1.8 Introduction to linear transformations (线性变换介绍)

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

目录

• Matrix Transformations 矩阵变换
• Linear Transformations 线性变换

The difference between a matrix equation $A\boldsymbol x=\boldsymbol b$Ax=b and the associated vector equation $x_1\boldsymbol a_2+...+x_n\boldsymbol a_n=\boldsymbol b$x1​a2​+...+xn​an​=b is merely a matter of notation.

However, a matrix equation $A\boldsymbol x=\boldsymbol b$Ax=b can arise in linear algebra (and in applications such as computer graphics and signal processing) in a way that is not directly connected with linear combinations of vectors. This happens when we think of the matrix $A$A as an object that “acts” on a vector $\boldsymbol x$x by multiplication to produce a new vector called $A\boldsymbol x$Ax.

For instance, the equations

say that multiplication by $A$A transforms $\boldsymbol x$x into $\boldsymbol b$b and transforms $\boldsymbol u$u into the zero vector. See Figure 1.

From this new point of view, solving the equation $A\boldsymbol x=\boldsymbol b$Ax=b amounts to finding all vectors $\boldsymbol x$x in $\mathbb R^4$R4 that are transformed into the vector $\boldsymbol b$b in $\mathbb R^2$R2 under the “action” of multiplication by $A$A

The correspondence from $\boldsymbol x$x to $A\boldsymbol x$Ax is a $function$function(函数) from one set of vectors to another.

A transformation(变换) (or function(函数) or mapping(映射)) $T$T from $\mathbb R^n$Rn to $\mathbb R^m$Rm is a rule that assigns to each vector $\boldsymbol x$x in $\mathbb R^n$Rn a vector $T(\boldsymbol x)$T(x) in $\mathbb R^m$Rm. The set $\mathbb R^n$Rn is called the domain(定义域) of $T$T , and $\mathbb R^m$Rm is called the codomain(余定义域 / 取值空间) of $T$T . For $\boldsymbol x$x in $\mathbb R^n$Rn, the vector $T(\boldsymbol x)$T(x) in $\mathbb R^m$Rm is called the image of $\boldsymbol x$x(像) (under the action of $T$T ). The set of all images $T(\boldsymbol x)$T(x)is called the range(值域) of $T$T . See Figure 2.

The range of $T$T is the set of all linear combinations of the columns of $A$A

Matrix Transformations 矩阵变换

EXAMPLE 2
If $A=\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}$A=⎣⎡​100​010​000​⎦⎤​, then the transformation $\boldsymbol x\rightarrow A\boldsymbol x$x→Ax projects points in $\mathbb R^3$R3 onto the $x_1x_2$x1​x2​-plane because

投影变换

EXAMPLE 3
Let $A=\begin{bmatrix}1&3\\0&1\end{bmatrix}$A=[10​31​]. The transformation $T:\mathbb R^2\rightarrow \mathbb R^2$T:R2→R2 is called a shear transformation(剪切变换). $T$T deforms the square as if the top of the square were pushed to the right while the base is held fixed. Shear transformations appear in physics, geology(地质学), and crystallography(晶体学).

Linear Transformations 线性变换

Theorem 5 in Section 1.4 shows that if $A$A is $m \times n$m×n, then the transformation $\boldsymbol x\rightarrow A\boldsymbol x$x→Ax has the properties

These properties identify the most important class of transformations in linear algebra.

DEFINITION

Every matrix transformation is a linear transformation. Important examples of linear transformations that are not matrix transformations will be discussed in Chapters 4 and 5.

Property (i) says that the result $T(\boldsymbol u+\boldsymbol v)$T(u+v) of first adding $\boldsymbol u$u and $\boldsymbol v$v in $\mathbb R^n$Rn and then applying $T$T is the same as first applying $T$T to $\boldsymbol u$u and to $\boldsymbol v$v and then adding $T(\boldsymbol u)$T(u) and $T(\boldsymbol v)$T(v) in $\mathbb R^m$Rm. These two properties lead easily to the following useful facts.

如果从群论角度来看，由性质(i)可知，$T$T 实际上可以看作群$<\mathbb R^n,+>$<Rn,+>到群$<\mathbb R^m,+>$<Rm,+>的一个群同态映射，群同态映射保持幺元，从而得到了上面的性质(3)

Observe that if a transformation satisfies (4) for all $\boldsymbol u$u, $\boldsymbol v$v and $c,d$c,d, it must be linear.
(Set $c = d = 1$c=d=1 for preservation of addition, and set $d = 0$d=0 for preservation of scalar multiplication.)

Repeated application of (4) produces a useful generalization(推广):

In engineering and physics, (5) is referred to as a $superposition\ principle$superposition principle(叠加原理). Think of $\boldsymbol v_1,...,\boldsymbol v_p$v1​,...,vp​ as signals that go into a system and $T(\boldsymbol v_1),...,T(\boldsymbol v_p)$T(v1​),...,T(vp​) as the responses of that system to the signals. The system satisfies the superposition principle if whenever an input is expressed as a linear combination of such signals, the system’s response is the same linear combination of the responses to the individual signals. We will return to this idea in Chapter 4.

An affine transformation (仿射变换) $T:\mathbb R^n \rightarrow \mathbb R^m$T:Rn→Rm has the form $T(\boldsymbol x)=A\boldsymbol x+\boldsymbol b$T(x)=Ax+b, with $A$A an $m \times n$m×n matrix and $\boldsymbol b$b in $\mathbb R^m$Rm. $T$T is not a linear transformation when $\boldsymbol b \neq \boldsymbol 0$b​=0. (Affine transformations are important in computer graphics.)

EXAMPLE 4
Given a scalar $r$r, define $T:\mathbb R^2\rightarrow \mathbb R^2$T:R2→R2 by $T(\boldsymbol x)=r\boldsymbol x$T(x)=rx. T is called a contraction(压缩变换) when $0 \leq r \leq 1$0≤r≤1 and a dilation(拉伸变换) when $r > 1$r>1. Let $r = 3$r=3, and show that $T$T is a linear transformation.
SOLUTION


EXAMPLE 5
Define a linear transformation $T:\mathbb R^2\rightarrow \mathbb R^2$T:R2→R2 by
$T(\boldsymbol x)=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}-x_2\\x_1\end{bmatrix}$T(x)=[01​−10​][x1​x2​​]=[−x2​x1​​]
Find the images under $T$T of $\boldsymbol u=\begin{bmatrix}4\\1\end{bmatrix}$u=[41​], $\boldsymbol v=\begin{bmatrix}2\\3\end{bmatrix}$v=[23​], and $\boldsymbol u + \boldsymbol v=\begin{bmatrix}6\\4\end{bmatrix}$u+v=[64​]
SOLUTION

It appears from Figure 6 that $T$T rotates $\boldsymbol u$u, $\boldsymbol v$v, and $\boldsymbol u + \boldsymbol v$u+v counterclockwise about the origin through $90\degree$90°.

EXAMPLE 6
A company manufactures two products, B and C. We construct a “unit cost” matrix, whose columns describe the “costs per dollar of output” for the products:

Let $\boldsymbol x = (x_1,x_2)$x=(x1​,x2​) be a “production” vector, corresponding to $x_1$x1​ dollars of product B and $x_2$x2​ dollars of product C, and define $T:\mathbb R^2\rightarrow \mathbb R^3$T:R2→R3 by

The mapping $T$T transforms a list of production quantities (measured in dollars) into a list of total costs. The linearity of this mapping is reflected in two ways:

1. If production is increased by a factor of, say, 4, from $x$x to $4x$4x, then the costs will increase by the same factor, from $T(\boldsymbol x)$T(x) to $4T(\boldsymbol x)$4T(x).
2. If $\boldsymbol x$x and $\boldsymbol y$y are production vectors, then the total cost vector associated with the combined production $\boldsymbol x+\boldsymbol y$x+y is precisely the sum of the cost vectors $T(\boldsymbol x)$T(x) and $T(\boldsymbol y)$T(y).

EXAMPLE 7

SOLUTION

线性相关与线性变换的联系：如果{$\boldsymbol v_1,\boldsymbol v_2,\boldsymbol v_3$v1​,v2​,v3​}线性相关，$T$T为一线性变换，则{$T(\boldsymbol v_1),T(\boldsymbol v_2),T(\boldsymbol v_3)$T(v1​),T(v2​),T(v3​)}线性相关