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

Whenever a linear transformation $T$T arises geometrically or is described in words, we usually want a “formula” for $T(\boldsymbol x)$T(x). The discussion that follows shows that every linear transformation from $\mathbb R^n$Rn to $\mathbb R^m$Rm is actually a matrix transformation $\boldsymbol x \mapsto A\boldsymbol x$x↦Ax.

The key to finding $A$A is to observe that $T$T is completely determined by what it does to the columns of the $n \times n$n×n identity matrix $I_n$In​.

EXAMPLE 1
The columns of $I_2=\begin{bmatrix}1&0\\0&1\end{bmatrix}$I2​=[10​01​] are $\boldsymbol e_1=\begin{bmatrix}1\\0\end{bmatrix}$e1​=[10​] and $\boldsymbol e_2=\begin{bmatrix}0\\1\end{bmatrix}$e2​=[01​]. Suppose $T$T is a linear transformation from $\mathbb R^2$R2 into $\mathbb R^3$R3 such that $\boldsymbol T(e_1)=\begin{bmatrix}5\\-7\\2\end{bmatrix}$T(e1​)=⎣⎡​5−72​⎦⎤​ and $\boldsymbol T(e_2)=\begin{bmatrix}-3\\8\\0\end{bmatrix}$T(e2​)=⎣⎡​−380​⎦⎤​. With no additional information, find a formula for the image of an arbitrary $\boldsymbol x$x in $\mathbb R^2$R2.
SOLUTION




The matrix $A$A in (3) is called the standard matrix for the linear transformation $T$T . (线性变换$T$T 的标准矩阵)

We know now that every linear transformation from $\mathbb R^n$Rn to $\mathbb R^m$Rm can be viewed as a matrix transformation, and vice versa (反之亦然).

EXAMPLE 3
Let $T$T $:$: $\mathbb R^2\rightarrow \mathbb R^2$R2→R2 be the transformation that rotates each point in $\mathbb R^2$R2 about the origin through an angle $\phi$ϕ, with counterclockwise rotation for a positive angle. We could show geometrically that such a transformation is linear. Find the standard matrix $A$A of this transformation.
SOLUTION

Geometric Linear Transformations of $\mathbb R^2$R2

Because the transformations are linear, they are determined completely by what they do to the columns of $I_2$I2​.

剪切变换

Existence and Uniqueness Questions

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

满射

“Does $T$T map $\mathbb R^n$Rn onto $\mathbb R^m$Rm?” is an existence question.

单射

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

The projection transformations shown in Table 4 are not one-to-one and do not map $\mathbb R^2$R2 onto $\mathbb R^2$R2. The transformations in Tables 1, 2, and 3 are one-to-one and do map $\mathbb R^2$R2 onto $\mathbb R^2$R2.

EXAMPLE 4
Let $T$T be the linear transformation whose standard matrix is
$A=\begin{bmatrix}1&-4&8&1\\0&2&-1&3\\0&0&0&5\end{bmatrix}$A=⎣⎡​100​−420​8−10​135​⎦⎤​Does $T$T map $\mathbb R^4$R4 onto $\mathbb R^3$R3? Is $T$T a one-to-one mapping?

SOLUTION
Since $A$A happens to be in echelon form, we can see at once that $A$A has a pivot position in each row. Therefore, $T$T maps $\mathbb R^4$R4 onto $\mathbb R^3$R3.
However, since the equation $A\boldsymbol x = \boldsymbol b$Ax=b has a free variable, each $\boldsymbol b$b is the image of more than one $\boldsymbol x$x. That is, $T$T is not one-to-one.

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

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

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

EXAMPLE 5
Let $T(x_1,x_2)=(3x_1+x_2,5x_1+7x_2,x_1+3x_2)$T(x1​,x2​)=(3x1​+x2​,5x1​+7x2​,x1​+3x2​). Show that $T$T is a one-to-one linear transformation. Does $T$T map $\mathbb R^2$R2 onto $\mathbb R^3$R3?
SOLUTION

So $T$T is indeed a linear transformation, with its standard matrix $A$A shown in (4).
The columns of $A$A are linearly independent because they are not multiples. By Theorem 12(b), $T$T is one-to-one.
To decide if $T$T is onto $\mathbb R^3$R3, examine the span of the columns of $A$A. Since $A$A is $3 \times 2$3×2, the columns of $A$A span $\mathbb R^3$R3 if and only if $A$A has 3 pivot positions. This is impossible, since $A$A has only 2 columns. So the columns of $A$A do not span $\mathbb R^3$R3, and the associated linear transformation is not onto $\mathbb R^3$R3.