Chapter 1 (Linear Equations in Linear Algebra): The matrix of a linear transformation

本文为《Linear algebra and its applications》的读书笔记

The matrix of a linear transformation

• Whenever a linear transformation $T$ arises geometrically or is described in words, we usually want a “formula” for $T(\mathbf{x})$. The discussion that follows shows that every linear transformation from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$ is actually a matrix transformation $\mathbf{x}\mapsto A\mathbf{x}$.
• The key to finding $A$ is to observe that $T$ is completely determined by what it does to the columns of the $n\times n$ identity matrix $I_n$.

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EXAMPLE 1

The columns of $I_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ are ${\mathbf{e}}_1=\left[\begin{array}{c}1\\ 0\end{array}\right]$ and ${\mathbf{e}}_2=\left[\begin{array}{c}0\\ 1\end{array}\right]$. Suppose $T$ is a linear transformation from ${\mathbb{R}}^2$ into ${\mathbb{R}}^3$ such that $T({\mathbf{e}}_1)=\left[\begin{array}{c}5\\ -7\\ 2\end{array}\right]$ and $T({\mathbf{e}}_2)=\left[\begin{array}{c}-3\\ 8\\ 0\end{array}\right]$. With no additional information, find a formula for the image of an arbitrary $\mathbf{x}$ in ${\mathbb{R}}^2$.

SOLUTION

$$\begin{array}{rl}T(\mathbf{x})& =T(x_1{\mathbf{e}}_1+x_2{\mathbf{e}}_2)=x_{1}T({\mathbf{e}}_1)+x_{2}T({\mathbf{e}}_2)\\ & =\left[\begin{array}{cc}T({\mathbf{e}}_1)& T({\mathbf{e}}_2)\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\\ & =A\mathbf{x}\end{array}$$

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• The matrix $A$ in (3) is called the standard matrix for the linear transformation $T$.

Geometric Linear Transformations of ${\mathbb{R}}^2$

• Because the transformations are linear, they are determined completely by what they do to the columns of $I_2$.

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EXAMPLE 3

Let $T$ $:$ ${\mathbb{R}}^2\to {\mathbb{R}}^2$ be the transformation that rotates each point in ${\mathbb{R}}^2$ about the origin through an angle $\varphi$, with counterclockwise rotation for a positive angle. We could show geometrically that such a transformation is linear. Find the standard matrix $A$ of this transformation.

SOLUTION

Existence and Uniqueness Questions

• The concept of a linear transformation provides a new way to understand the existence and uniqueness questions asked earlier.

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Existence question (满射)

• “Does $T$ map ${\mathbb{R}}^n$ onto ${\mathbb{R}}^m$?” is an existence question.

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Uniqueness question (单射)

• “Is $T$ one-to-one?” is a uniqueness question.

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EXAMPLE 4

• The projection transformations shown in Table 4 are not one-to-one and do not map ${\mathbb{R}}^2$ onto ${\mathbb{R}}^2$. The transformations in Tables 1, 2, and 3 are one-to-one and do map ${\mathbb{R}}^2$ onto ${\mathbb{R}}^2$.

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EXAMPLE 5

Let $T$ be the linear transformation whose standard matrix is
$$A=\left[\begin{array}{cccc}1& -4& 8& 1\\ 0& 2& -1& 3\\ 0& 0& 0& 5\end{array}\right]$$Does $T$ map ${\mathbb{R}}^4$ onto ${\mathbb{R}}^3$? Is $T$ a one-to-one mapping?

SOLUTION

• Since $A$ happens to be in echelon form, we can see at once that $A$ has a pivot position in each row. Therefore, $T$ maps ${\mathbb{R}}^4$ onto ${\mathbb{R}}^3$.
• However, since the equation $A\mathbf{x}=\mathbf{b}$ has a free variable, each $\mathbf{b}$ is the image of more than one $\mathbf{x}$. That is, $T$ is not one-to-one.

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PROOF

• 必要性: Since $T$ is linear, $T(0)=0$. If T is one-to-one, then the equation $T(\mathbf{x})=0$ has at most one solution and hence only the trivial solution.
• 充分性: If $T$ is not one-to-one, then there is a $\mathbf{b}$ that is the image of at least two different vectors in ${\mathbb{R}}^n$—say, $\mathbf{u}$ and $\mathbf{v}$. But then, since $T$ is linear, $T(\mathbf{u}-\mathbf{v})=T(\mathbf{u})-T(\mathbf{v})=\mathbf{b}-\mathbf{b}=0$. The vector $\mathbf{u}-\mathbf{v}$ is not zero. Hence the equation $T(\mathbf{x})=0$ has more than one solution.

也可以这样证明：设 $A$ 为线性变换 $T$ 对应的标准矩阵，则 $T$ 为单射 $\iff$ $A\mathbf{x}=\mathbf{b}$ 有唯一解 $\iff$ $A\mathbf{x}=0$ 只有平凡解 $\iff$ $T(\mathbf{x})=0$ 只有平凡解

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可以发现，只有当 $T$ 是方阵的时候 $T$ 才可能是双射 ( $T$ 的标准矩阵 $A$ 的每行每列都必须要有主元)

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EXAMPLE 6

Let $T(x_1,x_2)=(3x_1+x_2,5x_1+7x_2,x_1+3x_2)$. Show that $T$ is a one-to-one linear transformation. Does $T$ map ${\mathbb{R}}^2$ onto ${\mathbb{R}}^3$?

SOLUTION

• $T$ is indeed a linear transformation, with its standard matrix $A$ shown in (4).
  
• The columns of $A$ are linearly independent because they are not multiples. By Theorem 12(b), $T$ is one-to-one.
• To decide if $T$ is onto ${\mathbb{R}}^3$, examine the span of the columns of $A$. Since $A$ is $3\times 2$, the columns of $A$ span ${\mathbb{R}}^3$ if and only if $A$ has 3 pivot positions. This is impossible, since $A$ has only 2 columns. So the columns of $A$ do not span ${\mathbb{R}}^3$, and the associated linear transformation is not onto ${\mathbb{R}}^3$.