1
00:00:00,000 --> 00:00:00,706
我们看到

2
00:00:00,706 --> 00:00:02,271
我们可以采取2个方程组

3
00:00:02,271 --> 00:00:04,466
包含2个未知数并展现

4
00:00:04,466 --> 00:00:07,799
矩阵A是左边部分的

5
00:00:07,799 --> 00:00:10,472
系数的矩阵方程

6
00:00:10,472 --> 00:00:12,349
列向量x 有两个

7
00:00:12,349 --> 00:00:14,749
未知数 s 和 t

8
00:00:14,749 --> 00:00:16,525
然后 列向量b 基本上

9
00:00:16,525 --> 00:00:19,209
展现了这里的右边部分

10
00:00:19,209 --> 00:00:20,326
有趣的是

11
00:00:20,326 --> 00:00:22,066
那就表示了方程A

12
00:00:22,066 --> 00:00:23,910
矩阵A 乘以 列向量x

13
00:00:23,910 --> 00:00:26,285
等于 列向量b

14
00:00:26,285 --> 00:00:27,890
有趣的是

15
00:00:27,890 --> 00:00:29,320
你看，假如 A 是可逆的

16
00:00:29,320 --> 00:00:33,011
我们就可以把方程的两边

17
00:00:33,011 --> 00:00:34,220
都乘以

18
00:00:34,220 --> 00:00:36,090
我们要把两边的左边部分

19
00:00:36,090 --> 00:00:38,120
都乘以 逆A

20
00:00:38,120 --> 00:00:39,558
因为，记住矩阵

21
00:00:39,558 --> 00:00:41,803
当矩阵乘法顺序很重要时

22
00:00:41,803 --> 00:00:43,591
我们要把两边的

23
00:00:43,591 --> 00:00:45,346
左边部分都乘以

24
00:00:45,346 --> 00:00:47,877
如果我们这么做了，我们基本上可以

25
00:00:47,877 --> 00:00:50,498
解开未知的列向量了

26
00:00:50,498 --> 00:00:52,229
如果我们知道列向量是什么了

27
00:00:52,229 --> 00:00:54,660
我们就可以解开 s 和 t

28
00:00:54,660 --> 00:00:55,980
我们就基本上解决了

29
00:00:55,980 --> 00:00:58,059
这个方程组

30
00:00:58,059 --> 00:00:59,593
让我们行动起来吧

31
00:00:59,593 --> 00:01:02,211
让我们真正弄清楚 逆A 是什么

32
00:01:02,211 --> 00:01:04,213
然后将那个乘以 列向量b

33
00:01:04,213 --> 00:01:06,660
为了找出 列向量x

34
00:01:06,660 --> 00:01:08,799
和 s 与 t

35
00:01:08,799 --> 00:01:14,181
逆A矩阵， 逆A等于

36
00:01:14,181 --> 00:01:16,527
1 除以 A的行列式

37
00:01:16,527 --> 00:01:20,451
2x2 的 A的行列式

38
00:01:20,451 --> 00:01:25,460
等于 2 乘 4 减

39
00:01:25,460 --> 00:01:27,096
-2 乘 -5

40
00:01:27,096 --> 00:01:31,239
这等于 8 减 正10...

41
00:01:31,239 --> 00:01:33,061
8 减 正10

42
00:01:33,061 --> 00:01:34,711
等于 -2

43
00:01:34,711 --> 00:01:37,970
这就会等于 -2

44
00:01:38,023 --> 00:01:40,794
再一次，2 乘 4 等于 8 减

45
00:01:40,794 --> 00:01:43,380
-2 乘 -5

46
00:01:43,380 --> 00:01:46,614
所以 减 正10 就得到了 -2

47
00:01:46,614 --> 00:01:48,797
你可以将 行列式 分之 1

48
00:01:48,797 --> 00:01:53,551
乘以 有时候被称之为A的伴随

49
00:01:53,551 --> 00:01:56,585
本质上就是交换左上角

50
00:01:56,585 --> 00:01:59,527
和右下角，起码 2x2 的矩阵是这样的

51
00:01:59,527 --> 00:02:02,321
这会是 4

52
00:02:02,321 --> 00:02:04,319
这会是 2

53
00:02:04,319 --> 00:02:05,600
注意 我只是把这两个换了位置

54
00:02:05,600 --> 00:02:06,971
然后 使这两个数字为负

55
00:02:06,971 --> 00:02:09,123
把原本的数字再一次变负

56
00:02:09,123 --> 00:02:10,599
这是负2

57
00:02:10,599 --> 00:02:12,181
会变成正2

58
00:02:12,181 --> 00:02:13,307
这边的这个

59
00:02:13,307 --> 00:02:14,917
会变成正5

60
00:02:14,917 --> 00:02:17,643
如果你完全和这些东西不熟悉

61
00:02:17,643 --> 00:02:19,554
你可以去复习一下教程

62
00:02:19,554 --> 00:02:21,373
关于反转矩阵

63
00:02:21,373 --> 00:02:23,920
因为那就是我在这里做的

64
00:02:23,920 --> 00:02:27,231
所以 逆A 会等于...

65
00:02:27,231 --> 00:02:30,600
逆A会等于

66
00:02:30,600 --> 00:02:34,400
让我看看，这是 -1/2 乘 4

67
00:02:34,400 --> 00:02:35,453
是 -2

68
00:02:35,453 --> 00:02:41,385
-1/2，-1/2 乘 5

69
00:02:41,385 --> 00:02:46,989
是 -2.5，-2.5

70
00:02:46,989 --> 00:02:51,490
然后 -1/2 乘 2 是 -1...

71
00:02:51,490 --> 00:02:53,860
-1/2 乘 2 是 -1

72
00:02:53,860 --> 00:02:55,666
所以这就是 逆A

73
00:02:55,666 --> 00:02:58,051
现在让我们把 逆A 乘以

74
00:02:58,051 --> 00:03:00,400
列向量，7，-6

75
00:03:00,400 --> 00:03:01,768
开解吧

76
00:03:01,768 --> 00:03:03,855
这是 逆A，我会写下来

77
00:03:03,855 --> 00:03:08,113
-2，-2.5，-1

78
00:03:08,113 --> 00:03:13,837
-1 乘 7 和 -6

79
00:03:13,837 --> 00:03:16,643
乘以，我会把它们全部用白色写下来

80
00:03:16,643 --> 00:03:17,758
7，-6

81
00:03:17,758 --> 00:03:22,541
我们有过很多矩阵乘法练习

82
00:03:22,541 --> 00:03:25,075
所以这会等于？

83
00:03:25,075 --> 00:03:26,800
第一项会是-2

84
00:03:26,800 --> 00:03:32,200
乘 7 等于 -14 加

85
00:03:32,200 --> 00:03:37,228
-2.5 乘 -6

86
00:03:37,228 --> 00:03:39,460
让我看看，这会是个正数

87
00:03:39,460 --> 00:03:42,431
这会是 12 加 3

88
00:03:42,431 --> 00:03:44,346
这等于 正15...

89
00:03:44,346 --> 00:03:46,383
正15

90
00:03:46,383 --> 00:03:48,567
-2.5 乘 -6

91
00:03:48,567 --> 00:03:50,811
等于 正15

92
00:03:50,811 --> 00:03:52,823
然后我们会有 -1

93
00:03:52,823 --> 00:03:56,366
乘 7 是 -7 加

94
00:03:56,366 --> 00:03:58,810
-1 乘 -6

95
00:03:58,810 --> 00:04:01,377
这是 正6

96
00:04:01,377 --> 00:04:05,210
逆A 和 b 的结果

97
00:04:05,210 --> 00:04:07,492
相同于 列向量x

98
00:04:07,492 --> 00:04:08,679
等于

99
00:04:08,679 --> 00:04:10,520
我们现在应该得到一点鼓声

100
00:04:10,520 --> 00:04:14,396
列向量 1，-1

101
00:04:14,396 --> 00:04:17,594
我们刚才证明了这等于

102
00:04:17,594 --> 00:04:20,929
1，-1 或者 x 等于

103
00:04:20,929 --> 00:04:22,532
1，-1

104
00:04:22,532 --> 00:04:26,571
或者我们甚至可以说这个列向量

105
00:04:26,571 --> 00:04:31,400
列向量st

106
00:04:31,400 --> 00:04:35,452
有 s 和 t 作为项的列向量

107
00:04:35,452 --> 00:04:41,986
等于1，-1...

108
00:04:41,986 --> 00:04:45,276
等于1，-1

109
00:04:45,276 --> 00:04:46,273
是另外一种说法

110
00:04:46,273 --> 00:04:47,930
那 s 等于 1

111
00:04:47,930 --> 00:04:50,091
和 t 等于 -1

112
00:04:50,091 --> 00:04:51,150
我知道你在说

113
00:04:51,150 --> 00:04:52,219
我在上一个视频里看到了

114
00:04:52,219 --> 00:04:53,488
那好吧，我就再说一次

115
00:04:53,488 --> 00:04:54,852
你就好像：“你知道么，这会简单很多

116
00:04:54,852 --> 00:04:56,795
如果你简单粗暴地解开它

117
00:04:56,795 --> 00:04:59,807
直接用消元法或者置换法。”

118
00:04:59,807 --> 00:05:04,590
我同意，不过这是个很有用的技巧

119
00:05:04,590 --> 00:05:06,344
因为当你正在解决

120
00:05:06,344 --> 00:05:08,805
计算法中的问题时，会有情况发生

121
00:05:08,805 --> 00:05:10,791
那组合的左边部分

122
00:05:10,791 --> 00:05:13,480
是一样的

123
00:05:13,480 --> 00:05:15,042
然而左边部分却可以有

124
00:05:15,042 --> 00:05:16,954
其他很多不一样的值

125
00:05:16,954 --> 00:05:19,106
可能会简单一点，如果你

126
00:05:19,106 --> 00:05:22,587
只计算一次逆矩阵，然后

127
00:05:22,587 --> 00:05:24,946
将其乘与

128
00:05:24,946 --> 00:05:28,565
不一样的右边部分

129
00:05:28,565 --> 00:05:30,812
你可能和其他类型已经熟悉了

130
00:05:30,812 --> 00:05:32,549
你有图形处理器

131
00:05:32,549 --> 00:05:34,280
电脑上的显卡

132
00:05:34,280 --> 00:05:36,460
和他们所说的特殊的图形处理器

133
00:05:36,460 --> 00:05:38,077
这些都是真正关于

134
00:05:38,077 --> 00:05:40,555
有着特殊目的的硬件

135
00:05:40,555 --> 00:05:44,160
为了很快的矩阵乘法

136
00:05:44,160 --> 00:05:46,233
因为当你在处理图形时

137
00:05:46,233 --> 00:05:47,544
当你在三次元里

138
00:05:47,544 --> 00:05:48,559
建模事物时

139
00:05:48,559 --> 00:05:50,067
你就是在做所有这些转变

140
00:05:50,067 --> 00:05:51,521
你真的只是在做很多

141
00:05:51,521 --> 00:05:53,965
特别，特别，特别快的矩阵乘法

142
00:05:53,965 --> 00:05:56,373
在实时的情况下，所以在用户玩游戏时

143
00:05:56,373 --> 00:05:58,160
或者做其他事情时

144
00:05:58,160 --> 00:05:59,516
这就好像他们在一种

145
00:05:59,516 --> 00:06:02,546
3D实时现实

146
00:06:02,546 --> 00:06:04,441
无论如何，我只想指出这一点

147
00:06:04,441 --> 00:06:08,659
这不会是，假如我随便看到了这个

148
00:06:08,659 --> 00:06:11,977
我的直觉会是用消元法解开这题

149
00:06:11,977 --> 00:06:15,610
不过有把这个想成矩阵方程的能力

150
00:06:15,610 --> 00:06:20,322
是特别，特别有用的概念

151
00:06:20,322 --> 00:06:21,920
不只是在计算法里

152
00:06:21,920 --> 00:06:25,240
当你接触到深层次的科学

153
00:06:25,240 --> 00:06:27,558
特别是物理，你会看到很多

154
00:06:27,558 --> 00:06:30,506
这样的矩阵向量方程

155
00:06:30,506 --> 00:06:32,036
那种笼统地说

156
00:06:32,036 --> 00:06:33,251
你要知道这很重要，关于

157
00:06:33,251 --> 00:06:35,188
他们实际上代表着什么

158
00:06:35,188 --> 00:06:37,950
以及他们到底怎样才能被解开