1. Suppose $T\in\ca L(U, V)$ and $S\in\ca L(V, W)$ are both invertible linear maps. Prove that $ST\in\ca L(U, W)$ / is invertible and that $(ST)^{-1}=T^{-1}S^{-1}$.

Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 22. It is almost the same.

2. Suppose $V$ is finite-dimensional and $\dim V > 1$. Prove that the set of noninvertible operators on $V$ is not a subspace of $\ca L(V) $.

Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 25.

3. Suppose $V$ is finite-dimensional, $U$ is a subspace of $V$, and $S\in\ca L(U,V)$. Prove there exists an invertible operator $T\in\ca L(V)$ such that $Tu=Su$ for every $u\in U$ if and only if $S$ is injective.

Solution: If there exists an invertible operator $T\in\ca L(V)$ such that $Tu=Su$ for every $u\in U$, then $S$ is injective since $T$ is injective.

If $S$ is injective. Assume $u_1$, $\cdots$, $u_m$ is a basis of $U$, we can extend it to a basis of $V$ as $u_1$, $\cdots$, $u_m$, $v_{m+1}$, $\cdots$, $v_{n}$. Since $S$ is injective, by Problem 9 of Exercises 3B, we have $Su_1$, $\cdots$, $Su_m$ is linearly independent in $V$. Hence we can extend it to a basis of $V$ as $Su_1$, $\cdots$, $Su_m$, $w_{m+1}$, $\cdots$, $w_{n}$. Define $T\in \ca L(V)$ as below \[Tu_i=Su_i,\quad Tv_{j}=w_j, \quad 1\le i\le m, m+1\le j \le n.\]The existence of $T$ is guaranteed by 3.5(unique). Then for any $u=a_1u_1+\cdots+a_mu_m$, $a_i\in\mb F$, we have \begin{align*} Tu=&T(a_1u_1+\cdots+a_mu_m)\\ =&a_1Tu_1+\cdots+a_mTu_m\\ =&a_1Su_1+\cdots+a_mSu_m\\ =&S(a_1u_1+\cdots+a_mu_m)=Su. \end{align*}Moreover, $T$ is surjective by Problem 10 of Exercises 3B hence invertible by 3.69.

Compare this problem with Problem 11 of Exercises 3A.

4. Suppose $W$ is finite-dimensional and $T_1, T_2\in\ca L(V,W)$. Prove that null $T_1$= null $T_2$ if and only if there exists an invertible operator $S\in\ca L(W)$ such that $T_1=ST_2$.

Solution: If we assume $\m{null} T_1 =\m{null} T_2$. Since $W$ is finite-dimensional, so is range $T_2$. Let $w_1$, $\cdots$, $w_n$ be a basis of $\m{range} T_2$, then there exist $v_1$, $\cdots$, $v_n\in V$ such that \[ T_2v_i=w_i,\quad i=1,\cdots,n. \]Now we will show that $V=\m{null}T_2\oplus \m{span}(v_1,\cdots,v_n)$. For any $v\in V$, we \[ T_2v=a_1w_1+\cdots+a_nw_n \]for some $a_1$, $\cdots$, $a_n\in\mb F$. Hence \[ T_2(v-a_1v_1-\cdots-a_nv_n)=0, \]namely \[ v=(v-a_1v_1-\cdots-a_nv_n)+(a_1v_1+\cdots+a_nv_n), \]this implies $V=\m{null}T_2 +\m{span}(v_1,\cdots,v_n)$. Moreover, if $a_1v_1+\cdots+a_nv_n\in \m{null}T_2$, then we have \[ T_2(a_1v_1+\cdots+a_nv_n)=a_1w_1+\cdots+a_nw_n=0. \]Note that $w_1$, $\cdots$, $w_n$ is linearly independent, it follows $a_1=\cdots=a_n=0$. Thus we have\[V=\m{null}T_2 \oplus\m{span}(v_1,\cdots,v_n).\] Similarly, $T_1v_1$, $\cdots$, $T_1v_n$ is linearly independent. For if $a_1T_1v_1+\cdots+a_nT_1v_n=0$, we have \[ T_1(a_1v_1+\cdots+a_nv_n)=0. \]Note that $\m{null} T_1 =\m{null} T_2$, it follows that \[ 0=T_2(a_1v_1+\cdots+a_nv_n)=a_1w_1+\cdots+a_nw_n.\]Thus $a_1=\cdots=a_n=0$. Now extend $w_1$, $\cdots$, $w_n$ to a basis of $W$ as $w_1$, $\cdots$, $w_n$, $e_1$, $\cdots$, $e_m$ and $T_1v_1$, $\cdots$, $T_1v_n$ to a basis of $W$ as $T_1v_1$, $\cdots$, $T_1v_n$, $f_1$, $\cdots$, $f_m$. Define $S\in\ca L(W)$ by \[Sw_i=T_1v_i,Se_j=f_j,i=1,\cdots,n;j=1,\cdots,m.\]Note that \[ V=\m{null}T_2\oplus \m{span}(v_1,\cdots,v_n), \]any $v\in V$ can be expressed as \[v=v_{\m{null}}+a_1v_1+\cdots+a_nv_n,\]where $v_{\m{null}}\in\m{null}T_2=\m{null}T_1$ and $a_1$, $\cdots$, $a_n\in\mb F$. Hence we have \begin{align*} ST_2(v)=&ST_2(v_{\m{null}}+a_1v_1+\cdots+a_nv_n)\\ =&ST_2(a_1v_1+\cdots+a_nv_n)\\ =&S(a_1w_1+\cdots+a_nw_n)\\ =&a_1T_1v_1+\cdots+a_nT_1v_n\\ =&T_1(a_1v_1+\cdots+a_nv_n)\\ =&T_1(v_{\m{null}}+a_1v_1+\cdots+a_nv_n)=T_1(v) \end{align*}namely $ST_2=T_1$. Moreover, $S$ is surjective by Problem 10 of Exercises 3B hence invertible by 3.69.

If there exists an invertible operator $S\in\ca L(W)$ such that $ST_2=T_1$, then for any $\mu\in\m{null} T_1$, we have \[ST_2\mu=T_1\mu=0.\]As $S$ is invertible, we have $T_2\mu=0$. Hence $\mu\in \m{null} T_2$, it follows that $\m{null} T_1 \subset \m{null} T_2$. Similarly, consider $T_2=S^{-1}T_1$, $\m{null} T_2 \subset \m{null} T_1$. Thus we conclude null $T_1$= null $T_2$.

Compare this problem with Problem 24 of Exercises 3B.

5. Suppose $V$ is finite-dimensional and $T_1,T_2\in \ca L(V,W)$. Prove that range $T_1=$ range $T_2$ if and only if there exists an invertible operator $S\in\ca L(V) $ such that $T_1=T_2S$.

Solution: If we assume $\m{range} T_1 = \m{range} T_2$. Let $u_1$, $\cdots$, $u_m$ be a basis of null $T_1$, then we can extend it to a basis of $V$ as $u_1$, $\cdots$, $u_m$, $w_1$, $\cdots$, $w_n$. Then range $T_1$ is $\m{span}(T_1w_1,\cdots,T_1w_n)$ and $T_1w_1,\cdots,T_1w_n$ is linearly independent. There exist $v_1$, $\cdots$, $v_n\in V$ such that $T_1w_i=T_2v_i$ for $i=1,\cdots,n$ since $\m{range} T_1=\m{range} T_2$. As $T_1w_1,\cdots,T_1w_n$ is linearly independent, it follows that $v_1$, $\cdots$, $v_n$ is linearly independent by Problem 4 of Exercises 3A. Note that range $T_1=$ range $T_2$ implies null $T_1$ and null $T_2$ have the same dimension. Let $\zeta_1$, $\cdots$, $\zeta_m$ be a basis of null $T_2$, then $\zeta_1$, $\cdots$, $\zeta_m$, $v_1$, $\cdots$, $v_n$ is a basis of $V$ by the proof of 3.22. Define $S\in\ca L(V) $ by $Su_i=\zeta_i$ and $Sw_j=v_j$, then we have \[T_1w_j=T_2v_j=T_2Sw_j,\quad j=1,\cdots,n,\]and \[T_1u_i=0=T_2\zeta_i=T_2Su_i,\quad i =1,\cdots, m,\]hence $T_1=T_2S$ by uniqueness in 3.5. Moreover, $S$ is surjective by Problem 10 of Exercises 3B hence invertible by 3.69.

If there exists an invertible operator $S\in\ca L(V) $ such that $T_1=T_2S$, then for any $\mu\in V$, we have \[T_1\mu=T_2S\mu\in \m{range} T_2.\]Hence $\m{range} T_1 \subset \m{range} T_2$. As $S$ is invertible, we have $T_2=T_1S^{-1}$. Similarly, we conclude $\m{range} T_1 \subset \m{range} T_2$. Thus range $T_1=$ range $T_2$.

Compare this problem with Problem 25 of Exercises 3B.

6. Suppose $V$ and $W$ are finite-dimensional and $T_1,T_2\in\ca L(V,W)$. Prove that there exist invertible operators $R\in\ca L(V)$ and $S\in\ca L(W)$ such that $T_1=ST_2R$ if and only if $\dim$ null $T_1= \dim$ null $T_2$.

Solution: If there exist invertible operators $R\in\ca L(V)$ and $S\in\ca L(W)$ such that $T_1=ST_2R$, then $S^{-1}T_1=T_2R$. Hence null $T_1$= null $T_2R$ by Problem 4. Note that we have range $T_2R$= range $T_2$ by Problem 5, it follows that \[\dim\m{null} T_2R=\dim V-\dim\m{range} T_2R=\dim V-\dim\m{range} T_2=\dim\m{null} T_2.\]Hence $\dim\m{null} T_1=\dim\m{null} T_2$.

Conversely, if $\dim$ null $T_1= \dim$ null $T_2$. Let $u_1$, $\cdots$, $u_m$ be a basis of null $T_1$, then we can extend it to a basis of $V$ as $u_1$, $\cdots$, $u_m$, $w_1$, $\cdots$, $w_n$. Let $v_1$, $\cdots$, $v_m$ be a basis of null $T_2$, then we can extend it to a basis of $V$ as $v_1$, $\cdots$, $v_m$, $\zeta_1$, $\cdots$, $\zeta_n$. Then $T_1w_1$, $\cdots$, $T_1w_n$ is linearly independent in $W$, hence we can extend it to a basis of $V$ as $T_1w_1$, $\cdots$, $T_1w_n$, $\alpha_1$, $\cdots$, $\alpha_l$. Similarly, $T_2\zeta_1$, $\cdots$, $T_2\zeta_n$ is linearly independent in $W$, hence we can extend it to a basis of $V$ as $T_2\zeta_1$, $\cdots$, $T_2\zeta_n$, $\beta_1$, $\cdots$, $\beta_l$. Define $R\in\ca L(V)$ by \[Ru_i=v_i,Rw_j=\zeta_j,i=1,\cdots,m;j=1,\cdots,n.\]Define $S\in\ca L(W)$ by \[ST_2\zeta_j=T_1w_j,S\beta_k=\alpha_k,j=1,\cdots,n;k=1,\cdots,l.\]Since $S$ and $T$ map basis to basis, hence $S$ and $T$ are invertible(surjective). Moreover, it is easy to check $T_1u_i=0=ST_2Ru_i$ and \[ T_1w_j=ST_2\zeta_j=ST_2Rw_j, \]hence $T_1=ST_2R$.

Compare this problem with Problem 4 and Problem 5.

7. Suppose $V$ and $W$ are finite-dimensional. Let $v\in V$. Let \[E=\{T \in\ca L(V,W): T v= 0\}.\] (a) Show that $E$ is a subspace of $\ca L(V,W)$. (b) Suppose $v\ne 0$. What is $\dim E$?

Solution: a) Let $T,S\in E$, then \[(T+S)v=Tv+Sv=0+0=0,\]hence $T+S\in E$, namely $E$ is closed under addition. For every $\lambda\in \mb F$, \[(\lambda T)v=\lambda(Tv)=\lambda 0=0,\]hence $\lambda T\in E$, namely $E$ is closed under scalar multiplication. Therefore $E$ is a subspace of $\ca L(V,W)$.

(b) Since $v\ne 0$, we can extend it to a basis of $V$, namely $v$, $v_2$, $\cdots$, $v_n$. Let $w_1$, $\cdots$, $w_m$ be a basis of $W$. Under these bases, we have a isometric between $\ca L(V,W)$ and $\mb F^{m,n}$ by 3.60. Moreover, $Tv=0$ if and only if the first column vector of $\ca M(T)$ is zero. Hence $\dim E$ is exactly all matrices in $\mb F^{m,n}$ such the first column vector is zero. Now , it is easily seen that\[\dim E=m(n-1)=\dim W(\dim V-1).\]Or it is equivalent to the dimension all linear map from $\m{span}(v_2,\cdots,v_n)$ to $W$. Then we can use 3.61.

8. Suppose $V$ is finite-dimensional and $T: V \to W$ is a surjective linear map of $V$ onto $W$. Prove that there is a subspace $U$ of $V$ such that $T|_U$ is an isomorphism of $U$ onto $W$. (Here $T|_U$ means the function $T$ restricted to $U$. In other words, $T|_U$ is the function whose domain is $U$, with $T|_U$ defined by $T|_U(u)=Tu$ for every $u\in U$.)

Solution: Let $w_1$, $\cdots$, $w_n$ be a basis of $W$. Since $T$ is surjective, there exist $v_1$, $\cdots$, $v_n$ such that $Tv_i=w_i$. Moreover, by Problem 4 of Exercises 3A, it follows that $v_1$, $\cdots$, $v_n$ is linearly independent. Consider $U=\m{span}(v_1,\cdots,v_n)$, then $T|_U$ maps a basis of $U$ to a basis of $W$. Hence $T|_U$ is an isomorphism of $U$ onto $W$.

9. Suppose $V$ is finite-dimensional and $S, T \in\ca L(V)$. Prove that $ST$ is invertible if and only if both $S$ and $T$ are invertible.

Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 22.

10. Suppose $V$ is finite-dimensional and $S, T \in\ca L(V)$. Prove that $ST=I$ if and only if $TS=I$.

Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 23.

11. Suppose $V$ is finite-dimensional and $S, T ,U\in\ca L(V)$ and $ST U = I$. Show that $T$ is invertible and that $T^{-1}=US$.

Solution: By Problem 9, we have $TU$ is invertible. Again by Problem 9, we have $T$ is invertible. Multiply both side of $STU=I$ by $S^{-1}$ on left, we get $TU=S^{-1}$. Multiply both side of $TU=S^{-1}$ by $S$ on right, we have $TUS=I$. Multiply both side of $TUS=I$ by $T^{-1}$ on left, we get $US=T^{-1}$.

12. Show that the result in the previous exercise can fail without the hypothesis that $V$ is finite-dimensional.

Solution: Consider $V=\C^\infty$. Define $T$, $S$, $U$ by \[ T(z_1,z_2,z_3,\cdots)=(0,z_1,z_2,z_3,\cdots) \]\[ S(z_1,z_2,z_3,\cdots)=(z_2,z_3,z_4,\cdots) \]and $U=I$. Then $STU=I$. However $T$ is not surjective.

13. Suppose $V$ is a finite-dimensional vector space and $R,S, T\in\ca L(V)$ are such that $RST$ is surjective. Prove that $S$ is injective.

Solution: Since $V$ is finite-dimensional, the surjectivity of $RST$ implies $RST$ is invertible. Hence $S$ is invertible by Problem 11.

14. Suppose $v_1$, $\cdots$, $v_n$ is a basis of $V$. Prove that the map $T: V\to \mb F^{n,1}$ defined by \[Tv=\ca M(v)\] is an isomorphism of $V$ onto $\mb F^{n,1}$, here $\ca M(v)$ is the matrix of $v \in V$ with respect to the basis $v_1$, $\cdots$, $v_n$.

Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 20.

15. Prove that every linear map from $\mb F^{n,1}$ to $\mb F^{m,1}$ is given by a matrix multiplication. In other words, prove that if $T\in\ca L(\mb F^{n,1},\mb F^{m,1})$, then there exists an $m$-by-$n$ matrix $A$ such that $Tx=Ax$ for every $x\in \mb F^{n,1}$.

Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 21.

16. Suppose $V$ is finite-dimensional and $T\in \ca L(V)$. Prove that $T$ is a scalar multiple of the identity if and only if $ST=TS$ for every $S\in\ca L(V)$.

Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 24.

17. Suppose $V$ is finite-dimensional and $E$ is a subspace of $\ca L(V)$ such that $ST \in E$ and $TS \in E$ for all $S\in \ca L(V)$ and all $T \in E$. Prove that $E=\{0\}$ or $E=\ca L(V)$.

Solution: Let $e_1$, $\cdots$, $e_n$ be a basis of $V$. Suppose $E\ne \{0\}$, then there is nonzero $T\in E$. This implies there is some $s\in\{1,\cdots,n\}$ such that $Te_s\ne 0$. Let \[Te_s=a_1e_1+\cdots+a_ne_n,\quad a_j\in\mb F,j=1,\cdots,n.\]Since $Te_s\ne 0$, there is some $t\in\{1,\cdots,n\}$ such that $a_t\ne 0$. Define $E_{ij}\in \ca L(V)$ by $E_{ij}e_k=\delta_{ik}e_j$, where $\delta_{ik}=0$ if $i\ne k$ and $\delta_{ii}=0$ for every $i$.

By definition, for every $i\in\{1,\cdots,n\}$, we have \[ E_{ti}TE_{is}e_j=a_t\delta_{ij}e_i,\quad j=1,\cdots,n. \]By assumption, $E_{ti}TE_{is}\in E$. Note that $E$ is a subspace of $\ca L(V)$. It follows that \[ (E_{t1}TE_{1s}+\cdots+E_{tn}TE_{ns})e_j=a_te_j,\quad j=1,\cdots,n, \]and $(E_{t1}TE_{1s}+\cdots+E_{tn}TE_{ns})\in E$. This implies $a_jI\in E$, hence $I\in E$. Therefore for any $S\in \ca L(V)$, $S=SI\in E$, namely $E=\ca L(V)$.

18. Show that $V$ and $\ca L(\mb F, V)$ are isomorphic vector spaces.

Solution: For given $v\in V$, define $\vp_v:\mb F\to V$ by $\vp_v(\lambda)=\lambda v$. Then $\vp_v\in \ca L(\mb F, V)$ (check it). Hence we can define $\vp: V\to \ca L(\mb F, V)$ by $v\mapsto \vp_v$. It suffices to show $\vp$ is an isomorphic from $V$ to $\ca L(\mb F, V)$. First, we should check $\vp$ is linear. For every $v_1,v_2\in V$ and $\lambda,a\in\mb F$, we have \begin{align*} \vp_{v_1+\lambda v_2}(a)=&a(v_1+\lambda v_2)=av_1+\lambda av_2\\=&\vp_{v_1}(a)+(\lambda\vp_{v_2})(a)=(\vp_{v_1}+\lambda\vp_{v_2})(a). \end{align*}Hence $\vp$ is linear. Then $\vp_v\equiv 0$ imples $av=0$ for every $a\in\mb F$, thus $v=0$. We conclude injectivity. For any $f\in \ca L(\mb F, V)$, if $f(1)=v$ then $f(\lambda)=\lambda v=\vp_v(\lambda)$. Hence every $f\in \ca L(\mb F, V)$ can be express as $\vp_{f(1)}$, namely $\vp$ is surjective.

19. Suppose $T\in\ca L(\ca P(\R))$ is such that $T$ is injective and $\deg Tp\le \deg p$ for every nonzero polynomial $p\in\ca P(\R)$.

(a) Prove that $T$ is surjective.

(b) Prove that $\deg Tp= \deg p$ for every nonzero $p\in\ca P(\R)$.

Solution: (a) Since $\deg Tp\le \deg p$ for every nonzero polynomial $p\in\ca P(\R)$, we have $T|_{\ca P_n(\R)}:\ca P_n(\R)\to \ca P_n(\R)$ for any $n\in \mb N^+$. Note that $T$ is injective, hence $T|_{\ca P_n(\R)}:\ca P_n(\R)\to \ca P_n(\R)$ is injective. Note that $\ca P_n(\R)$ is finite-dimensional, we conclude $T|_{\ca P_n(\R)}:\ca P_n(\R)\to \ca P_n(\R)$ is surjective by 3.56. Hence $T$ is surjective as any polynomial must be contained in some $\ca P_n(\R)$.

(b) We argue it by induction on $\deg p$. It is true for $\deg =0$. Suppose it is true for $n$. As $T|_{\ca P_n(\R)}:\ca P_n(\R)\to \ca P_n(\R)$ is surjective, every polynomial $p$ with degree $\le n$ can be attained by $Tq$ for some $q\in \ca P_n(\R)$. Moreover, $T|_{\ca P_{n+1}(\R)}:\ca P_{n+1}(\R)\to \ca P_{n+1}(\R)$ is surjective, every polynomial $p$ with degree $\le n+1$ can be attained by $Tq$ for some $q\in \ca P_{n+1}(\R)$. If there exist $p\in \ca P_{n+1}(\R)$ with degree $n+1$ such that $\deg Tp<n+1$. Then there exist some $q\in \ca P_n(\R)$ such that $Tq=Tp$ since $T|_{\ca P_n(\R)}:\ca P_n(\R)\to \ca P_n(\R)$ is surjective. Hence $Tp=Tq$. Note that $T$ is injective and $p\ne q$ (different degrees), we get a contradiction. Hence $\deg Tp= \deg p$ for every nonzero $p\in\ca P_{n+1}(\R)$ with degree $n+1$. The proof is complete.

20. Suppose $n$ is a positive integer and $A_{i,j}\in\mb F$ for $i, j = 1,\cdots,n$. Prove that the following are equivalent (note that in both parts below, the number of equations equals the number of variables):

(a) The trivial solution $x_1=\cdots=x_n=0$ is the only solution to the homogeneous system of equations \[\sum_{k=1}^nA_{1,k}x_k=0\] \[\cdots\] \[\sum_{k=1}^nA_{n,k}x_k=0.\](b) For every $c_1,\cdots,c_n\in\mb F$, there exists a solution to the system of equations\[\sum_{k=1}^nA_{1,k}x_k=c_1\] \[\cdots\] \[\sum_{k=1}^nA_{n,k}x_k=c_n.\] Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 26.