域、 共域和范围

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Given a linear transformation of the form
::给定窗体的直线变换

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$\begin{array}{r}T(\stackrel{\to }{x})=A\stackrel{\to }{x}\end{array}$  where  $\begin{array}{r}A\end{array}$  is a standard matrix
::T(x)=Ax,其中A为标准矩阵

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The domain of the transformation is the set of all inputs of the variable  $\begin{array}{r}\stackrel{\to }{x}\end{array}$ . So, for example, if  $\begin{array}{r}T\end{array}$  is a linear transformation 
::变换的域是变量 x 的所有输入的集合。例如,如果 T 是线性变换

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$\begin{array}{r}T:{\mathbb{R}}^n\to {\mathbb{R}}^m\end{array}$
::T:RnRm

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The set  $\begin{array}{r}{\mathbb{R}}^n\end{array}$  is the input space, or the domain of the linear transformation  $\begin{array}{r}T\end{array}$ . So, let's look at an example of a linear transformation and try to find the domain.
::设置 Rn 是输入空间, 或者线性变换 T 的域 。 所以, 让我们看看线性变换的示例, 并试着找到域 。

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$\begin{array}{r}T:{\mathbb{R}}^3\to {\mathbb{R}}^3\\ T(\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right])=\left[\begin{array}{ccc}3& -1& 2\\ 0& -2& 1\\ -1& 2& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\end{array}$
::T:R3QR3T([x1x2x3]) =[3-120-21-120][x1x2x3]

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And looking at this transformation we see that the input space has every vector in  $\begin{array}{r}{\mathbb{R}}^3\end{array}$ .
::我们看到输入空间 含有R3中的每一个矢量

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Now, the codomain of a linear transformation is the space in which the linear transformation maps to.
::现在,线性变换的共域 就是线性变换映射到的空间。

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So, for example, taking
::例如,举例说,以

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$\begin{array}{r}T:{\mathbb{R}}^2\to {\mathbb{R}}^3\\ T(\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right])=\left[\begin{array}{cc}1& -1\\ 2& 0\\ 3& -2\end{array}\right]\cdot \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\end{array}$  
::T:R2R3T([x1x2]) =[1-1203-2] [x1x2]

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Here, we see that the domain of this linear transformation is  $\begin{array}{r}{\mathbb{R}}^2\end{array}$  and the codomain of this matrix is  $\begin{array}{r}{\mathbb{R}}^3\end{array}$  because that is the output space of this linear transformation as the resulting vector will be a vector in  $\begin{array}{r}{\mathbb{R}}^3\end{array}$ .
::在这里,我们可以看到,这种线性变换的域是R2,而这个矩阵的共域是R3,因为这就是这种线性变换的输出空间,因为由此产生的矢量将是R3中的矢量。

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Now, the codomain of a linear transformation is different than the range of a linear transformation.
::现在,线性变换的共域与线性变换的范围不同。

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The range of a linear transformation is the set of all of the images of a linear transformation.
::线性变换的范围是线性变换的所有图像的集合。

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So, staying with the last example we looked at, we have that the codomain is the reals in 3 dimensions. However, the set of all of the images is
::所以,与我们所看到的最后一个例子一样, 我们发现, 共域是3维的真数。 然而, 所有图像的集是

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$\begin{array}{r}\text{range}(T)=\{\left[\begin{array}{c}{x}_1-x_2\\ 2x_1\\ 3x_1-2x_2\end{array}\right]|\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\in {\mathbb{R}}^2\}\end{array}$
::区域( T) [x1 - x22x13x1 - 2x2] [x1x2] R2}

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which is a subset of the codomain, however, it is not equal to that.
::这是共域的子集, 但是,它不等于它。

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Next, let's look at a couple examples and the diagrams of those examples:
::下面,让我们看看几个例子和这些例子的图表:

 

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Let  $\begin{array}{r}T:{\mathbb{R}}^3\to {\mathbb{R}}^3\\ T(\stackrel{\to }{x})=A\stackrel{\to }{x}\\ A=\left[\begin{array}{ccc}1& 0& -1\\ 2& 3& 0\\ -1& 5& 2\end{array}\right]\\ A\stackrel{\to }{x}=\left[\begin{array}{ccc}1& 0& -1\\ 2& 3& 0\\ -1& 5& 2\end{array}\right]\cdot \left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\end{array}$
::让 T: R3QR3T( x%) =AxIA = [10- 1230-152] Ax{[10- 1230-152] {[10- 1230-152] {[x1x2x3]

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The domain of this transformation is all of  $\begin{array}{r}{\mathbb{R}}^3\end{array}$  and the entire output space of this linear transformation is  $\begin{array}{r}{\mathbb{R}}^3\end{array}$ , so the codomain is the same set as the domain which is  $\begin{array}{r}{\mathbb{R}}^3\end{array}$ .
::这种变换的域为R3, 线性变换的整个输出空间为R3, 因此共域与R3的域相同。

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Now, searching for the set of all images, or in other words, the range, we get that it is equal to 
::现在,搜索全部图像集, 换言之, 范围, 我们得到它等于

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$\begin{array}{r}\text{Range}(T(\stackrel{\to }{x}))=\{\left[\begin{array}{c}{x}_1-x_3\\ 2x_1+3x_2\\ -1x_1+5x_2+2x_3\end{array}\right]|x_1,x_2,x_3\in \mathbb{R}\to \left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\in {\mathbb{R}}^3\}\end{array}$
::区域( T( x) ) [x1 - x32x1+3x2 - 1x1+5x2+2x3] x1,x2,x3}[x1x2x3] [x1x2x3] {R3}

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Due to technical constraints it will be harder to visualize this linear transformation, but let's try another example and try and apply our principles of domain, codomain and range.
::由于技术限制,很难想象这种线性转变, 但是让我们再试一个例子, 尝试运用我们的域、 共域和范围原则。

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Now, let 
::现在,让我们

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$\begin{array}{r}T:{\mathbb{R}}^2\to {\mathbb{R}}^2\\ T(\stackrel{\to }{x})=A\stackrel{\to }{x}\\ A=\left[\begin{array}{cc}5& 2\\ -2& 3\end{array}\right]\\ A\stackrel{\to }{x}=\left[\begin{array}{cc}5& 2\\ -2& 3\end{array}\right]\cdot \left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\end{array}$
::T:R2-R2T(x__)=Ax_A=[52-23]Ax[52-23]和[x1x2]

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From this, we see that the input and output space, domain and codomain, are both  $\begin{array}{r}{\mathbb{R}}^2\end{array}$ .
::我们从中看到,输入和输出空间, 域和共同域, 既是R2,又是R2。

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Looking at this to find the range, we get that the range, or the set of all images of the transformations is
::或所有变换图像的集

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$\begin{array}{r}\{\left[\begin{array}{c}5x_1+2x_2\\ -2x_1+3x_2\end{array}\right],x_1,x_2\in \mathbb{R}\}\end{array}$
::{5x1+2x2-2x1+3x2,x1,x2R}

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Looking at this visually, we see that this linear transformation transforms the space
::我们看到这种直线转换 改变了空间

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to end up getting
::最终得到