3.1 Linear Transformations

从这一章开始，就讨论本书的主题线性变换了。定义很像subspace的定义，线性变换是一个满足 $T(c\alpha+\beta)=cT(\alpha)+T(\beta)$的函数。常见的例子有zero transformation，differentiation transformation，矩阵乘法的transformation，积分transformation等。
如果 $T$是一个linear transformation，那么 $T(0)=0$，且 $T$保持linear combination。Theorem 1说明 $V$的一组基可以用唯一的线性变换指到 $W$的任何一组 $\dim V$数量的向量上，特别地，任意一个linear transformation $T$ 是唯一地由其 images of standard basis 决定。并且可以使用由 images of standard basis 为行组成的矩阵来显式表示出来。
接下来讨论的是range和null space，二者都是subspace，由定义可以证。range的dimension称为rank of T，null space的dimension称为nullity of T。Theorem 2是线性代数中一个很重要的定理： $rank(T)+nullity(T)=\dim V$。由此可以得到关于矩阵的Theorem 3：矩阵的行rank和列rank相同。证明中要使用到方程组解空间的维数是 $n-r$，其中 $r$为系数矩阵行空间dimension的结论。

Exercises

1. Which of the following functions $T$ from $R^2$ into $R^2$ are linear transformations?

( a ) $T(x_1,x_2)=(1+x_1,x_2)$

( b ) $T(x_1,x_2)=(x_2,x_1)$

( c ) $T(x_1,x_2)=(x_1^2,x_2)$

( d ) $T(x_1,x_2)=(\sin x_1,x_2)$

( e ) $T(x_1,x_2)=(x_1-x_2,0)$

Solution:
( a ) No, since $T(2(1,0))=T(2,0)=(1+2,0)=(3,0)\neq 2(2,0)=2T(1,0)$.
( b ) Yes, since
$\begin{aligned}T(c(x_1,x_2)+(y_1,y_2))&=(cx_2+y_2,cx_1+y_1)\\&=c(x_2,x_1)+(y_2,y_1)\\&=cT(x_1,x_2)+T(y_1,y_2)\end{aligned}$
( c ) No.
( d ) No.
( e ) Yes, since
$\begin{aligned}T(c(x_1,x_2)+(y_1,y_2))&=(cx_1+y_1-cx_2-y_2,0)\\&=c(x_1-x_2,0)+(y_1-y_2,0)\\&=cT(x_1,x_2)+T(y_1,y_2)\end{aligned}$

2. Find the range, rank, null space, and nullity for the zero transformation and the identity transformation on a finite-dimensional space $V$.

Solution: Let $Z$ be the linear transformation, and $I$ be the identity transformation, then
the range of $Z$ is 0, the rank of $Z$ is 0, the null space of $Z$ is $V$, the nullity of $Z$ is $\dim V$.
the range of $I$ is $V$, the rank of $I$ is $\dim V$, the null space of $I$ is 0, the nullity of $I$ is 0.

3. Describe the range and the null space for the differentiation transformation of Example 2. Do the same for the integration transformation of Example 5.

Solution:
For Example 2, range $D=V$, null $D=\{f(x):f(x)=k,k\in F\}$.
For Example 5, null $T=\{0\}$, range $T=\{f:f \text{ has a continuous first derivative}\}$.

4. Is there a linear transformation $T$ from $R^3$ into $R^2$ such that $T(1,-1,1)=(1,0)$ and $T(1,1,1)=(0,1)$?

Solution: Let $T$ be defined as
$T(1,-1,1)=(1,0),\quad T(1,1,1)=(0,1),\quad T(0,0,1)=(0,0)$
Since $(1,-1,1),(1,1,1),(0,0,1)$ is a basis for $R^3$, $T$ is well defined.

5. If

$\alpha_1=(1,-1),\quad \beta_1=(1,0) \\ \alpha_2=(2,-1),\quad \beta_2=(0,1) \\\alpha_3=(-3,2),\quad \beta_3=(1,1)$

is there a linear transformation $T$ from $R^2$ into $R^2$ such that $T\alpha_i=\beta_i$ for $i=1,2,3$?

Solution: Assume such a $T$ exists, then $T(-\alpha_1-\alpha_2 )=-T\alpha_1-T\alpha_2$, notice that $-\alpha_1-\alpha_2=\alpha_3$, we have
$\beta_3=T\alpha_3=-T\alpha_1-T\alpha_2=-\beta_1-\beta_2=-(1,1)=-\beta_3$
this means $\beta_3=(0,0)$, a contradiction.

6. Describe explicitly (as in Exercises 1 and 2) the linear transformation $T$ from $F^2$ into $F^2$ such that $T\epsilon_1=(a,b),T\epsilon_2=(c,d)$.

Solution:
$\begin{aligned}T(x_1,x_2 )&=x_1 T\epsilon_1+x_2 T\epsilon_2=x_1 (a,b)+x_2 (c,d)\\&=(x_1 a+x_2 c,x_1 b+x_2 d)\end{aligned}$
If $a=b=c=d=0$, then $T$ is the zero transformation.
If at least one of $a,b,c,d$ is not 0, but $ad-bc=0$, then the rank of $T$ and the nullity of $T$ are both 1. The range of $T$ is $\{(0,y):y\in R\}$ if $a=c=0$, $\{(x,0):x\in R\}$ if $b=d=0$, $\{k(c,d):k\in R\}$ if $a=b=0$, $\{k(a,b):k\in R\}$ if $c=d=0$, and $\{k(c,d):k\in R\}$ otherwise.
If $ad-bc\neq 0$, then the rank of $T$ is 2 and the nullity of $T$ is 0, The range of $T$ is $R^2$, the null space of $T$ is $\{0\}$.

7. Let $F$ be a subfield of the complex numbers and let $T$ be the function from $F^3$ into $F^3$ defined by

$T(x_1,x_2,x_3)=(x_1-x_2+2x_3,2x_1+x_2,-x_1-2x_2+2x_3).$

( a ) Verify that $T$ is a linear transformation.

( b ) If $(a,b,c)$ is a vector in $F^3$, what are the conditions on $a,b,c$ that the vectors be in the range of $T$? What is the rank of $T$?

( c ) What are the conditions on on $a,b,c$ that $(a,b,c)$ be in the null space of $T$? What is the nullity of $T$?

Solution:
( a ) Let $x=(x_1,x_2,x_3),y=(y_1,y_2,y_3)$, then it’s easy to see
$T(cx+y)=T(cx_1+y_1,cx_2+y_2,cx_3+y_3 )=cTx+Ty$
( b ) The conditions are $a-b=c$, the rank of $T$ is 2.
( c ) $(a,b,c)=(2t,-4t,-3t),t\in R$, the nullity of $T$ is 1.

8. Describe explicitly a linear transformation from $R^3$ into $R^3$ which has as its range the subspace spaned by $(1,0,-1)$ and $(1,2,2)$.

Solution: One possible choice of $T$ can be
$T(x_1,x_2,x_3 )=(x_1+x_2,2x_2+2x_3,-x_1+2x_2+3x_3)$
Since $T\epsilon_1=(1,0,-1),T\epsilon_2=(1,2,2),T\epsilon_3=(0,2,3)=T\epsilon_2-T\epsilon_1$, the condition is satisfied.

9. Let $V$ be the vector space of all $n\times n$ matrices over the field $F$, and let $B$ be a fixed $n\times n$ matrix. If

$T(A)=AB-BA$

verify that $T$ is a linear transformation from $V$ into $V$.

Solution: Let $A,D$ be any matrix in $V$, then
$\begin{aligned}T(cA+D)&=(cA+D)B-B(cA+D)\\&=cAB+DB-cBA-BD\\&=c(AB-BA)+(DB-BD)\\&=cT(A)-T(D)\end{aligned}$

10. Let $V$ be the set of all complex numbers regarded as a vector space over the field of real numbers (usual operations). Find a function from $V$ into $V$ which is a linear transformation on the above vector space, but which is not a linear transformation on $C^1$, i.e., which is not complex linear.

Solution: Let f be defined as $f(a+bi)=a$ for $a+bi\in C$ (in which $a,b\in R$), then $f$ is a linear transformation from $V$ into $V$. To see it’s not a transformation on $C^1$, we have $f(1+i)=1$, and $f(i)=0$, but for $c=1+i$ we have
$f(c(1+i)+i)=f(3i)=0\neq cf(1+i)+f(i)$

11. Let $V$ be the space of $n\times 1$ matrices over $F$ and let $W$ be the space of $m\times 1$ matrices over $F$. Let $A$ be a fixed $m\times n$ matrix over $F$ and let $T$ be the linear transformation from $V$ into $W$ defined by $T(X)=AX$. Prove that $T$ is the zero transformation if and only if $A$ is the zero matrix.

Solution: If $A$ is the zero matrix, then obviously $T$ is the zero transformation. Now suppose $T$ is the zero transformation and assume $A\neq 0$, then at least one $a_{ij}\neq 0$, here $1\leq i\leq m,1\leq j\leq n$, we let $X=a_{ij} \epsilon_j$ in $V$, the $j$-th coordinate of $X$ is $a_{ij}$ and the other coordinates are $0$, then the $i$-th coordinate of $AX$ is $a_{ij}^2\neq 0$, thus $T(X)=AX\neq 0$, a contradiction.

12. Let $V$ be an $n$-dimensional vector space over the field $F$ and let $T$ be a linear transformation from $V$ into $V$ such that the range and null space of $T$ are identical. Prove that $n$ is even. (Can you give an example of such a linear transformation $T$)?

Solution: We have $rank(T)=nullity(T)$, since the range of $T$ and the null space of $T$ are equal, thus
$n=\dim V=rank(T)+nullity(T)=2rank(T)$
so $n$ is even. One example may be $T(x_1,x_2 )=(x_2,0)$.

13. Let $V$ be a vector space over the field $F$ and $T$ a linear transformation from $V$ into $V$. Prove that the following two statements about $T$ are equivalent.

( a ) The intersection of the range of $T$ and the null space of $T$ is the zero subspace of $V$.

( b ) If $T(T\alpha)=0$, then $T(\alpha)=0$.

Solution: Let $A$ be the range of $T$ and $B$ be the null space of $T$.
First suppose (a) is true, then $A\cap B=\{0\}$, if $T(T\alpha)=0$, then $T\alpha\in B$, but obviously $T\alpha\in A$, thus $T\alpha\in A\cap B$ and $T\alpha=0$.
Conversely, suppose $T(T\alpha)=0$ means $T\alpha=0$, and let $x\in A\cap B$, then $Tx=0$, and there’s $\beta\in V$ s.t. $T\beta=x$, so we have $T(T\beta)=Tx=0$, so $T\beta=0=x$, thus $A\cap B=\{0\}$.