3 - 2 - Addition and Scalar Multiplication (7 min).srt

1
00:00:00,250 --> 00:00:01,612
In this video we'll talk about
在这段视频中
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

2
00:00:01,612 --> 00:00:03,503
matrix addition and subtraction,
我们将讨论矩阵的加法和减法运算

3
00:00:03,503 --> 00:00:04,950
as well as how to
以及如何进行

4
00:00:04,950 --> 00:00:06,582
multiply a matrix by a
数和矩阵的乘法

5
00:00:06,582 --> 00:00:09,292
number, also called Scalar Multiplication.
也就是标量乘法

6
00:00:09,292 --> 00:00:11,825
Let's start an example.
让我们从下面这个例子开始

7
00:00:11,825 --> 00:00:14,725
Given two matrices like these,
假设有这样两个矩阵

8
00:00:14,725 --> 00:00:16,735
let's say I want to add them together.
如果想对它们做求和运算

9
00:00:16,735 --> 00:00:18,038
How do I do that?
应该怎么做呢？

10
00:00:18,038 --> 00:00:20,538
And so, what does addition of matrices mean?
或者说 矩阵的加法到底是如何进行的？

11
00:00:20,538 --> 00:00:21,632
It turns out that if you
答案是

12
00:00:21,632 --> 00:00:24,312
want to add two matrices, what
如果你想将两个矩阵相加

13
00:00:24,312 --> 00:00:25,762
you do is you just add
你只需要将这两个矩阵的

14
00:00:25,762 --> 00:00:28,076
up the elements of these matrices one at a time.
每一个元素都逐个相加

15
00:00:28,076 --> 00:00:30,363
So, my result of adding
因此 两个矩阵相加

16
00:00:30,363 --> 00:00:31,480
two matrices is going to
所得到的结果

17
00:00:31,480 --> 00:00:33,415
be itself another matrix and
就是一个新的矩阵

18
00:00:33,415 --> 00:00:34,972
the first element again just by
它的第一个元素

19
00:00:34,972 --> 00:00:36,732
taking one and four and
是1和4相加的结果

20
00:00:36,732 --> 00:00:39,470
multiplying them and adding them together, so I get five.
因此我们得到5

21
00:00:39,470 --> 00:00:41,578
The second element I get
接下来是第二个元素

22
00:00:41,578 --> 00:00:43,092
by taking two and two
用2和2相加

23
00:00:43,092 --> 00:00:44,169
and adding them, so I get
因此得到4

24
00:00:44,169 --> 00:00:47,240
four; three plus three
然后是3加0得到3

25
00:00:47,255 --> 00:00:49,568
plus zero is three, and so on.
以此类推

26
00:00:49,570 --> 00:00:51,442
I'm going to stop changing colors, I guess.
这里我用不同颜色区别一下

27
00:00:51,442 --> 00:00:52,768
And, on the right is open
接下来右边这一列元素

28
00:00:52,768 --> 00:00:54,820
five, ten and two.
就是0.5 10和2

29
00:00:56,140 --> 00:00:57,182
And it turns out you can
这里大家不难发现

30
00:00:57,182 --> 00:01:00,408
add only two matrices that are of the same dimensions.
只有相同维度的两个矩阵才能相加

31
00:01:00,408 --> 00:01:02,789
So this example is
对于这个例子而言

32
00:01:02,789 --> 00:01:05,595
a three by two matrix,
这是一个3行2列的矩阵

33
00:01:07,120 --> 00:01:09,029
because this has 3
也就是说矩阵的行数为3

34
00:01:09,029 --> 00:01:11,917
rows and 2 columns, so it's 3 by 2.
列数是2 因此是3行2列

35
00:01:11,917 --> 00:01:13,451
This is also a 3
第二个矩阵

36
00:01:13,451 --> 00:01:15,113
by 2 matrix, and the
也是一个3行2列的矩阵

37
00:01:15,113 --> 00:01:16,202
result of adding these two
因此这两个矩阵相加的结果

38
00:01:16,202 --> 00:01:19,415
matrices is a 3 by 2 matrix again.
也是一个3行2列的矩阵

39
00:01:19,415 --> 00:01:20,468
So you can only add
所以你只能将相同维度的矩阵

40
00:01:20,470 --> 00:01:21,837
matrices of the same
进行相加运算

41
00:01:21,837 --> 00:01:23,533
dimension, and the result
同时 所得到的结果

42
00:01:23,550 --> 00:01:24,959
will be another matrix that's of
将会是一个新的矩阵

43
00:01:24,959 --> 00:01:28,057
the same dimension as the ones you just added.
这个矩阵与相加的两个矩阵维度相同

44
00:01:29,180 --> 00:01:30,785
Where as in contrast, if you
反过来

45
00:01:30,785 --> 00:01:31,803
were to take these two matrices, so this
如果你想将这样两个矩阵相加

46
00:01:31,803 --> 00:01:32,894
one is a 3 by
这是一个3行2列的矩阵

47
00:01:32,894 --> 00:01:36,208
2 matrix, okay, 3 rows, 2 columns.
行数为3 列数为2

48
00:01:36,230 --> 00:01:38,659
This here is a 2 by 2 matrix.
而这一个是2行2列的矩阵

49
00:01:39,190 --> 00:01:41,190
And because these two matrices
那么由于这两个矩阵

50
00:01:41,200 --> 00:01:42,837
are not of the same dimension,
维度是不相同的

51
00:01:43,160 --> 00:01:44,635
you know, this is an error,
这就出现错误了

52
00:01:44,635 --> 00:01:46,400
so you cannot add these
所以我们不能将它们相加

53
00:01:46,430 --> 00:01:48,508
two matrices and, you know,
也就是说

54
00:01:48,508 --> 00:01:52,184
their sum is not well-defined.
这两个矩阵的和是没有意义的

55
00:01:52,642 --> 00:01:54,561
So that's matrix addition.
这就是矩阵的加法运算

56
00:01:54,561 --> 00:01:58,382
Next, let's talk about multiplying matrices by a scalar number.
接下来 我们讨论矩阵和标量的乘法运算

57
00:01:58,382 --> 00:02:00,069
And the scalar is just a,
这里所说的标量

58
00:02:00,069 --> 00:02:02,028
maybe a overly fancy term for,
可能是一个复杂的结构

59
00:02:02,028 --> 00:02:04,342
you know, a number or a real number.
或者只是一个简单的数字 或者说实数

60
00:02:04,760 --> 00:02:07,075
Alright, this means real number.
标量在这里指的就是实数

61
00:02:07,076 --> 00:02:10,280
So let's take the number 3 and multiply it by this matrix.
如果我们用数字3来和这个矩阵相乘

62
00:02:10,280 --> 00:02:13,182
And if you do that, the result is pretty much what you'll expect.
那么结果是显而易见的

63
00:02:13,182 --> 00:02:14,926
You just take your elements
你只需要将矩阵中的所有元素

64
00:02:14,926 --> 00:02:16,184
of the matrix and multiply
都和3相乘

65
00:02:16,184 --> 00:02:18,114
them by 3, one at a time.
每一个都逐一与3相乘

66
00:02:18,114 --> 00:02:19,428
So, you know, one
因此 1和3相乘

67
00:02:19,428 --> 00:02:21,708
times three is three.
结果是3

68
00:02:21,708 --> 00:02:24,011
What, two times three is
2和3相乘

69
00:02:24,011 --> 00:02:25,988
six, 3 times 3
结果是6

70
00:02:25,988 --> 00:02:28,181
is 9, and let's see, I'm
最后3乘以3得9

71
00:02:28,181 --> 00:02:30,152
going to stop changing colors again.
我再换一下颜色

72
00:02:30,157 --> 00:02:31,654
Zero times 3 is zero.
0乘以3得0

73
00:02:31,654 --> 00:02:35,992
Three times 5 is 15, and 3 times 1 is three.
3乘以5得15 最后3乘以1得3

74
00:02:35,992 --> 00:02:37,849
And so this matrix is the
这样得到的这个矩阵

75
00:02:37,849 --> 00:02:40,702
result of multiplying that matrix on the left by 3.
就是左边这个矩阵和3相乘的结果

76
00:02:40,702 --> 00:02:42,173
And you notice, again,
我们再次注意到

77
00:02:42,173 --> 00:02:43,443
this is a 3 by 2
这是一个3行2列的矩阵

78
00:02:43,443 --> 00:02:44,903
matrix and the result is
得到的结果矩阵

79
00:02:44,903 --> 00:02:47,505
a matrix of the same dimension.
维度也是相同的

80
00:02:47,505 --> 00:02:48,634
This is a 3 by
也就是说这两个矩阵

81
00:02:48,634 --> 00:02:49,920
2, both of these are
都是3行2列

82
00:02:49,920 --> 00:02:52,607
3 by 2 dimensional matrices.
这也是3行2列

83
00:02:52,634 --> 00:02:54,334
And by the way,
顺便说一下

84
00:02:54,334 --> 00:02:57,050
you can write multiplication, you know, either way.
你也可以写成另一种方式

85
00:02:57,050 --> 00:02:59,491
So, I have three times this matrix.
这里是3和这个矩阵相乘

86
00:02:59,491 --> 00:03:01,468
I could also have written this
你也可以把这个矩阵写在前面

87
00:03:01,470 --> 00:03:05,256
matrix and 0, 2, 5, 3, 1, right.
1 0 2 5 3 1

88
00:03:05,256 --> 00:03:07,672
I just copied this matrix over to the right.
把左边这个矩阵照抄过来

89
00:03:07,672 --> 00:03:11,228
I can also take this matrix and multiply this by three.
我们也可以用这个矩阵乘以3

90
00:03:11,228 --> 00:03:12,040
So whether it's you know, 3
也就是说

91
00:03:12,060 --> 00:03:13,388
times the matrix or the
3乘以这个矩阵

92
00:03:13,388 --> 00:03:14,983
matrix times three is
和这个矩阵乘以3

93
00:03:14,983 --> 00:03:18,771
the same thing and this thing here in the middle is the result.
结果都是一回事 都是中间的这个矩阵

94
00:03:19,380 --> 00:03:22,869
You can also take a matrix and divide it by a number.
你也可以用矩阵除以一个数

95
00:03:22,869 --> 00:03:24,275
So, turns out taking
那么 我们可以看到

96
00:03:24,275 --> 00:03:25,716
this matrix and dividing it by
用这个矩阵除以4

97
00:03:25,716 --> 00:03:27,140
four, this is actually the
实际上就是

98
00:03:27,172 --> 00:03:29,055
same as taking the number
用四分之一

99
00:03:29,055 --> 00:03:32,819
one quarter, and multiplying it by this matrix.
来和这个矩阵相乘

100
00:03:32,819 --> 00:03:35,318
4, 0, 6, 3 and
4 0 6 3

101
00:03:35,318 --> 00:03:36,803
so, you can figure
不难发现

102
00:03:36,820 --> 00:03:38,593
the answer, the result of
相乘的结果是

103
00:03:38,593 --> 00:03:40,365
this product is, one quarter
1/4和4相乘为1

104
00:03:40,365 --> 00:03:43,274
times four is one, one quarter times zero is zero.
1/4和0相乘得0

105
00:03:43,282 --> 00:03:46,570
One quarter times six is,
1/4乘以6

106
00:03:46,590 --> 00:03:49,353
what, three halves, about six over
结果是3/2

107
00:03:49,353 --> 00:03:50,369
four is three halves, and
6/4也就是3/2

108
00:03:50,369 --> 00:03:53,862
one quarter times three is three quarters.
最后1/4乘以3得3/4

109
00:03:54,410 --> 00:03:55,880
And so that's the results
这样我们就得到了

110
00:03:55,920 --> 00:03:59,207
of computing this matrix divided by four.
这个矩阵除以4的结果

111
00:03:59,207 --> 00:04:01,677
Vectors give you the result.
结果就是是右边这个矩阵

112
00:04:01,697 --> 00:04:03,805
Finally, for a slightly
最后

113
00:04:03,805 --> 00:04:05,714
more complicated example, you can
我们来看一个稍微复杂一点的例子

114
00:04:05,714 --> 00:04:09,460
also take these operations and combine them together.
我们可以把所有这些运算结合起来

115
00:04:09,513 --> 00:04:11,448
So in this calculation, I
在这个运算中

116
00:04:11,448 --> 00:04:12,801
have three times a vector
需要用3来乘以这个向量

117
00:04:12,801 --> 00:04:16,370
plus a vector minus another vector divided by three.
然后加上一个向量 再减去另一个向量除以3的结果

118
00:04:16,370 --> 00:04:18,344
So just make sure we know where these are, right.
让我们先来整理一下这几项运算

119
00:04:18,344 --> 00:04:20,031
This multiplication.
首先第一个运算

120
00:04:20,031 --> 00:04:23,648
This is an example of
很明显这是标量乘法的例子

121
00:04:23,680 --> 00:04:27,986
scalar multiplication because I am taking three and multiplying it.
因为这里是用3来乘以一个矩阵

122
00:04:27,986 --> 00:04:30,240
And this is, you know, another
然后这一项

123
00:04:30,240 --> 00:04:32,067
scalar multiplication.
很显然这是另一个标量乘法

124
00:04:32,067 --> 00:04:34,182
Or more like scalar division, I guess.
或者可以叫标量除法

125
00:04:34,182 --> 00:04:36,503
It really just means one zero times this.
其实也就是1/3乘以这个矩阵

126
00:04:36,503 --> 00:04:39,445
And so if we evaluate
因此

127
00:04:39,509 --> 00:04:43,044
these two operations first, then
如果我们先考虑这两项运算

128
00:04:43,044 --> 00:04:44,612
what we get is this thing
那么我们将得到的是

129
00:04:44,612 --> 00:04:47,127
is equal to, let's see,
我们看一下

130
00:04:47,127 --> 00:04:49,902
so three times that vector is three,
3乘以这个矩阵

131
00:04:49,912 --> 00:04:53,200
twelve, six, plus
结果是3 12 6

132
00:04:53,200 --> 00:04:55,088
my vector in the middle which
然后和中间的矩阵相加

133
00:04:55,088 --> 00:04:58,552
is a 005 minus
也就是0 0 5

134
00:04:59,850 --> 00:05:03,733
one, zero, two-thirds, right?
最后再减去1 0 2/3

135
00:05:03,740 --> 00:05:05,318
And again, just to make
同样地 为了便于理解

136
00:05:05,318 --> 00:05:07,064
sure we understand what is going on here,
我们再来梳理一下这几项

137
00:05:07,064 --> 00:05:11,504
this plus symbol, that is
这里的这个加号

138
00:05:11,520 --> 00:05:15,690
matrix addition, right?
表明这是一个矩阵加法 对吧？

139
00:05:15,690 --> 00:05:16,973
I really, since these are
当然这里是向量

140
00:05:16,973 --> 00:05:20,204
vectors, remember, vectors are special cases of matrices, right?
别忘了 向量是特殊的矩阵 对吧？

141
00:05:20,204 --> 00:05:21,538
This, you can also call
或者你也可以称之为

142
00:05:21,538 --> 00:05:25,106
this vector addition This
向量加法运算

143
00:05:25,110 --> 00:05:27,148
minus sign here, this is
同样 这里的减号表明

144
00:05:27,160 --> 00:05:30,162
again a matrix subtraction,
这是一个矩阵减法运算

145
00:05:30,162 --> 00:05:32,249
but because this is an
但由于这是一个n行1列的矩阵

146
00:05:32,249 --> 00:05:33,432
n by 1, really a three
实际上是3行1列

147
00:05:33,432 --> 00:05:35,547
by one matrix, that this
因此这个矩阵

148
00:05:35,547 --> 00:05:36,494
is actually a vector, so this is
实际上是也一个向量

149
00:05:36,494 --> 00:05:39,822
also vector, this column.
一个列向量

150
00:05:39,850 --> 00:05:43,677
We call this matrix a vector subtraction, as well.
因此也可以把它称作向量的减法运算

151
00:05:43,677 --> 00:05:44,392
OK?
好了！

152
00:05:44,392 --> 00:05:46,073
And finally to wrap this up.
最后再整理一下

153
00:05:46,110 --> 00:05:48,103
This therefore gives me a
最终的结果依然是一个向量

154
00:05:48,118 --> 00:05:49,952
vector, whose first element is
向量的第一个元素

155
00:05:49,952 --> 00:05:53,632
going to be 3+0-1,
是3+0-1

156
00:05:53,632 --> 00:05:56,150
so that's 3-1, which is 2.
就是3-1 也就是2

157
00:05:56,150 --> 00:06:01,204
The second element is 12+0-0, which is 12.
第二个元素是12+0-0 也就是12

158
00:06:01,214 --> 00:06:03,970
And the third element
最后第三个元素

159
00:06:03,970 --> 00:06:07,222
of this is, what, 6+5-(2/3),
6+5-(2/3)

160
00:06:07,222 --> 00:06:10,678
which is 11-(2/3), so
也就是11-(2/3)

161
00:06:10,678 --> 00:06:14,021
that's 10 and one-third
结果是10又三分之一

162
00:06:14,021 --> 00:06:16,029
and see, you close this square bracket.
关闭右括号

163
00:06:16,029 --> 00:06:17,983
And so this gives me a
我们得到了最终的结果

164
00:06:17,983 --> 00:06:21,671
3 by 1 matrix, which is
这是一个3行1列的矩阵

165
00:06:21,671 --> 00:06:23,901
also just called a 3
或者也可以说是

166
00:06:23,901 --> 00:06:29,005
dimensional vector, which
一个维度为3的向量

167
00:06:29,030 --> 00:06:32,847
is the outcome of this calculation over here.
这就是这个运算式的计算结果

168
00:06:32,847 --> 00:06:34,984
So that's how you
所以

169
00:06:34,984 --> 00:06:36,698
add and subtract matrices and
你学会了矩阵或向量的加减运算

170
00:06:36,698 --> 00:06:41,488
vectors and multiply them by scalars or by row numbers.
以及矩阵或向量跟标量 或者说实数 的乘法运算

171
00:06:41,488 --> 00:06:42,767
So far I have only talked
到目前为止

172
00:06:42,767 --> 00:06:44,718
about how to multiply matrices and
我只介绍了如何进行

173
00:06:44,718 --> 00:06:46,994
vectors by scalars, by row numbers.
矩阵或向量与数的乘法运算

174
00:06:46,994 --> 00:06:48,128
In the next video we will
在下一讲中

175
00:06:48,128 --> 00:06:49,418
talk about a much more
我们将讨论一个更有趣的话题

176
00:06:49,418 --> 00:06:51,035
interesting step, of taking
那就是如何进行

177
00:06:51,035 --> 00:06:54,112
2 matrices and multiplying 2 matrices together.
两个矩阵的乘法运算