01.000
therefore, tracking down all the infinite elements

324
00:18:01.000 --> 00:18:03.592
and analyzing each correspondence

325
00:18:03.592 --> 00:18:05.663
is actually close to impossible

326
00:18:05.663 --> 00:18:09.376
However, if we analyze the movement of the basis vector

327
00:18:09.376 --> 00:18:13.356
which is the foundation holding up the space

328
00:18:13.356 --> 00:18:17.267
then even if we don't check every single element

329
00:18:17.267 --> 00:18:19.017
we can visually predict

330
00:18:19.017 --> 00:18:21.000
how the space will change

331
00:18:21.000 --> 00:18:23.250
Through those predictions, we can have the wanted effect

332
00:18:23.250 --> 00:18:25.713
by tracking backwards

333
00:18:25.713 --> 00:18:29.525
So the original space had two foundations

334
00:18:29.525 --> 00:18:33.634
That was, the standard basis vector, E1 and E2

335
00:18:33.634 --> 00:18:35.584
And it was changed

336
00:18:35.584 --> 00:18:39.624
The point in the changed space

337
00:18:39.624 --> 00:18:42.713
was changed like this

338
00:18:42.713 --> 00:18:47.713
The first standard basis vector corresponding to (1, 0) is (a,c)

339
00:18:47.713 --> 00:18:54.663
and the second standard basis vector is (b, d)

340
00:18:54.663 --> 00:18:57.763
Then the vector (1, 1) made by

341
00:18:57.763 --> 00:19:02.178
combining these with 1 to 1 proportion

342
00:19:02.178 --> 00:19:04.228
changes into this form

343
00:19:04.228 --> 00:19:07.386
All the vectors in the space change this way

344
00:19:07.386 --> 00:19:12.396
Of course, the elements are infinite, so even if the space is changed

345
00:19:12.396 --> 00:19:14.446
this flat surface

346
00:19:14.446 --> 00:19:18.743
will eventually create infinite flat surfaces

347
00:19:18.743 --> 00:19:25.000
But the process of making that

348
00:19:25.000 --> 00:19:28.750
is the same as applying

349
00:19:28.750 --> 00:19:31.000
this matrix we talked about earlier

350
00:19:31.000 --> 00:19:33.832
It's the same

351
00:19:33.832 --> 00:19:38.406
So ax+by, cx+dy

352
00:19:38.406 --> 00:19:42.000
Applying this linear transformation is the same

353
00:19:42.000 --> 00:19:45.000
This is how it's applied

354
00:19:45.000 --> 00:19:50.158
On x, multiply (a, c)

355
00:19:50.158 --> 00:19:53.408
Then on y, multiply (b, d)

356
00:19:53.408 --> 00:19:56.238
and the results are the same

357
00:19:56.238 --> 00:20:02.198
On an arbitrary point

358
00:20:02.198 --> 00:20:04.548
we multiplied x on the first basis vector

359
00:20:04.548 --> 00:20:07.000
and multiplied y on the second basis vector

360
00:20:07.000 --> 00:20:10.400
Just like that, on a changed vector of (1, 0)

361
00:20:10.400 --> 00:20:15.000
which is (a, c), we multiply x

362
00:20:15.000 --> 00:20:16.604
We applied the same proportion

363
00:20:16.604 --> 00:20:20.354
And on the changed vector of (0, 1), which is (b, d)

364
00:20:20.354 --> 00:20:23.000
we apply y

365
00:20:23.000 --> 00:20:29.208
and it proceeds the same way as the linear transformation from earlier

366
00:20:29.208 --> 00:20:33.564
Then here, this value of the matrix

367
00:20:33.564 --> 00:20:35.762
which we mentioned as corresponding to linear transformation

368
00:20:35.762 --> 00:20:40.772
when we look at the column vector, not the row

369
00:20:40.772 --> 00:20:44.000
there are two column vectors

370
00:20:44.000 --> 00:20:49.600
This 2x2 matrix, corresponding to the function

371
00:20:49.600 --> 00:20:50.881
can be seen as the column vector

372
00:20:50.881 --> 00:20:54.594
Here, the first column vector is (a, c)

373
00:20:54.594 --> 00:20:57.680
It stands for the changed value of

374
00:20:57.680 --> 00:20:59.238
the first basis vector

375
00:20:59.238 --> 00:21:03.388
And (b, d), corresponding to the second column

376
00:21:03.388 --> 00:21:07.475
is the changed value of the second basis vector

377
00:21:07.475 --> 00:21:11.000
Therefore, we're tracking the difference of this basis vector

378
00:21:11.000 --> 00:21:12.218
That's what using a matrix is

379
00:21:12.218 --> 00:21:15.564
Using this matrix designed as abcd

380
00:21:15.564 --> 00:21:19.426
is actually looking at the column vector

381
00:21:19.426 --> 00:21:22.683
and identifying the two standard basis vectors

382
00:21:22.683 --> 00:21:27.083
that each change into (a, c) and (b, d)

383
00:21:27.083 --> 00:21:29.168
in this process

384
00:21:29.168 --> 00:21:32.068
When we think of it this way, then we can easily design

385
00:21:32.068 --> 00:21:34.000
the transformation we want

386
00:21:34.000 --> 00:21:37.000
For example, we'll take a look at the scale transformation matrix

387
00:21:37.000 --> 00:21:42.742
Let's say there's this basic object in the original space

388
00:21:42.742 --> 00:21:45.356
and I want to stretch this horizontally

389
00:21:45.356 --> 00:21:49.000
We'll stretch it "a" times

390
00:21:49.000 --> 00:21:53.990
And vertically, we'll shrink it "b" times

391
00:21:53.990 --> 00:21:56.940
Then each basis vector turns out this way

392
00:21:56.940 --> 00:22:00.366
(1, 0) increases to (a, 0)

393
00:22:00.366 --> 00:22:04.554
and (0, 1) decreases to (0, b)

394
00:22:04.554 --> 00:22:06.000
When this is expressed as a matrix

395
00:22:06.000 --> 00:22:10.584
each of the changed basis vector, (a, 0) and (0, b)

396
00:22:10.584 --> 00:22:14.307
just needs to be inserted as the column vector

397
00:22:14.307 --> 00:22:17.337
Then it's made into a 2x2 matrix

398
00:22:17.337 --> 00:22:21.802
that actually conducts the transformation

399
00:22:21.802 --> 00:22:26.000
To actually try it out, when x, y vector is multiplied here

400
00:22:26.000 --> 00:22:35.139
it becomes the matrix ax, by

401
00:22:35.139 --> 00:22:42.148
Here, the x was multiplied by a, and y was multiplied by b

402
00:22:42.148 --> 00:22:45.000
and it can be seen as a linear transformation

403
00:22:45.000 --> 00:22:48.445
It corresponds to scale transformation

404
00:22:48.445 --> 00:22:50.594
We'll now think of translation

405
00:22:50.594 --> 00:22:54.000
I'll translate a space to the side like this

406
00:22:54.000 --> 00:22:58.000
When we leave one side as-is, and push the other side

407
00:22:58.000 --> 00:23:01.000
since x-axis is fixed, it stays as-is

408
00:23:01.000 --> 00:23:04.000
Then the first basis vector has no change

409
00:23:04.000 --> 00:23:06.525
Just use the (1, 0) as-is

410
00:23:06.525 --> 00:23:10.000
When we slant it one unit to the side

411
00:23:10.000 --> 00:23:13.257
the (0, 1) vector changes into (1, 1)

412
00:23:13.257 --> 00:23:16.852
Then, we just insert (1, 1) into the second column

413
00:23:16.852 --> 00:23:22.552
This is the translation of

414
00:23:22.552 --> 00:23:25.416
pushing y-axis one unit toward the x-axis

415
00:23:25.416 --> 00:23:28.228
If we pushed it "a" times

416
00:23:28.228 --> 00:23:33.000
then just replace it with (a, 1)

417
00:23:33.000 --> 00:23:38.218
and we'll end up with a matrix with the wanted amount of translation

418
00:23:38.218 --> 00:23:44.218
That's how you can design the matrix

419
00:23:44.218 --> 00:23:47.218
This time, we'll think about rotation, 90 degrees

420
00:23:47.218 --> 00:23:48.188
clockwise

421
00:23:48.188 --> 00:23:52.139
For rotation, we mentioned earlier but

422
00:23:52.139 --> 00:23:57.000
the size of the two basis vectors are the same as the original

423
00:23:57.000 --> 00:24:01.050
and it maintains 90 degrees, and the direction of the 90 degrees

424
00:24:01.050 --> 00:24:04.000
also stay the same

425
00:24:04.000 --> 00:24:08.000
This way, the shape of the object is maintained

426
00:24:08.000 --> 00:24:14.297
Therefore, rotation keeps the shape of the object as-is

427
00:24:14.297 --> 00:24:19.000
but changes the direction

428
00:24:19.000 --> 00:24:21.455
When we rotate it 90 degrees clockwise

429
00:24:21.455 --> 00:24:26.852
x-axis would to go negative y-axis, and y-axis would become x-axis

430
00:24:26.852 --> 00:24:29.702
It would become like this

431
00:24:29.702 --> 00:24:32.614
Then since the difference of x-axis is 0, -1

432
00:24:32.614 --> 00:24:34.693
make this the first column

433
00:24:34.693 --> 00:24:36.693
And put (1, 0) in the second column

434
00:24:36.693 --> 00:24:38.901
Then you get a 90 degrees rotation

435
00:24:38.901 --> 00:24:43.614
It becomes a rotation that rotates 90 degrees clockwise

436
00:24:43.614 --> 00:24:48.396
If we actually multiply this

437
00:24:48.396 --> 00:24:54.941
If we multiply the actual vectors, x and y

438
00:24:54.941 --> 00:24:58.465
the result becomes x

439
00:24:58.465 --> 00:25:02.792
When we multiply the matrix of x and y

440
00:25:02.792 --> 00:25:04.950
the first element would become y

441
00:25:04.950 --> 00:25:09.000
and the second element would become -x

442
00:25:09.000 --> 00:25:12.000
So 90 degrees rotation means

443
00:25:12.000 --> 00:25:14.307
flipping x and y

444
00:25:14.307 --> 00:25:18.743
and putting a negative sign on the y value

445
00:25:18.743 --> 00:25:21.993
Put the opposite sign on it

446
00:25:21.993 --> 00:25:23.574
and it gets solved right away, actually

447
00:25:23.574 --> 00:25:25.544
Even if we don't use a matrix

448
00:25:25.544 --> 00:25:29.356
we can obtain it right away by swapping

449
00:25:29.356 --> 00:25:31.556
This time, we'll learn about the

450
00:25:31.556 --> 00:25:33.297
90 degrees counterclockwise rotation

451
00:25:33.297 --> 00:25:38.673
As we've seen earlier, we can predict it right away

452
00:25:38.673 --> 00:25:42.386
Counterclockwise means it's turning this way

453
00:25:42.386 --> 00:25:46.000
That means the y-axis became -x

454
00:25:46.000 --> 00:25:48.238
and the x-axis became the y-axis

455
00:25:48.238 --> 00:25:51.099
So since y-axis became -x

456
00:25:51.099 --> 00:25:53.416
and x-axis became the y-axis

457
00:25:53.416 --> 00:25:58.663
it means a rotation of (0, 1), (-1, 0)

458
00:25:58.663 --> 00:26:00.564
For this, it's the same

459
00:26:00.564 --> 00:26:04.000
Swap the two

460
00:26:04.000 --> 00:26:07.307
and put the opposite sign on the first element

461
00:26:07.307 --> 00:26:10.495
Then you can calculate it right away

462
00:26:10.495 --> 00:26:13.733
With that, we learned about the 90 degrees rotation

463
00:26:13.733 --> 00:26:17.386
How would the rotation of an arbitrary value, theta, work?

464
00:26:17.386 --> 00:26:20.286
For this, it's the same method as

465
00:26:20.286 --> 00:26:22.742
when we used trigonometric functions last time

466
00:26:22.742 --> 00:26:25.327
For each of the two standard basis vectors

467
00:26:25.327 --> 00:26:28.663
When the first standard basis vector is rotated θ degrees

468
00:26:28.663 --> 00:26:34.277
this moves to point (cosθ, sinθ)

469
00:26:34.277 --> 00:26:37.594
And when (0, 1) is rotated θ degrees

470
00:26:37.594 --> 00:26:44.228
the vector moves to (-sinθ, cosθ)

471
00:26:44.228 --> 00:26:46.178
Then how would we make

472
00:26:46.178 --> 00:26:47.822
the rotation matrix that corresponds to this?

473
00:26:47.822 --> 00:26:50.663
Set (cosθ, sinθ) as the first column

474
00:26:50.663 --> 00:26:56.000
and (-sinθ, cosθ) as the second column

475
00:26:56.000 --> 00:26:59.200
This way, you can implement a rotation

476
00:26:59.200 --> 00:27:03.000
on an arbitrary value of θ

477
00:27:03.000 --> 00:27:07.396
When you make a matrix like this and try multiplying x and y

478
00:27:07.396 --> 00:27:13.000
the result gives you xcosθ-ysinθ as the first element

479
00:27:13.000 --> 00:27:17.139
and xsinθ+ycosθ as the second element

480
00:27:17.139 --> 00:27:21.000
This is the same result as

481
00:27:21.000 --> 00:27:25.000
the calculated value from trigonometric functions earlier

482
00:27:25.000 --> 00:27:28.550
Therefore, from now on, if we're rotating a vector

483
00:27:28.550 --> 00:27:31.000
with a value of θ

484
00:27:31.000 --> 00:27:35.396
think of the coordinates of these two basis vectors

485
00:27:35.396 --> 00:27:39.495
Since they're similar right triangles

486
00:27:39.495 --> 00:27:43.000
the y value here becomes x value, the opposite way

487
00:27:43.000 --> 00:27:50.000
and here, the x value becomes the y value

488
00:27:50.000 --> 00:27:54.683
Therefore, you can see that the first (cosθ, sinθ)

489
00:27:54.683 --> 00:28:01.227
becomes (-sinθ, cosθ) for the y value

490
00:28:01.227 --> 00:28:06.000
As long as you clearly know these two vectors

491
00:28:06.000 --> 00:28:08.505
you just need to insert them here one by one

492
00:28:08.505 --> 00:28:12.555
cosθ, sinθ, -sinθ, cosθ

493
00:28:12.555 --> 00:28:15.000
This becomes the rotation matrix

494
00:28:15.000 --> 00:28:19.000
With that, we learned about the three basic transformations

495
00:28:19.000 --> 00:28:22.000
that we'll need from now on

496
00:28:22.000 --> 00:28:26.198
Those were, scaling, translation, and rotation

497
00:28:26.198 --> 00:28:29.000
We learned about these three types

498
00:28:29.000 --> 00:28:30.750
and how each of these

499
00:28:30.750 --> 00:28:33.752
are expressed as matrix

500
00:28:33.752 --> 00:28:36.902
Then finally, we're going to talk about

501
00:28:36.902 --> 00:28:38.940
the multiplication of matrix

502
00:28:38.940 --> 00:28:41.290
They say that matrix is simple but

503
00:28:41.290 --> 00:28:46.000
we don't necessarily use matrices just because they're simple

504
00:28:46.000 --> 00:28:48.850
There's a very important characteristic

505
00:28:48.850 --> 00:28:51.000
that the matrix has

506
00:28:51.000 --> 00:28:54.000
We can also call it the characteristic of matrix multiplication

507
00:28:54.000 --> 00:28:56.614
Let's learn about it

508
00:28:56.614 --> 00:29:01.000
Let's say that there are three vector spaces, like below

509
00:29:01.000 --> 00:29:06.406
Let's say that each is real vector space R2

510
00:29:06.406 --> 00:29:09.317
Then if we apply linear transformation twice

511
00:29:09.317 --> 00:29:11.817
the vector space will be transformed twice

512
00:29:11.817 --> 00:29:17.000
and it will ultimately proceed into a new vector space

513
00:29:17.000 --> 00:29:20.356
We'll call the first transformation Matrix B

514
00:29:20.356 --> 00:29:25.000
and we'll call the second transformation Matrix A

515
00:29:25.000 --> 00:29:30.700
If we say that it will go through two transformations

516
00:29:30.700 --> 00:29:33.653
and if we express this in an equation

517
00:29:33.653 --> 00:29:37.436
multiply the element V, of the first vector space

518
00:29:37.436 --> 00:29:40.643
with matrix B

519
00:29:40.643 --> 00:29:44.343
then, you'll ultimately get the element of the second vector space

520
00:29:44.343 --> 00:29:46.109
which is vector u

521
00:29:46.109 --> 00:29:49.654
When you multiply vector u with matrix A

522
00:29:49.654 --> 00:29:53.404
it proceeds to the last vector w

523
00:29:53.404 --> 00:29:55.465
That is expressed in this equation

524
00:29:55.465 --> 00:29:58.665
But for matrix, I've mentioned earlier that

525
00:29:58.665 --> 00:30:00.396
the associative property stands

526
00:30:00.396 --> 00:30:04.520
Therefore, we'll try calculating this equation after switching the order

527
00:30:04.520 --> 00:30:11.640
Then w has the same result as

528
00:30:11.640 --> 00:30:14.640
v multiplied to

529
00:30:14.640 --> 00:30:18.799
the product of matrix A and B

530
00:30:18.799 --> 00:30:21.840
Then let me explain what this is

531
00:30:21.840 --> 00:30:27.919
Matrix of A and B multiplied is also a square matrix, after all

532
00:30:27.919 --> 00:30:30.651
It's a 2x2 square matrix

533
00:30:30.651 --> 00:30:35.599
What the square matrix itself means is

534
00:30:35.599 --> 00:30:41.599
it serves the same role as a composite function

535
00:30:41.599 --> 00:30:45.880
which goes directly from vector space V to vector space W

536
00:30:45.880 --> 00:30:47.780
As mentioned earlier, a matrix is

537
00:30:47.780 --> 00:30:50.239
a linear transformation, and transformation is morphism

538
00:30:50.239 --> 00:30:52.320
and morphism is ultimately a function

539
00:30:52.320 --> 00:30:57.520
Therefore, for the result of multiplying these two matrices

540
00:30:57.520 --> 00:31:01.200
we can ultimately say that matrix corresponds to function

541
00:31:01.200 --> 00:31:04.200
and this brings the same result

542
00:31:04.200 --> 00:31:06.550
as the concept of composite function

543
00:31:06.550 --> 00:31:08.960
that we learned earlier

544
00:31:08.960 --> 00:31:11.960
Normally, when we learn about trigonometric functions

545
00:31:11.960 --> 00:31:13.599
we learn a lot about the addition formula

546
00:31:13.599 --> 00:31:14.640
and memorize them

547
00:31:14.640 --> 00:31:21.000
cos(α+β) is CCSS, that's how I learned it

548
00:31:21.000 --> 00:31:24.550
And the addition formula for the sine function is

549
00:31:24.550 --> 00:31:28.400
SCCS, that's how I learned it

550
00:31:28.400 --> 00:31:32.750
Rather than just memorizing it

551
00:31:32.750 --> 00:31:36.039
we're going to actually act it out using matrix multiplication

552
00:31:36.039 --> 00:31:39.799
Here, let's say that we first rotate it by α degrees

553
00:31:39.799 --> 00:31:44.080
and then rotate it by β degrees

554
00:31:44.080 --> 00:31:46.980
Rotating it by α degrees first

555
00:31:46.980 --> 00:31:49.239
and then rotating it by β degrees

556
00:31:49.239 --> 00:31:53.189
is obviously the same as

557
00:31:53.189 --> 00:31:55.440
rotating it by α+β degrees

558
00:31:55.440 --> 00:31:59.190
Then taking the result from α degree rotation

559
00:31:59.190 --> 00:32:02.760
and the result from β degree rotation

560
00:32:02.760 --> 00:32:07.239
and multiplying the two transformations

561
00:32:07.239 --> 00:32:10.889
is the same as

562
00:32:10.889 --> 00:32:12.719
rotating once by α+β degrees

563
00:32:12.719 --> 00:32:17.359
Therefore, I wrote down the α degree rotation here

564
00:32:17.359 --> 00:32:19.880
Multiplying these together is the same as what?

565
00:32:19.880 --> 00:32:24.719
The same as applying α+β

566
00:32:24.719 --> 00:32:30.640
Then here, we just need to take a look at the first column

567
00:32:30.640 --> 00:32:33.919
Cos(α+β) is the same as

568
00:32:33.919 --> 00:32:42.239
CCSS, thus (cos α·cos β)+(-sin α·sin β)

569
00:32:42.239 --> 00:32:48.080
And for sin(α+β), we usually call it SCCS

570
00:32:48.080 --> 00:32:52.760
(sin α·cos β) and then (cos α·sin β)

571
00:32:52.760 --> 00:32:56.560
The order has changed, but since the associative property stands

572
00:32:56.560 --> 00:32:59.520
we can call it SCCS

573
00:32:59.520 --> 00:33:03.970
So it's the same as this, and I left this as a question mark but

574
00:33:03.970 --> 00:33:05.919
the rest are actually the same as well

575
00:33:05.919 --> 00:33:10.969
Ultimately, the second element here

576
00:33:10.969 --> 00:33:15.280
The first and second elements of column 2 are all the same

577
00:33:15.280 --> 00:33:18.960
This corresponds to -sin(α+β)

578
00:33:18.960 --> 00:33:23.799
and this corresponds to cos(α+β)

579
00:33:23.799 --> 00:33:26.399
So later, by any chance

580
00:33:26.399 --> 00:33:29.799
if you can't remember the addition formula of trigonometric functions

581
00:33:29.799 --> 00:33:34.387
use the rotation of trigonometric functions

582
00:33:34.387 --> 00:33:39.280
then you'll be able to proceed right away

583
00:33:39.280 --> 00:33:42.239
I hope you use it usefully

584
00:33:42.239 --> 00:33:45.389
Finally, we'll talk about the advantages of

585
00:33:45.389 --> 00:33:47.760
multiplication of matrices

586
00:33:47.760 --> 00:33:51.310
As mentioned earlier, the associative property holds true for matrix

587
00:33:51.310 --> 00:33:54.320
so the order doesn't matter

588
00:33:54.320 --> 00:33:57.570
Doing this first, as long as the direction is the same

589
00:33:57.570 --> 00:34:00.367
the order doesn't matter

590
00:34:00.367 --> 00:34:07.599
Normally, the transformation is applied on the vector

591
00:34:07.599 --> 00:34:10.919
and another transformation is applied on the new vector

592
00:34:10.919 --> 00:34:13.369
and then another transformation on the new vector

593
00:34:13.369 --> 00:34:15.359
That's how we should proceed

594
00:34:15.359 --> 00:34:20.559
But after all, it's the same as

595
00:34:20.559 --> 00:34:25.159
multiplying the transformation matrices first

596
00:34:25.159 --> 00:34:27.409
and then multiplying

597
00:34:27.409 --> 00:34:30.039
the original vector

598
00:34:30.039 --> 00:34:33.239
From now on, when we

599
00:34:33.239 --> 00:34:35.599
basically express something on the computer

600
00:34:35.599 --> 00:34:38.449
we go through a lot of linear transformation processes

601
00:34:38.449 --> 00:34:42.000
We go through at least these 5 processes

602
00:34:42.000 --> 00:34:45.632
Scale, rotation, translation, view, and projection

603
00:34:45.632 --> 00:34:48.000
are the five processes

604
00:34:48.000 --> 00:34:49.599
It means there are five types of these

605
00:34:49.599 --> 00:34:53.727
The transformation itself is A, B, C, D, and E

606
00:34:53.727 --> 00:34:55.427
Rather than calling it A, B, C, D, E

607
00:34:55.427 --> 00:34:57.799
we take the first letter of each word

608
00:34:57.799 --> 00:35:00.299
S for scale, R for rotation

609
00:35:00.299 --> 00:35:03.400
T for translation, and

610
00:35:03.400 --> 00:35:08.119
V for view and P for projection

611
00:35:08.119 --> 00:35:14.080
We'll basically have these five transformations

612
00:35:14.080 --> 00:35:16.960
Let's say that we're expressing a character

613
00:35:16.960 --> 00:35:18.910
and the character is composed of

614
00:35:18.910 --> 00:35:20.359
100,000 dots

615
00:35:20.359 --> 00:35:23.719
Then in order to show this character of 100,000 dots

616
00:35:23.719 --> 00:35:28.039
on the screen, I need to go through these five matrices

617
00:35:28.039 --> 00:35:32.679
Then basically, we'll need 500,000 calculations

618
00:35:32.679 --> 00:35:36.440
As you can see here, we need 500,000 calculations

619
00:35:36.440 --> 00:35:40.479
But the transformation itself is always the same

620
00:35:40.479 --> 00:35:45.080
That means we don't need to calculate them separately

621
00:35:45.080 --> 00:35:48.680
We'll multiply the matrices PVTRS

622
00:35:48.680 --> 00:35:52.359
and make it beforehand

623
00:35:52.359 --> 00:35:57.400
Once we create a matrix with the five transformations applied

624
00:35:57.400 --> 00:36:01.719
then all we need to do is apply that matrix for 100,000 dots

625
00:36:01.719 --> 00:36:04.200
and we will get the same result

626
00:36:04.200 --> 00:36:06.350
Therefore, we can decrease 500,000 calculations

627
00:36:06.350 --> 00:36:09.719
to 100,000 calculations

628
00:36:09.719 --> 00:36:12.119
Because of these characteristics of a matrix

629
00:36:12.119 --> 00:36:14.719
computer graphics

630
00:36:14.719 --> 00:36:17.200
especially game graphics that are sensitive to sound

631
00:36:17.200 --> 00:36:21.000
must use matrices

632
00:36:21.000 --> 00:36:24.159
In all the base processes of computer graphic transformations

633
00:36:24.159 --> 00:36:29.159
these formulas and equations of the matrix are used

634
00:36:29.159 --> 00:36:31.760
as what you should know

635
00:36:31.760 --> 00:36:35.479
Until now, we learned about the matrix, why we use the matrix

636
00:36:35.479 --> 00:36:37.679
and the three types of transformation

637
00:36:37.679 --> 00:36:41.312
that we'll often use, and their mechanisms

638
00:36:41.680 --> 00:36:45.462
Inverse matrix

639
00:36:45.919 --> 00:36:50.000
First, inverse matrix corresponds

640
00:36:50.000 --> 00:36:53.479
to the concept of inverse function that we learned before

641
00:36:53.479 --> 00:36:56.379
In the big perspective of linear transformation

642
00:36:56.379 --> 00:36:58.159
matrix is ultimately a concept of function

643
00:36:58.159 --> 00:37:02.159
Therefore, we can say that inverse matrix is also the same concept as an inverse function

644
00:37:02.159 --> 00:37:05.160
Then there's a special function

645
00:37:05.160 --> 00:37:08.400
that always came along with inverse function

646
00:37:08.400 --> 00:37:10.760
That was the identity function

647
00:37:10.760 --> 00:37:15.920
Identity function referred to a function where

648
00:37:15.920 --> 00:37:20.560
the results of the domain and the range correspond to the same value

649
00:37:20.560 --> 00:37:22.910
In the same way as this concept

650
00:37:22.910 --> 00:37:26.439
the identity matrix exists for matrices as well

651
00:37:26.439 --> 00:37:29.479
We'll now learn about how this is composed

652
00:37:29.479 --> 00:37:33.660
An identity matrix ultimately means that

653
00:37:33.660 --> 00:37:37.460
the original space and the transformed space after linear transformation

654
00:37:37.460 --> 00:37:39.880
have the same values

655
00:37:39.880 --> 00:37:45.400
That means, the two standard basis vectors that formed up the original space

656
00:37:45.400 --> 00:37:50.040
have the same values even after transformation

657
00:37:50.040 --> 00:37:54.079
Therefore, when designing an identity matrix

658
00:37:54.079 --> 00:37:57.529
the first column of the first changed standard basis vector

659
00:37:57.529 --> 00:37:59.439
uses the standard basis vector as-is

660
00:37:59.439 --> 00:38:00.800
It uses (1, 0)

661
00:38:00.800 --> 00:38:04.350
And for the second column, (0, 1) is used

662
00:38:04.350 --> 00:38:07.880
to design the identity matrix

663
00:38:07.880 --> 00:38:12.857
Then to see if the results of applying the identity matrix

664
00:38:12.857 --> 00:38:14.520
really has no change

665
00:38:14.520 --> 00:38:17.720
we'll name an arbitrary matrix abcd

666
00:38:17.720 --> 00:38:21.280
When we multiply this identity matrix

667
00:38:21.280 --> 00:38:23.920
we can see that the result value is the same

668
00:38:23.920 --> 00:38:27.880
When we multiply any vector to an identity matrix

669
00:38:27.920 --> 00:38:31.800
an identical vector is made

670
00:38:31.800 --> 00:38:36.760
Therefore, with this special identity matrix

671
00:38:36.760 --> 00:38:42.119
we can add a concept of inverse matrix

672
00:38:42.119 --> 00:38:47.560
It takes the result of a linear transformation, transformed by matrix

673
00:38:47.560 --> 00:38:52.160
and turns it back to the original

674
00:38:52.160 --> 00:38:54.360
It's a linear transformation

675
00:38:54.360 --> 00:38:56.439
that turns back a linear-transformed result

676
00:38:56.439 --> 00:39:00.680
Therefore, when we combine these two, it becomes an identity transformation

677
00:39:00.680 --> 00:39:07.040
Since it created the same value as the original, it is an identity transformation

678
00:39:07.040 --> 00:39:11.640
If you see here, (1, 0) and (0, 1)

679
00:39:11.640 --> 00:39:15.400
of this standard basis vector of the original space

680
00:39:15.400 --> 00:39:19.680
was changed into ac and bd

681
00:39:19.680 --> 00:39:22.800
In order to turn this back to its original

682
00:39:22.800 --> 00:39:27.319
so for the transformation to turn back into A

683
00:39:27.319 --> 00:39:32.760
it is conceptually settled that an inverse matrix of A exists

684
00:39:32.760 --> 00:39:36.860
And the result of combining these two

685
00:39:36.860 --> 00:39:39.479
comes back to the original

686
00:39:39.479 --> 00:39:43.560
Therefore, in the concept of a function, it comes back to identity function

687
00:39:43.560 --> 00:39:48.680
In the concept of matrix, combining inverse matrix and the matrix

688
00:39:48.719 --> 00:39:53.839
becomes the identity matrix

689
00:39:54.680 --> 00:39:58.439
Then how can we calculate this inverse matrix?

690
00:40:01.000 --> 00:40:04.440
It's good to know about the concept of a determinant beforehand

691
00:40:04.440 --> 00:40:08.090
A determinant is

692
00:40:08.090 --> 00:40:10.160
a property of a matrix

693
00:40:10.160 --> 00:40:15.460
Let's say we have a two dimensional square matrix

694
00:40:15.460 --> 00:40:17.280
called abcd

695
00:40:17.280 --> 00:40:20.680
the determinant, which is the property of this matrix

696
00:40:20.680 --> 00:40:24.649
is obtained through the calculation ad-bc

697
00:40:24.649 --> 00:40:29.080
We'll learn about what ad-bc means

698
00:40:29.080 --> 00:40:31.163
Under here, I wrote down

699
00:40:31.163 --> 00:40:35.040
a system of equations

700
00:40:35.040 --> 00:40:39.340
If we say that this system of equations is abcd

701
00:40:39.340 --> 00:40:45.416
and set the x, y, coefficients of 2, 1, 1, 0.5

702
00:40:45.416 --> 00:40:50.766
as abcd, and then when we do 2x0.5 - 1x1

703
00:40:50.766 --> 00:40:52.439
we can see that it becomes 0

704
00:40:52.439 --> 00:40:56.021
I'll explain what happens when the value of ad-bc

705
00:40:56.021 --> 00:40:58.639
becomes 0 like this

706
00:40:58.639 --> 00:41:02.639
If you look here, when we multiply the second equation by 2

707
00:41:02.639 --> 00:41:06.279
it becomes 2x+y=8

708
00:41:06.279 --> 00:41:11.080
Then this system of equations has no solution

709
00:41:11.080 --> 00:41:14.080
This condition of having no solutions

710
00:41:14.080 --> 00:41:17.919
is distinguished through this equation of a determinant

711
00:41:17.919 --> 00:41:22.480
That's why it's called a determinant

712
00:41:22.480 --> 00:41:31.639
It's a value that can analyze and determine something

713
00:41:31.639 --> 00:41:35.139
When we express this system of equations as a matrix

714
00:41:35.139 --> 00:41:38.639
it can be expressed like this

715
00:41:38.639 --> 00:41:41.389
We can use these coefficients

716
00:41:41.389 --> 00:41:44.199
and make it into a matrix like this

717
00:41:44.199 --> 00:41:49.080
In this matrix too, ad-bc would give us 0

718
00:41:49.080 --> 00:41:53.830
Then we'll see what kind of a transformation

719
00:41:53.830 --> 00:41:55.680
this matrix forms

720
00:41:55.680 --> 00:42:00.320
In this matrix, the first standard basis vector transformed to (2, 1)

721
00:42:00.320 --> 00:42:02.720
and the second standard basis vector (0, 1)

722
00:42:02.720 --> 00:42:06.000
has been transformed into (1, 0.5)

723
00:42:06.000 --> 00:42:12.450
When the second vector is multiplied by 2

724
00:42:12.450 --> 00:42:14.080
we get (2, 1)

725
00:42:14.080 --> 00:42:17.839
therefore, they have the same slope

726
00:42:17.839 --> 00:42:21.339
That means, the two vectors (1, 0) and (0, 1)

727
00:42:21.339 --> 00:42:24.536
that were linearly independent

728
00:42:24.536 --> 00:42:28.936
ended up with same slopes through the linear transformation process

729
00:42:28.936 --> 00:42:33.800
and changed into a linearly dependent relationship

730
00:42:33.800 --> 00:42:36.600
The following expresses this as a picture

731
00:42:36.600 --> 00:42:39.600
The vectors with different slopes like this

732
00:42:39.600 --> 00:42:43.039
changed into a space having the same slope

733
00:42:43.039 --> 00:42:46.539
Then here, we were able to create all dots

734
00:42:46.539 --> 00:42:49.119
of the two dimensional space through a linear combination

735
00:42:49.119 --> 00:42:52.169
but here, even when we combine the two vectors

736
00:42:52.169 --> 00:42:54.219
we can only create the dots

737
00:42:54.219 --> 00:42:56.919
on top of this slope

738
00:42:56.919 --> 00:43:02.869
Therefore, a 2D space has been changed into a one-dimensional space

739
00:43:02.869 --> 00:43:06.240
A bad transformation, we could say

740
00:43:06.240 --> 00:43:09.690
Therefore, when this transformation is applied

741
00:43:09.690 --> 00:43:13.759
the result of the linear transformation brought down 2D into 1D

742
00:43:13.759 --> 00:43:16.509
so ultimately, there's no path back

743
00:43:16.509 --> 00:43:20.639
from one-dimension to two-dimension

744
00:43:20.639 --> 00:43:22.939
This means

745
00:43:22.939 --> 00:43:26.160
we cannot make the inverse matrix

746
00:43:26.160 --> 00:43:28.560
Therefore, before obtaining the inverse matrix

747
00:43:28.560 --> 00:43:31.110
we must know the determinant first because

748
00:43:31.110 --> 00:43:34.639
the inverse matrix may or may not exist

749
00:43:34.639 --> 00:43:36.789
How do we determine that?

750
00:43:36.789 --> 00:43:40.789
Use the determinant ad-bc

751
00:43:40.789 --> 00:43:44.804
to first determine whether the inverse matrix exists

752
00:43:44.804 --> 00:43:48.160
or doesn't exist

753
00:43:48.160 --> 00:43:52.110
Then how come the dimension disappears

754
00:43:52.110 --> 00:43:54.559
when the value of ad-bc is 0?

755
00:43:54.559 --> 00:43:59.320
Let's check the area of the parallelogram formed by the two vectors

756
00:43:59.360 --> 00:44:03.160
First, the square formed by the two standard basis vectors

757
00:44:03.160 --> 00:44:06.720
in the original space had an area of 1

758
00:44:06.720 --> 00:44:12.920
But when e1 is transformed into (a, c)

759
00:44:12.920 --> 00:44:16.320
and when e2 is transformed into (b, d)

760
00:44:16.320 --> 00:44:19.920
a parallelogram like this is formed

761
00:44:19.920 --> 00:44:25.370
When we try obtaining the final value here

762
00:44:25.370 --> 00:44:29.040
with the x value of a+b

763
00:44:29.040 --> 00:44:35.000
and y value of c+d, we multiply them to obtain a rectangular area

764
00:44:35.000 --> 00:44:38.750
When we obtain this gray area

765
00:44:38.750 --> 00:44:41.750
by subtracting these triangle and rectangle areas

766
00:44:41.750 --> 00:44:45.640
we get the value ad-bc

767
00:44:45.640 --> 00:44:50.390
That means, ad-bc

768
00:44:50.390 --> 00:44:54.279
is the area of the parallelogram

769
00:44:54.279 --> 00:44:57.229
formed by two linearly independent vectors

770
00:44:57.229 --> 00:45:01.920
as they go through linear transformation

771
00:45:01.920 --> 00:45:05.839
The fact that this is 0, means the area of the parallelogram is 0

772
00:45:05.839 --> 00:45:10.539
Therefore, that means the two vectors

773
00:45:10.539 --> 00:45:14.079
are along the same straight line

774
00:45:15.799 --> 00:45:22.079
With that, we learned about the meaning of a determinant

775
00:45:22.079 --> 00:45:24.829
Then let's say that the value of the determinant is not 0

776
00:45:24.829 --> 00:45:27.720
Then how can we use that?

777
00:45:27.720 --> 00:45:34.070
First of all, the value of ad-bc

778
00:45:34.070 --> 00:45:39.959
is ultimately going from the area of 1, formed by the two standard vectors

779
00:45:39.959 --> 00:45:46.799
to the parallelogram of the form ad-bc

780
00:45:46.799 --> 00:45:51.099
The changed area

781
00:45:51.099 --> 00:45:56.200
the size has been changed by ad-bc

782
00:45:56.200 --> 00:45:58.850
Then how do we turn it back?

783
00:45:58.850 --> 00:46:03.200
We can restore it by multiplying the inverse of the changed size

784
00:46:03.239 --> 00:46:06.289
So the determinant

785
00:46:06.289 --> 00:46:11.160
can simply determine whether there's an inverse matrix

786
00:46:11.160 --> 00:46:15.310
but it can also be used when the space has been changed

787
00:46:15.310 --> 00:46:18.960
to tell how much size change has occurred

788
00:46:18.960 --> 00:46:21.258
after a transformation

789
00:46:25.040 --> 00:46:27.940
Now that we can obtain this inverse matrix

790
00:46:27.940 --> 00:46:31.160
I'll briefly talk about how we can utilize it

791
00:46:31.160 --> 00:46:34.660
Normally, it's useful for

792
00:46:34.660 --> 00:46:39.519
obtaining the solution of a system of equations that we've seen earlier

793
00:46:39.519 --> 00:46:41.919
For obtaining the solution as well

794
00:46:41.919 --> 00:46:45.279
we showed this visually because we use graphics

795
00:46:45.279 --> 00:46:49.079
but like this, we can use matrices very usefully

796
00:46:49.079 --> 00:46:51.000
to obtain the solution

797
00:46:51.000 --> 00:46:55.727
Let's say that this matrix is A, and (x, y) is v

798
00:46:55.727 --> 00:46:59.600
and let's say that the result value is v'

799
00:46:59.600 --> 00:47:02.250
We would get an equation like this

800
00:47:02.250 --> 00:47:06.160
Here, we know v', the result

801
00:47:06.160 --> 00:47:07.810
We know the result and

802
00:47:07.810 --> 00:47:11.060
we also know what transformation we're applying

803
00:47:11.060 --> 00:47:13.710
But let's say that we don't know

804
00:47:13.710 --> 00:47:16.399
what the original value is

805
00:47:16.399 --> 00:47:19.199
Then this is how we solve this

806
00:47:19.199 --> 00:47:23.063
If we obtain the inverse matrix of A and multiply it to both sides

807
00:47:23.063 --> 00:47:27.239
we would do A^-1 times A

808
00:47:27.239 --> 00:47:32.640
and here, we would multiply the transformed v' to A^-1

809
00:47:32.640 --> 00:47:34.840
Then for this value

810
00:47:34.840 --> 00:47:37.740
since the associative property stands here

811
00:47:37.740 --> 00:47:41.200
this eventually becomes the identity matrix I, and we would be left with v

812
00:47:41.200 --> 00:47:44.650
Therefore, if we can obtain the inverse matrix

813
00:47:44.650 --> 00:47:49.600
multiply the inverse matrix on the transformed vector

814
00:47:49.600 --> 00:47:52.200
and we would be able to find the solution

815
00:47:52.200 --> 00:47:55.119
of the original value right away

816
00:47:55.119 --> 00:47:57.669
Just like that, matrix can be useful

817
00:47:57.669 --> 00:48:00.279
for solving equations

818
00:48:00.279 --> 00:48:03.929
It can be used for various, complex problems

819
00:48:03.929 --> 00:48:07.629
Just now, we used it for a very simple form of

820
00:48:07.629 --> 00:48:10.559
system of equations with just two equations

821
00:48:10.559 --> 00:48:13.609
but even with 10 equations or

822
00:48:13.609 --> 00:48:17.119
10 complex coefficients tied together

823
00:48:17.119 --> 00:48:19.219
as long as we know the principle of the matrix

824
00:48:19.219 --> 00:48:23.069
and as long as we have enough samples for each

825
00:48:23.069 --> 00:48:26.119
we can ultimately find the solution

826
00:48:26.119 --> 00:48:28.919
We can use the Gaussian elimination

827
00:48:28.919 --> 00:48:32.719
and the Cramer's rule to obtain the inverse matrix

828
00:48:32.719 --> 00:48:38.069
For the main transformations that we use for graphics

829
00:48:38.069 --> 00:48:42.799
although they would be good to know, you don't really need to know these

830
00:48:42.799 --> 00:48:48.200
That's because we can obtain the inverse matrix intuitively

831
00:48:48.200 --> 00:48:54.039
We'll first look at the scale matrix among the matrices

832
00:48:54.039 --> 00:48:57.589
We'll think about how the inverse matrix

833
00:48:57.589 --> 00:48:59.679
of a scale matrix works

834
00:48:59.679 --> 00:49:05.786
There's a matrix where e1 was increased by a times

835
00:49:05.786 --> 00:49:10.408
and e2 was decreased by b times, from a standard basis vector

836
00:49:10.408 --> 00:49:12.258
In order to make that matrix

837
00:49:12.258 --> 00:49:14.750
come back to (1, 0) and (0, 1)

838
00:49:14.750 --> 00:49:17.440
what should we do?

839
00:49:17.440 --> 00:49:21.280
It's so simple, we just need to decrease by the amount we increased

840
00:49:21.280 --> 00:49:25.630
So if we multiply the inverse of the increased coefficient

841
00:49:25.630 --> 00:49:29.919
it comes back to (1, 0) and (0, 1)

842
00:49:29.919 --> 00:49:33.269
If we multiply the inverse of the increased amount

843
00:49:33.269 --> 00:49:35.119
and make it into a scale matrix

844
00:49:35.119 --> 00:49:38.969
we can say that this becomes the inverse matrix of the scale matrix

845
00:49:38.969 --> 00:49:41.280
that comes back to the original size

846
00:49:41.280 --> 00:49:43.479
We'll also take a look at translation matrix

847
00:49:43.479 --> 00:49:45.200
We translated it

848
00:49:45.200 --> 00:49:52.520
If we only translate on one side, this part increases by a times

849
00:49:52.520 --> 00:49:55.280
Then do should we make it come back to the original?

850
00:49:55.280 --> 00:49:57.599
We would just translate to the opposite direction

851
00:49:57.599 --> 00:49:59.679
Since we just need to push to the opposite side

852
00:49:59.679 --> 00:50:03.919
it would come back to the original if we translate by -a

853
00:50:03.919 --> 00:50:07.679
Therefore, for the inverse matrix of a translation matrix translated by "a"

854
00:50:07.679 --> 00:50:13.960
translating it by "-a" allows us to easily design the inverse matrix

855
00:50:13.960 --> 00:50:19.080
Next, I'll talk about the inverse matrix of a rotation matrix

856
00:50:19.080 --> 00:50:25.039
The first standard basis vector is (cos, sinθ)

857
00:50:25.039 --> 00:50:31.159
and the second standard basis vector is (-sinθ, cosθ)

858
00:50:31.159 --> 00:50:36.000
How can we change this back to the original?

859
00:50:36.000 --> 00:50:40.150
The way of turning it back to its original

860
00:50:40.150 --> 00:50:42.919
is rotating it in the opposite direction by the same amount

861
00:50:42.919 --> 00:50:48.119
In other words, rotating it -θ degrees

862
00:50:48.119 --> 00:50:51.669
If this is a normal rotation

863
00:50:51.669 --> 00:50:54.719
we would substitute -θ degrees into the rotation

864
00:50:54.719 --> 00:50:58.280
Then based on the property of a cos function

865
00:50:58.280 --> 00:51:02.030
where the cos function symmetrical

866
00:51:02.030 --> 00:51:05.440
about the y-axis

867
00:51:05.440 --> 00:51:09.840
-cosθ and cosθ have the same values

868
00:51:09.840 --> 00:51:13.919
Therefore, we can see that the diagonal elements are the same

869
00:51:13.919 --> 00:51:19.799
In the case of a sin function, it's symmetrical

870
00:51:19.799 --> 00:51:21.400
but the sign changes vertically

871
00:51:21.400 --> 00:51:28.880
Therefore, obtaining -sinθ

872
00:51:28.880 --> 00:51:32.320
is the same as sin(-θ)

873
00:51:32.320 --> 00:51:36.840
Ultimately, this goes from negative to positive

874
00:51:36.840 --> 00:51:39.290
and the positive becomes a negative

875
00:51:39.290 --> 00:51:41.679
in this matrix

876
00:51:41.679 --> 00:51:45.479
This is the inverse matrix of a rotation

877
00:51:45.479 --> 00:51:49.919
which turns a rotation matrix back to its original

878
00:51:49.919 --> 00:51:54.269
If we closely examine R_θ and R_(-θ) here

879
00:51:54.269 --> 00:51:57.679
what relationship would they have?

880
00:51:57.679 --> 00:52:00.379
First of all, the value of cosθ is the same

881
00:52:00.379 --> 00:52:04.640
and we can see that only the sinθ values have switched

882
00:52:04.640 --> 00:52:08.919
Therefore, this can also be seen as a transpose relationship

883
00:52:08.919 --> 00:52:14.119
To organize it, a rotation matrix and its inverse matrix

884
00:52:14.119 --> 00:52:17.880
have a transpose relationship with each other

885
00:52:17.880 --> 00:52:20.030
And the inverse matrix of a rotation matrix

886
00:52:20.030 --> 00:52:22.730
is rotation toward the opposite direction

887
00:52:22.730 --> 00:52:26.679
as what we can get from this

888
00:52:26.679 --> 00:52:30.629
With that, we learned about the essential inverse matrices

889
00:52:30.629 --> 00:52:34.840
needed for graphics that we'll perform later

890
00:52:34.840 --> 00:52:37.140
That's all for this lecture

891
00:52:37.140 --> 00:52:38.990
Great job for listening to this lesson

892
00:52:38.990 --> 00:52:39.901
Thank you

893
00:52:40.281 --> 00:52:42.131
Matrix
Matrix
Numbers arranged inside a rectangle as rows and columns
Basic operations of matrix: addition of matrices, multiplication of a matrix...

894
00:52:42.131 --> 00:52:43.881
Characteristics of matrix multiplication
Doesn't satisfy the commutative property
Satisfies the associative property
Satisfies (A·B)^T=B^T·A^T
Tran...

895
00:52:43.881 --> 00:52:45.597
Linear transformation and matrix
Square matrix is linear transformation where the same dimensional space correspond to each other
Column major...

896
00:52:45.597 --> 00:52:46.797
Visualization of linear transformation
Linear transformation of the original vector space:
[a b]
[c d]
Using a matrix means tracking the change of...

897
00:52:46.797 --> 00:52:47.997
Scale transformation matrix:
[a 0]
[0 b]
Translation matrix:
[1 a]
[0 1]

898
00:52:47.997 --> 00:52:49.197
90 degrees clockwise rotation matrix:
[ 0 1]
[-1 0]
90 degrees counterclockwise rotation matrix:
[0 -1]
[1  0]

899
00:52:49.197 --> 00:52:50.393
Rotation matrix of an arbitrary θ:
[xcosθ - ysinθ]
[xsinθ + ycosθ]

900
00:52:50.393 --> 00:52:51.678
Multiplication of matrix
Associative property stands
Product of square matrix has the same result as composite function

901
00:52:51.678 --> 00:52:52.785
Product of matrix can be used for the addition formula of trigonometric function
Matrix operation with a valid associative property guarantees the...

902
00:52:52.785 --> 00:52:54.069
Major linear transformation: Scale(S), Rotation(R), Translation(T), View(V), Projection(P)

903
00:52:54.069 --> 00:52:55.306
When 5 linear transformations are the same, the product of PVTRS can be applied to lessen the number of operations

904
00:52:55.306 --> 00:52:56.157
Inverse matrix
Inverse matrix
Identity matrix: Matrix with no change in the linear transformation I =
[1 0]
[0 1]

905
00:52:56.157 --> 00:52:56.957
Inverse matrix: Linear transformation that reverts the linearly transformed result
Combining inverse matrix and matrix results in identity matrix
H...

906
00:52:56.957 --> 00:52:57.806
Determinant: Expression that determines a condition with no solution
A = [a b]
[c d], det(A) = ad - bc

907
00:52:57.806 --> 00:52:58.606
Using determinant ad-bc to determine whether the inverse matrix exists
Utilizing inverse matrix: useful for solving a system of equations

908
00:52:58.606 --> 00:52:59.456
Inverse matrix of scale matrix:
[1/a 0]
[0 1/b]
Inverse matrix of translation matrix:
[1 -a]
[0  1]

909
00:52:59.456 --> 00:53:00.306
Inverse matrix of rotation:
[ cos(θ) sin(θ)]
[-sin(θ) cos(θ)]