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  循环空间

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  In this lecture we will learn about another type of vector space known as the null space of a matrix.
  ::在这次演讲中,我们将了解另一种类型的矢量空间,即矩阵空格的空格。

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  Definition: The Null Space (or the kernel) of a Matrix is the set of all vectors in  $\begin{array}{r}{\mathbb{R}}^n\end{array}$  such that  $\begin{array}{r}A\stackrel{\to }{v}=\stackrel{\to }{0}\end{array}$  given some matrix in the set  $\begin{array}{r}{M}_{m\times n}(\mathbb{R})\end{array}$
  ::定义:母体的内核空间(或内核)是Rn所有矢量的组合,因此Av0给定的 mmxn(R) 中的一些矩阵

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  To show this more concretely, let's say that  $\begin{array}{r}A\end{array}$  is an m by n matrix and $\begin{array}{r}x\in {\mathbb{R}}^n\end{array}$ then the matrix-vector product becomes a vector and we have to solve for when that vector is 0. Take the example of
  ::要更具体地显示这一点,让我们假设 A 是一个 m 乘 n 矩阵和x {{{{{{{{{{{{{{{{{{{{{}}} 矩阵矢量产品变成向量,我们必须解决该向量是0时的向量。举个例子

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  $\begin{array}{r}A=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right]\end{array}$
  ::A=[2412]

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  Then the matrix-vector product represented by some vector in the space  $\begin{array}{r}{\mathbb{R}}^2\end{array}$  can be represented here:
  ::然后空间R2中某些矢量所代表的矩阵矢量-矢量产品可以在这里代表:

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  Here, you can see that this matrix spans a line in  $\begin{array}{r}{\mathbb{R}}^2\end{array}$  and all of the points labeled are the ones that map to zero.
  ::这里,您可以看到这个矩阵横跨 R2 的线条, 所有标记的点都是地图为零的点 。

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  So you can see here that the null space of the matrix is just any vector of the form  $\begin{array}{r}\stackrel{\to }{v}=\left[\begin{array}{c}a\\ -\frac{a}{2}\end{array}\right]\end{array}$
  ::所以您可以看到,矩阵的空空格只是窗体的任意矢量 v[a-a2]。

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  More formally, given the matrix  $\begin{array}{r}A=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right]\end{array}$ , the null space of  $\begin{array}{r}A\end{array}$ , denoted  $\begin{array}{r}Null(A)\end{array}$  we get 
  ::更正式的说,鉴于矩阵A=[2412],A的空域(指Null(A))

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  $$\begin{array}{r}\\ Null(A)=\{\stackrel{\to }{v}\mid A\stackrel{\to }{v}=\stackrel{\to }{0}\}\\ =\{\stackrel{\to }{v}=\left[\begin{array}{c}a\\ -\frac{a}{2}\end{array}\right],\mathrm{\forall }a\in \mathbb{R}\}\end{array}$$

  
  ::努尔(A)

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  Let's look at another example:
  ::让我们看看另一个例子:

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  Let  $\begin{array}{r}A=\left[\begin{array}{cc}1& 2\\ 4& 7\end{array}\right]\end{array}$
  ::让我们 A = [1247]

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  Now, let's calculate the null space of A.
  ::现在,让我们计算一下A的空域

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  $$\begin{array}{r}A\stackrel{\to }{x}=\stackrel{\to }{0}\\ \left[\begin{array}{cc}1& 2\\ 4& 7\end{array}\right]\cdot \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]\end{array}$$

  Next, let's multiply the matrix and vector and solve the system of equations, yielding
  ::Ax0[1247][x1x2]=[00] 下一步,让我们乘以矩阵和矢量并解析方程系统,产生

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  $$\begin{array}{r}{x}_1+2x_2=0\\ 4x_1+7x_2=0\end{array}$$

  
  ::x1+2x2=04x1+7x2=0

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  Solving, we get the only solutions to this system of linear equations as $\begin{array}{r}{x}_1=0\text{ and }{x}_2=0\end{array}$ .
  ::正在解析, 我们找到这个线性方程式系统的唯一解决方案 asx1=0 和 x2=0 。

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  Hence,  $\begin{array}{r}Null(A)=\{\left[\begin{array}{c}0\\ 0\end{array}\right]\}\end{array}$
  ::因此,Null(A)[00]}

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  Visualizing the transformation geometrically, we get the following picture:
  ::从几何角度观察变形情况,我们得到以下图象:

   

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  Here, we see that this matrix-vector product looks like it will span all of  $\begin{array}{r}{\mathbb{R}}^2\end{array}$ , because it is not just spanning a line. Recall that the only two options for a linear combination in  $\begin{array}{r}{\mathbb{R}}^2\end{array}$  is to either span a line or a plane or just a point. So for a matrix vector product that takes a vector in  $\begin{array}{r}{\mathbb{R}}^2\end{array}$ and maps it to another vector in  $\begin{array}{r}{\mathbb{R}}^2\end{array}$  we get that the null space either has dimension 0,1 or 2. This brings us to introduce a new concept, nullity. 
  ::在这里,我们看到这个矩阵矢量产品看起来将覆盖所有R2, 因为它不仅横跨一条线。 回顾R2线性组合的唯一两个选项是横跨一条线条或一平面, 或仅仅是一个点。 所以对于在 R2 中将矢量带入一个矢量的矩阵矢量产品, 在 R2 中将其映射为另一个矢量的矩阵矢量, 我们得到的是空格空间有维度 0, 1 或 2 。 这让我们引入一个新的概念, 无效性 。

   

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  Nullity definition: The nullity of a matrix A is the dimension of the null space of a matrix. 
  ::无效性定义:矩阵A的无效性是矩阵空格的空格。

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  Let's look at a new example: Call  $\begin{array}{r}A=\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right]\end{array}$  and the matrix vector product is  $\begin{array}{r}A\stackrel{\to }{x}=\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right]\cdot \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\end{array}$
  ::让我们看看一个新例子:呼叫A=[1326],矩阵矢量产品为Ax*[1326]}[x1x2]。

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  So  $\begin{array}{r}A\stackrel{\to }{x}\end{array}$  represents the linear combination of  $\begin{array}{r}\left[\begin{array}{c}1\\ 2\end{array}\right]\end{array}$  and  $\begin{array}{r}\left[\begin{array}{c}3\\ 6\end{array}\right]\end{array}$ , but because 
  ::So Ax是[12]和[36]的线性组合,但由于

  $\begin{array}{r}\left[\begin{array}{c}3\\ 6\end{array}\right]=3\cdot \left[\begin{array}{c}1\\ 2\end{array}\right]\end{array}$

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  From this we see that this matrix vector product is a line and we can prove it by visualizing it:
  ::从这一点可以看出,这种矩阵矢量产品是一条线,我们可以通过直观的描述来证明这一点:

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  And we can see that every point that maps to 0 is of the form  $\begin{array}{r}c\left[\begin{array}{c}-3\\ 1\end{array}\right],c\in \mathbb{R}\end{array}$
  ::我们可以看到,地图到0的每一个点都是表c[-31],cR

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  So the null space is a line and has dimension 1.
  ::因此,空域是一条线,具有维度1。

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  If we look back at our earlier example with the matrix  $\begin{array}{r}\left[\begin{array}{cc}1& 2\\ 4& 7\end{array}\right]\end{array}$  only has 0 in the null space so the dimension of the null space is 0.
  ::如果我们回顾一下我们早先使用矩阵的例子[1247],则空域中只有0个,因此空域的尺寸为0。

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  Rank nullity theorem:
  ::无效定理排序 :

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  Given a matrix with n columns we can see that the rank of the matrix plus the nullity is the number of columns, in other words,
  ::根据带有n列的矩阵,我们可以看到,矩阵的等级加上无效性是列数,换句话说,

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  $\begin{array}{r}\text{dim}(\text{nul}(A))+\text{dim}(\text{col}(A))=n\end{array}$  where  $\begin{array}{r}n\end{array}$  is the number of columns of  $\begin{array}{r}A\end{array}$
  ::dim(nul( A))+dim(col( A))=n, n是 A 列数的 n

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  Applications to the Invertible Matrix Theorem
  ::不可逆矩阵定理的应用

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  - Applying any elementary row operation onto a matrix A does not change it's null space
  ::- 将任何基本行操作应用到矩阵A上,不会改变它的空格

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  - If  $\begin{array}{r}{x}_0\end{array}$   is any solution of  $\begin{array}{r}A\stackrel{\to }{x}=\stackrel{\to }{b}\end{array}$   and  $\begin{array}{r}\stackrel{\to }{v_1},\stackrel{\to }{v_2},\cdots ,\stackrel{\to }{v_r}\end{array}$   for a basis for the null space of A, then any solution of  $\begin{array}{r}A\stackrel{\to }{x}=\stackrel{\to }{b}\end{array}$   can be written as  $\begin{array}{r}\stackrel{\to }{x}=c_1\stackrel{\to }{x_1}+\cdots +c_r\stackrel{\to }{x_r}\end{array}$   and conversely, any vector written this way for any scalars  $\begin{array}{r}{c}_1,\cdots ,c_r\end{array}$   is a solution of  $\begin{array}{r}A\stackrel{\to }{x}=\stackrel{\to }{b}\end{array}$ .
  ::- 如果 x0 是 Axb 和 v1,v2, ,vr 为 A 无效空间的基础的任何解决方案, 那么任何 Axb 的解决方案都可以写成 xc1x1crxr , 反之, 任何 calars c1, cr 以这种方式写成的任何矢量, 则是 Axb 的解决方案 。

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  Let's watch a few videos to better understand the null space:
  ::让我们看几段影片, 以便更好地了解空域: