3 - 4 - Matrix Matrix Multiplication (11 min).srt

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In this video we talk about
在这段视频中我们将会讨论
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

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matrix, matrix multiplication or
矩阵 矩阵的乘法以及

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how to multiply two matrices together.
如何将两个矩阵相乘

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When we talk about the method
我们会使用这样一种方法

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in linear regression for how
在线性回归中用以解决

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to solve for the parameters,
参数计算的问题

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theta zero and theta one, all in one shot.
这种方法会把θ0、θ1等参数都放在一起来计算

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So, without needing an iterative algorithm like gradient descent.
也就是说 我们不需要一个迭代的梯度下降算法

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When we talk about that algorithm,
当我们谈到这个算法的时候

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it turns out that matrix, matrix
就会发现矩阵以及矩阵间的乘法运算

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multiplication is one of the key steps that you need to know.
是你必须理解的关键步骤之一

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So, let's, as usual, start with an example.
所以让我们像往常那样 从一个例子开始

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Let's say I have two matrices
比方说 我有两个矩阵

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and I want to multiply them together.
我想将它们相乘

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Let me again just reference this
让我先只是按照这个例子做一遍（乘法）

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example and then I'll tell you in a little bit what happens.
然后告诉你这其中运算的细节

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So, the first thing
那么  我要做的第一件事是

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I'm gonna do is, I'm going
我先把

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to pull out the first
右边这个矩阵的第一列

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column of this matrix on the right.
提取出来

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And I'm going to take this
然后我将会把

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matrix on the left and
左边的这个矩阵和

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multiply it by, you know, a vector.
之前取出来的这一列（前面提过的，向量）相乘

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That's just this first column, OK?
这只是第一列  是吧？

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And it turns out if I
然后我们可以看到 如果我

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do that I am going to get the vector 11, 9.
这么做 我就会得到向量（11,9）

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So, this is the same matrix
所以这是与上个视频的矩阵

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vector multiplication as you saw in the last videos.
和向量的乘法是一样的

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I worked this out in advance so, I know it's 11, 9.
我已经提前算出了这个结果 是（11,9）

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And, then, the second thing
那么 之后的第二件事

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I'm going to do is, I'm going
我要做的就是

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to pull out the second column,
我将把第二列再单独提出出来

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this matrix on the right and
右边这个矩阵的第二列

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I am then going to
然后我将要把它和

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take this matrix on the left,
左边这个矩阵相乘

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right, so, it will be that matrix,
是的吧  所以 这就是那个矩阵

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and multiply it by
用右边的第二列

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that second column on the right.
来乘以这个矩阵

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So, again, this is a matrix
因此 同样的 这是一个矩阵和

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vector multiplication set, which
向量的乘法运算  这

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you saw from the previous video, and
就是你从上一个视频所学到的

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it turns out that if you
如果你这么做

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multiply this matrix and this
把这个矩阵和这个向量相乘

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vector, you get 10,
你会得到

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14 and by
（10,14）这个结果

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the way, if you want to practice
顺便说一下 如果你想练习

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your matrix vector multiplication, feel
矩阵和向量的乘法运算

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free to pause the video and check this product yourself.
那么就先暂停下视频  自己算一算结果对不对

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Then, I'm just going
好吧 现在我仅仅需要

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to take these two results and
将得到的这两个结果放在一起

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put them together, and that will be my answer.
那么这就是我的答案了

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So, turns out the
那么 我们可以看到

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outcome of this product is going
计算结果是

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to be a 2 by 2 matrix, and
一个2 x 2的矩阵

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The way I am going to fill
我用来填充这个矩阵的方法

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in this matrix is just by
就是

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taking my elements 11,
把我的（11,9）

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9 and plugging them here, and
填在这里

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taking 10, 14 and plugging
把（10，14）填在

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them into the second column.
第二列

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Okay?
是的吧？

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So, that was the mechanics of
所以 这就是如何

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how to multiply a matrix by
将两个矩阵相乘的

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another matrix.
详细方法与过程

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You basically look at the
每次你只需要看

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second matrix one column at a time, and you assemble the answers.
第二个矩阵的一列    然后把你的答案拼凑起来

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And again, we will step
再次强调下 我们将一步步的来计算

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through this much more carefully in
几秒中的时间里需要非常仔细

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a second, but I just
但我也要指出

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want to point out also, this
我也要指出的是

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first example is a 2x3 matrix matrix.
第一个例子是一个2X3矩阵

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Multiplying that by a
乘以一个

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3x2 matrix, and the
3x2的矩阵 他们相乘

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outcome of this product, it
得到的结果

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turns out to be a 2x2
是一个2x2的

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matrix.
矩阵

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And again, we'll see in a second why this was the case.
我们将很快知道为什么是这个结果

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All right.
好的

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That was the mechanics of the calculation.
这是计算的技巧

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Let's actually look at the
让我们再看看

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details and look at what
这其中的细节

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exactly happened.
看看究竟发生了什么

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Here are details.
下面就是详细的过程

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I have a matrix A and
我有一个矩阵A

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I want to multiply that
我要把它乘以

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with a matrix B, and the result
矩阵B  其结果

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will be some new matrix C. And
会是一个新的矩阵C

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it turns out you can only
并且你会发现你只能

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multiply together matrices whose
相乘那些维度

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dimensions match so A
匹配的矩阵

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is an m by n matrix,
因此如果A是一个m×n的矩阵

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so m columns, n columns and
就是说m行n列

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I am going to multiply
我将要用它与

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that with an n by o
一个n×o的矩阵相乘

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and it turns out this n
并且实际上这里的n

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here must match this n
必须匹配这里的这个n

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here, so the number of columns
所以第一个矩阵的列的数目

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in first matrix must equal to the number of rows in second matrix.
必须等于第二矩阵中的行的数目

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And the result of this
并且相乘得到的结果

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product will be an M
结果会是一个m×o的矩阵

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by O matrix, like the the matrix C here.
就像这个矩阵C这样

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And, in the previous
并且 在前面的视频中

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video, everything we did corresponded
我们所做的一切都符合这个规则

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to this special case of OB
这是一种当矩阵B的o值

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equal to 1.
等于1的特殊情况（指的是矩阵和向量相乘）

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Okay?
明白了吗？

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That was, that was in case of B being a vector.
这是在B是一个向量的情况下

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But now, we are going to
但是现在 我们要处理

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view of the case of values of O larger than 1.
O的值大于1的情况

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So, here's how you
所以 这里就是你怎样

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multiply together the two matrices.
把两个矩阵相乘

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In order to get, what
为了得到结果

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I am going to do is
我要做的就是

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I am going to take the
我将要取

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first column of B
B矩阵的第一列

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and treat that as a vector,
把取出的这列看成一个向量

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and multiply the matrix A,
并乘以矩阵A

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with the first column of B,
用B矩阵的第一列

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and the result of that will
这个计算结果将是

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be a M by 1 vector,
m×1的矩阵（也就是一个向量）

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and we're going to put that over here.
我们把结果先放在这里

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Then, I'm going to
然后 我将要取

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take the second column
B矩阵的

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of B, right, so,
第二列

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this is another n by
那么我会又得到一个n×1的向量

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one vector, so, this column
也就是 这里的这一列

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here, this is right, n
这是正确的

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by one, those are n dimensional
n×1的矩阵 也就是n维的向量

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vector, gonna multiply this
我将要把这个矩阵

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matrix with this n by one vector.
和这些n乘1的向量相乘

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The result will be
其结果将是

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a M dimensional vector,
一个m维的向量

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which we'll put there.
然后我会把结果先放在那里

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And, so on.
依此类推

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Okay?
对吧？

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And, so, you know, and then
那么 你知道的

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I'm going to take the third
我开始取第三列

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column, multiply it by
把它和这个矩阵相乘

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this matrix, I get a M dimensional vector.
我又得到了一个M维向量

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And so on, until you get
依此类推 直到你计算到了

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to the last column times,
最后一列

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the matrix times the
矩阵乘以

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lost column gives you
你取到的最后一列

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the lost column of C.
就是C的最后一列

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Just to say that again.
再说一遍

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The ith column of the
矩阵C的第i列

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matrix C is attained
是根据把

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by taking the matrix A and
矩阵A与

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multiplying the matrix A with
矩阵B的第i列

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the ith column of the
相乘得到的

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matrix B for the values
结果

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of I equals 1, 2
依次相加

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up through O. Okay ?
从1,2到o依次相加的 对吧？

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So, this is just a summary
那么 我们在这里做一个总结

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of what we did up there
我们总结了我们为了

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in order to compute the matrix
计算矩阵C所做的步骤

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C.  Let's look at just one more example.
让我们再看一个例子

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Let 's say, I want to multiply together these two matrices.
比方说我想把这两个矩阵相乘

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So, what I'm going to
那么我首先要做的是

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do is, first pull
先取出

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out the first column
我的第二个矩阵的

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of my second matrix, that
第一列

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was matrix B, that was
就是这个矩阵B 这就是

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my matrix B on the previous slide.
上一张幻灯片上出现的矩阵B

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And, I therefore, have this
因此 我就这么

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matrix times my vector and
用矩阵和我取的向量相乘

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so, oh, let's do this calculation quickly.
所以    让我们快速的计算这个结果

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There's going to be equal to,
这等于

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right, 1, 3 times 0,
没错  （1,3）乘以（0,3）

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3 so that gives 1
所以 就是

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times 0, plus 3 times 3.
1x0+3x3（9）

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And, the second element
此外 第二元素

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is going to be 2,
就是（2，5）

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5 times 0, 3 so, that's going to
乘以（0，3）

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be two times 0 plus 5
就是0x2+5x3（15）

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times 3 and that is
那么结果出来了

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9,15, actually didn't
（9,15）实际上该用绿色的颜色标记

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write that in green, so this
所以这就是（9.15）

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is nine fifteen, and then mix.
那么

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I am going to pull out
我将同样的取出

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the second column of this,
这个的第二列

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and do the corresponding
做相同的计算

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calculation so there's this
所以

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matrix times this vector 1, 2.
这是这个矩阵乘以（1,2）

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Let's also do this
让我们快点算吧

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quickly, so that's one times
所以这是

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one plus three times two.
1x1 + 3x2

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So that deals with that
那么这就处理了这一行

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row, let's do the
让我们计算另一行

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other one, so let's see,
让我们来看看

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that gives me two times
这次是

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one plus times two,
2x1 + 5x2

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so that is going to
因此这就等于

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be equal to, let's see,
我们看一下

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one times one plus three times
1x1 + 3x1结果是4

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one is four and two
2x1 + 5x2

197
00:06:50,378 --> 00:06:52,282
times one plus five times two
结果是

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is twelve.

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00:06:55,570 --> 00:06:56,660
So now I have these two
所以现在我有两个这个了

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00:06:56,660 --> 00:06:58,448
you, and so my
因此我得到的

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outcome, so the product
这两个矩阵

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of these two matrices is going
相乘的结果就是

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to be, this goes
这个在这儿

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here and this
那个放那边

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goes here, so I
所以我得到了

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get nine fifteen and
9,15和

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four twelve and you
4,12

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may notice also that the result
你也可能会注意到这个结果

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of multiplying a 2x2 matrix
一个2×2的矩阵

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with another 2x2 matrix.
乘以另一个2x2的矩阵

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The resulting dimension is going
这个维度会是

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to be that first two times
第一个矩阵的2乘以第二的矩阵的2

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that second two, so the result
所以这个结果本身

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is itself also a two by two matrix.
也是一个2x2的矩阵

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Finally let me show you
最后让我告诉你

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one more neat trick you can
一个更加具体的技巧    你可以

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do with matrix matrix multiplication.
在矩阵和矩阵的乘法中使用

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Let's say as before that we
比方说 在这之前 我们

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have four houses whose
有四间房子

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prices we want to predict,
我们要预测其价格

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only now we have three
但是现在我们有三个

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competing hypothesis shown here
不同的竞争假设集在这儿

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on the right, so if
在右侧 因此 如果

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you want to So apply all
你想要去把

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3 competing hypotheses to
这三个竞争假设集用来

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all four of the houses, it
适应这4个房屋的数据

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turns out you can do that
那么你可以这样做

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very efficiently using a
这将非常高效

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matrix matrix multiplication so here
我们使用这里的矩阵乘法来计算

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on the left is my usual
左边是我通常使用的矩阵

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00:08:07,370 --> 00:08:08,626
matrix, same as from the
这与我上个视频

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00:08:08,626 --> 00:08:11,063
last video where these values
一样  这些值就是

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are my housing prices and I put ones there on the left as well.
我的住房价格   我把这些值也放在左边

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And, what I'm going to
那么 我要去做的就是

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do is construct another matrix, where
构造另一个矩阵

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here these, the first
这个矩阵的第一列

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column, is this minus
是-40

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40 and two five and
0.25

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00:08:26,070 --> 00:08:28,372
the second column is this two
第二列是

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hundred open one and so
（200.0.1）

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on and it
以此类推

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turns out that if you
事实证明  如果你

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multiply these two matrices
把这两个矩阵相乘

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what you find is that, this
你就会发现得到了结果的

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first column, you know,
第一列  你知道的

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oh, well how do you get this first column, right?
那么你怎么得到这个第一列呢？

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A procedure from matrix
这就要用到我们讲过的

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00:08:48,850 --> 00:08:50,565
matrix multiplication is the way
矩阵和矩阵相乘的过程

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you get this first column, is
你得到的这个矩阵的第一列

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00:08:51,960 --> 00:08:53,360
you take this matrix and you
通过你用这个矩阵

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multiply it by this
乘以

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first column, and we
这个矩阵的第一列

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00:08:56,724 --> 00:08:58,540
saw in the previous video that this
这是我们从之前的视频中看到过的

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is exactly the predicted
这就是从第一个假设

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00:09:00,490 --> 00:09:02,050
housing prices of the
预测出的

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00:09:02,150 --> 00:09:05,701
first hypothesis, right?
住房价格 对吗？

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00:09:05,701 --> 00:09:08,775
Of this first hypothesis here.
就是这里的这个假设集

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00:09:08,790 --> 00:09:10,794
And, how about a second column?
那么 第二列是什么呢？

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Well, how do setup the second column?
那么 我们应该怎么计算第二列呢？

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The way you get the second column
用来得到第二列的方法

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00:09:14,332 --> 00:09:15,548
is, well, you take this
就是

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matrix and you multiply by this second column.
用这个矩阵乘以这个矩阵的第二列

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00:09:19,270 --> 00:09:21,293
And so this second column turns
那么得到的第二列就是

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00:09:21,293 --> 00:09:24,651
out to be the predictions of
基于第二个假设

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00:09:24,651 --> 00:09:27,728
the second hypothesis of
做出的预测结果

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00:09:27,750 --> 00:09:30,228
the second hypothesis up there,
第二个假设集是在那里

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and similarly for the third column.
对于第三列 我们也能得到类似的结果

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And so, I didn't step
那么 我并没有

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00:09:35,810 --> 00:09:38,058
through all the details but hopefully
把详细的细节列出

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00:09:38,058 --> 00:09:39,139
you just, feel free to
不过 我还是希望你们能够把

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00:09:39,140 --> 00:09:40,448
pause the video and check
视频暂停下 自己算一算

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00:09:40,448 --> 00:09:41,786
the math yourself and check
检查下结果对不对

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00:09:41,786 --> 00:09:43,972
that what I just claimed really is true.
检验下我刚才计算的结果的正确性

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00:09:43,990 --> 00:09:45,611
But it turns out that by
那么 实际上通过

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constructing these two matrices, what
构建这两个矩阵

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00:09:47,454 --> 00:09:48,937
you can therefore do is very
你就可以

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00:09:48,940 --> 00:09:51,180
quickly apply all three
快速的把这三个假设集

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00:09:51,180 --> 00:09:52,602
hypotheses to all four
应用到所有四个

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00:09:52,602 --> 00:09:54,455
house sizes to get,
房子的尺寸中来计算价格了

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00:09:54,455 --> 00:09:56,452
you know, all twelve predicted
你看 所有的12种预测到的价格是

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00:09:56,452 --> 00:09:57,721
prices output by your
通过你的假设集

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00:09:57,721 --> 00:10:00,928
three hypotheses on your four houses.
以及你的四个房屋数据集得到的

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00:10:00,928 --> 00:10:03,366
So one matrix multiplications
所以 一次矩阵乘法操作

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00:10:03,366 --> 00:10:05,072
that you manage to make 12
就使你做出了12种预测

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00:10:05,080 --> 00:10:07,130
predictions and, even
更好的是

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00:10:07,130 --> 00:10:08,446
better, it turns out that
事实证明

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00:10:08,446 --> 00:10:09,937
in order to do that matrix
为了做到

288
00:10:09,937 --> 00:10:11,408
multiplication and there are
矩阵间的乘法

289
00:10:11,408 --> 00:10:13,130
lots of good linear algebra libraries
有很多很好的线性代数库函数

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00:10:13,150 --> 00:10:14,767
in order to do this
都是为了做到这一点

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00:10:14,767 --> 00:10:16,676
multiplication step for you,
为你实现矩阵乘法

292
00:10:16,676 --> 00:10:18,250
and no matter so pretty
而且不管你用的是

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00:10:18,250 --> 00:10:22,025
much any reasonable programming language that you might be using.
多么合理的编程语言

294
00:10:22,025 --> 00:10:24,005
Certainly all the top ten
当然 当下最流行的

295
00:10:24,005 --> 00:10:27,898
most popular programming languages will have great linear algebra libraries.
编程语言中的前十名都有很棒的线性代数函数库

296
00:10:27,898 --> 00:10:29,554
And they'll be good thing are
这是很好的事情

297
00:10:29,554 --> 00:10:31,463
highly optimized in order
我们能够在高度优化下

298
00:10:31,463 --> 00:10:33,415
to do that, matrix matrix
做到矩阵

299
00:10:33,440 --> 00:10:36,531
multiplication very efficiently, including
和矩阵间高效的乘法 包括

300
00:10:36,531 --> 00:10:38,501
taking, taking advantage of
采取了一些优化处理的方式

301
00:10:38,501 --> 00:10:41,119
any parallel computation that
如并行计算

302
00:10:41,130 --> 00:10:42,886
your computer may be capable
如果你的电脑支持的话

303
00:10:42,886 --> 00:10:46,297
of, when your computer has multiple
当你的计算机有多个

304
00:10:46,330 --> 00:10:48,016
calls or lots of
调度或者

305
00:10:48,016 --> 00:10:49,866
multiple processors, within a processor sometimes
多个处理器  一个处理器内有时

306
00:10:49,866 --> 00:10:53,285
there's there's parallelism as well called symdiparallelism [sp].
存在并行的计算 我们称之为SIMD Parallelism

307
00:10:53,285 --> 00:10:55,242
The computer take care of
在计算机管理机制下

308
00:10:55,242 --> 00:10:56,727
and you should, there are
你应该有

309
00:10:56,730 --> 00:10:58,826
very good free libraries
非常不错的免费类库

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00:10:58,826 --> 00:11:00,146
that you can use to do
你可以用来做

311
00:11:00,146 --> 00:11:02,326
this matrix matrix multiplication very
高效的矩阵间的乘法计算

312
00:11:02,326 --> 00:11:04,104
efficiently so that you
因此你就能

313
00:11:04,110 --> 00:11:05,908
can very efficiently, you
你知道的

314
00:11:05,930 --> 00:11:08,738
know, makes lots of predictions of lots of hypotheses.
方便地计算有很多假设集时的预测数据