--- **Problem 1:** Give an example of a linear map $T$ such that $\dim \operatorname{null} T = 3$ and $\dim \operatorname{range} T = 2$. Suppose $T\in \mathcal{L}(\mathbb{R}^{5}, \mathbb{R}^{5})$ with $T(x_{1},x_{2},x_{3},x_{4},x_{5})=(0,0,0,x_{4},x_{5})$. --- **Problem 2:** Suppose $V$ is a vector space and $S, T \in \mathcal{L}(V, V)$ are such that $\operatorname{range} S \subset \operatorname{null} T.$ Prove that $(ST)^2 = 0$. Consider $STSTv$. **Case 1**: $Tv=0$, then $STSTv=STS(0)=0$. **Case 2:** $Tv\neq{}0$. **Case 2A**: $Tv\in \text{null } S$. Then $S(Tv)=0$ and $STSTv=ST(0)=0$. **Case 2B**: $Tv\not\in \text{null } S$. Then $STv\neq{}0$ and $STv\in \text{range }S\subset \text{null }T$. Thus $T(STv)\in \text{null }T$. Thus, $STSTv=S(TSTv)=S(0)=0.$ --- **Problem 3:** Suppose $v_1, \ldots, v_m$ is a list of vectors in $V$. Define $T \in \mathcal{L}(\mathbf{F}^m, V)$ by $T(z_1, \ldots, z_m) = z_1 v_1 + \cdots + z_m v_m.$ **(a)** What property of $T$ corresponds to $v_1, \ldots, v_m$ spanning $V$? **(b)** What property of $T$ corresponds to $v_1, \ldots, v_m$ being linearly independent? **(a)** If $T$ is surjective. **(b)** if $T$ is injective. Then only $z_{1}=\ldots=z_{m}=0$ maps to 0. --- **Problem 4:** Show that ${T \in \mathcal{L}(\mathbf{R}^5, \mathbf{R}^4) : \dim \operatorname{null} T > 2}$ is not a subspace of $\mathcal{L}(\mathbf{R}^5, \mathbf{R}^4)$. Suppose $T\in \mathcal{L}(\mathbb{R}^{5}, \mathbb{R}^{4})$ with $T(x_{1},x_{2},x_{3},x_{4},x_{5})=(0,0,x_{3},x_{4})$. Suppose $S\in \mathcal{L}(\mathbb{R}^{5}, \mathbb{R}^{4})$ with $S(x_{1},x_{2},x_{3},x_{4},x_{5})=(x_{1},x_{2},0,0)$. Consider $(S+T)v$ with $v=(x_{1},x_{2},x_{3},x_{4},x_{5})$. $(S+T)v=Sv+Tv=(x_{1},x_{2},x_{3},x_{4})$. Thus $\text{dim null }(S+T)=1$ and is not in the set $\{T \in \mathcal{L}(\mathbf{R}^5, \mathbf{R}^4) : \dim \operatorname{null} T > 2\}$. --- **Problem 5:** Give an example of a linear map $T : \mathbf{R}^4 \to \mathbf{R}^4$ such that $\operatorname{range} T = \operatorname{null} T.$ Suppose $T\in \mathcal{L}(\mathbb{R}^{4}, \mathbb{R}^{4})$ with $T(x_{1},x_{2},x_{3},x_{4})=(0,0,x_{1},x_{2})$. - Range $T = (0,0,x,y) : x,y\in \mathbb{R}$ - Null $T = (0,0,x,y) : x,y\in \mathbb{R}$ --- **Problem 6:** Prove that there does not exist a linear map $T : \mathbf{R}^5 \to \mathbf{R}^5$ such that $\operatorname{range} T = \operatorname{null} T.$ Suppose there exists a map $T : \mathbb{R}^{5}\to \mathbb{R}^{5}$ such that range $T$ = null $T$. Then, substituting into $\text{dim }\mathbb{R}^{5}=\text{range }T +\text{null }T$ we get $5=x+x\implies x=2.5\not\in \mathbb{Z}$. Thus is not possible. --- **Problem 7:** Suppose $V$ and $W$ are finite-dimensional with $2 \leq \dim V \leq \dim W$. Show that $U=\{T \in \mathcal{L}(V, W) : T \text{ is not injective}\}$ is not a subspace of $\mathcal{L}(V, W)$. Let $V=W=\mathbb{R}^{2}$ . Suppose $T\in U$ with $T(x,y)=(0,y)$ and $S\in U$ with $S(x,y)=(x,0)$. Then $(S+T)(x,y)=S(x,y)+T(x,y)=(x,0) +(0,y)=(x,y)$, which is injective. Thus $U$ is not a subspace. --- **Problem 8:** Suppose $V$ and $W$ are finite-dimensional with $\dim V \geq \dim W \geq 2$. Show that $U=\{T \in \mathcal{L}(V, W) : T \text{ is not surjective}\}$ is not a subspace of $\mathcal{L}(V, W)$. Let $V=W=\mathbb{R}^{2}$. Suppose $T\in U$ with $T(x,y)=(0,y)$ and $S\in U$ with $S(x,y)=(x,0)$. Then $(S+T)(x,y)=S(x,y)+T(x,y)=(x,0) +(0,y)=(x,y)$, which is surjective. Thus $U$ is not a subspace. --- **Problem 9:** Suppose $T \in \mathcal{L}(V, W)$ is injective and $v_1, \ldots, v_n$ is linearly independent in $V$. Prove that $Tv_1, \ldots, Tv_n$ is linearly independent in $W$. Consider $ \begin{align} 0 & = a_{1}v_{1}+\ldots+a_{n}v_{n} \\ T(0)& =T(a_{1}v_{1}+\ldots+a_{n}v_{n}) \\ 0& =a_{1}Tv_{1}+\ldots+a_{n}Tv_{n} \end{align} $ Then $a_{1}=\ldots=a_{n}=0$ and $Tv_{1},\ldots,Tv_{n}$ are linearly independent in $W.$ --- **Problem 10:** Suppose $v_1, \ldots, v_n$ spans $V$ and $T \in \mathcal{L}(V, W)$. Prove that the list $Tv_1, \ldots, Tv_n$ spans $\operatorname{range} T$. The range $T$ is $Tv : v\in V$. Suppose an arbitrary $v\in V$, then $v=a_{1}v_{1}+\ldots+a_{n}v_{n}$. Consider $ \begin{align} Tv & = T(a_{1}v_{1}+\ldots+a_{n}v_{n}) \\ & = a_{1}Tv_{1}+\ldots a_{n}Tv_{n} \end{align} $ So $Tv_{1},\ldots,Tv_{n}$ spans $\text{range }T$. --- **Problem 11:** Suppose $S_1, \ldots, S_n$ are injective linear maps such that $S_1 S_2 \cdots S_n$ makes sense. Prove that $S_1 S_2 \cdots S_n$ is injective. Step 1: Consider $S_{n}v$. Then since it is injective, the range $S_{n}$ is distinct and the domain of $S_{n-1}$ is distinct. Step i: Consider $S_{i}v$. Then, since $S_{i}$ is injective and the domain is distinct, the range $S_{i}$ is distinct. And $S_{i},\ldots,S_{n}$ is injective. Thus on step n, $S_{1},\ldots,S_{n}$ is injective. --- **Problem 12:** Suppose that $V$ is finite-dimensional and that $T \in \mathcal{L}(V, W)$. Prove that there exists a subspace $U$ of $V$ such that $U \cap \operatorname{null} T = {0}$ and $\operatorname{range} T = {Tu : u \in U}$. Suppose that $V$ is finite-dimensional and that $T \in \mathcal{L}(V, W)$. Since $V$ is finite dimensional, by [[2B1 - Bases]], since null $T$ is a subspace of $V$, there exists a subspace $U$ of $V$ such that $\text{null }T\oplus U=V.$ Then $U$ satisfies $U\cap \text{null }T=\{ 0 \}$. Thus since null V makes up the vectors which $T$ maps to 0, $U$ makes up the vectors which map to range $T$. ==TODO== Properly prove the range statement. --- **Problem 13:** Suppose $T$ is a linear map from $\mathbf{F}^4$ to $\mathbf{F}^2$ such that $\operatorname{null} T = {(x_1, x_2, x_3, x_4) \in \mathbf{F}^4 : x_1 = 5x_2 \text{ and } x_3 = 7x_4}.$ Prove that $T$ is surjective. $\text{dim null }T=2.$ Thus, range $T=2$. Then we can form a basis of range $T$ with $(x_{1},x_{2})\in F^{2}$. However, since $\text{dim }F^{2}=2$ and has a length two basis. It must be that $(x_{1},x_{2})$ is also a basis for $F^{2}$. --- **Problem 14:** Suppose $U$ is a 3-dimensional subspace of $\mathbf{R}^8$ and that $T$ is a linear map from $\mathbf{R}^8$ to $\mathbf{R}^5$ such that $\operatorname{null} T = U$. Prove that $T$ is surjective. Dimension of $U$ is 3. Thus by the fundamental theorem of linear maps, 8 = 3 + dim null $T$. So $\text{dim null }T =5$. Thus, by same logic as problem 13, spans $\mathbb{R}^{5}.$ --- **Problem 15:** Prove that there does not exist a linear map from $\mathbf{F}^5$ to $\mathbf{F}^2$ whose null space equals ${(x_1, x_2, x_3, x_4, x_5) \in \mathbf{F}^5 : x_1 = 3x_2 \text{ and } x_3 = x_4 = x_5}.$ The dimension of the null space is $2$, which forces the dim range $T$ to be 3. However, codomain has dim $F^{2}=2$. Thus, there is no linear map from $F^{5}\to F^{2}$ with that null space. --- **Problem 16:** Suppose there exists a linear map on $V$ whose null space and range are both finite-dimensional. Prove that $V$ is finite-dimensional. Suppose $T$ is a linear map on $V.$ Since the null space is finite dimensional, we know theres a $u_{1},\ldots,u_{n}$ which spans $\text{null }T.$ Since the range is finite dimensional, there is a $Tv_{1},\ldots,Tv_{m}$ which is a basis for range $T$. Then for any $v\in V$, $Tv=Tb_{1}v_{1}+\ldots+Tb_{m}v_{m}$. Rearranging we get, $T(v-b_{1}v_{1}+\ldots+\ldots b_{m}v_{m})=0$. Thus, $ \begin{align} v-b_{1}v_{1}+\ldots+\ldots b_{m}v_{m} & =0 \\ & =a_{1}u_{1}+\ldots+a_{n}u_{n} & \text{Null space basis} \\ v & =b_{1}v_{1}+\ldots+\ldots b_{m}v_{m}+a_{1}u_{1}+\ldots+a_{n}u_{n} \end{align} $ Thus any $v\in V$ in span$(v_{1},\ldots,v_{m},u_{1},\ldots,u_{n})$. --- **Problem 17:** Suppose $V$ and $W$ are both finite-dimensional. Prove that there exists an injective linear map from $V$ to $W$ if and only if $\dim V \leq \dim W$. $(\Rightarrow)$Suppose $T$ is injective. Then $\text{null }V=0$. $ \begin{align} \text{dim }V & =\text{null }V + \text{range }T \\ & =0 + \text{range }T \\ & \leq{} \text{dim }W \end{align} $ $(\Leftarrow)$ Suppose that dim $V=n$ $\leq{}$ dim $W=m$. Let $v_{1},\ldots,v_{n}$ and $w_{1},\ldots,w_{m}$ be a basis for $V$ and $W$, respectively. Consider $T\in \mathcal{L}(V,W)$ with $Tv_{i}=w_{i}$ for $i=1,\ldots,n$. Then $Tv_{1},\ldots,Tv_{n}$ is linearly independent in $W.$ Thus $ \begin{align} 0 & =Tv \\ &=T(a_{1}v_{1}+\ldots+a_{n}v_{n}) \\ &= a_{1}Tv_{1}+\ldots+a_{n}Tv_{n} \\ \end{align} $ Then $a_{1}=\ldots=a_{n}=0$ and $v=0$. Thus null $T=\{ 0 \}$. And $T$ is injective. --- **Problem 18:** Suppose $V$ and $W$ are both finite-dimensional. Prove that there exists a surjective linear map from $V$ onto $W$ if and only if $\dim V \geq \dim W$. $(\Rightarrow)$ Suppose $T\in \mathcal{L}(V,W)$ is surjective. Then, $ \begin{align} \dim V & = \dim \text{null } T + \text{range } T \\ & =\text{dim null }T + \dim W \\ & \geq{} \dim W \end{align} $ $(\Leftarrow)$ Suppose $\text{dim }V \geq{} \text{dim }W$. Let $v_{1},\ldots,v_{n}$ and $w_{1},\ldots,w_{m}$ be a basis for $V$ and $W$, respectively, with $n\geq{}m$. Then consider $T\in \mathcal{L}(V,W)$ with $Tv_{i}=w_{i}$ for $i=1,\ldots,m$ and $Tv_{i>m}=0$. Suppose an arbitrary $w\in W \text{ with }w=a_{1}w_{1}+\ldots+a_{m}w_{m}$. Then construct a $v\in V$ with $v=a_{1}v_{1}+\ldots+a_{m}v_{m}+0v_{m+1}+\ldots+0v_{n}$. $ \begin{align} Tv & =T(a_{1}v_{1}+\ldots+a_{m}v_{m} + \ldots) \\ & =a_{1}Tv_{1}+\ldots+a_{m}Tv_{m} + 0 + \ldots\\ & =a_{1}w_{1}+\ldots+a_{m}w_{m} \\ & =w \end{align} $ Thus every $w\in W$ can be mapped by a $v\in V$. --- **Problem 19:** Suppose $V$ and $W$ are finite-dimensional and that $U$ is a subspace of $V$. Prove that there exists $T \in \mathcal{L}(V, W)$ such that $\operatorname{null} T = U$ if and only if$\dim U \geq \dim V - \dim W$. _Problem is saying that the minimum dimension of the null space is equal to the dimension gap between V and W._ $(\Rightarrow)$ Suppose there exists $T \in \mathcal{L}(V, W)$ such that $\operatorname{null} T = U$. Then $ \begin{align} \dim V & = \dim U + \text{range } T \\ & \leq{} \dim U + \dim W \\ \dim V - \dim W & \leq{} \dim U \end{align} $ $(\Leftarrow)$ Suppose $\dim U \geq \dim V - \dim W$. Let $u_{1},\ldots,u_{n}$ be a basis for $U$ with $Tu_{i}=0$ for $i=1,\ldots,n$. Extend this basis to one of $V$, as $u_{1},\ldots,u_{n},v_{1},\ldots,v_{m}$. $m=\dim V - \dim U$. Let $w_{1},\ldots,w_{k}$ be a basis for $W$ where $\dim W \geq{} \dim V - \dim U=m$. Then map $Tv_{i}=w_{i}$ for $i=1,\ldots,m$. Thus $\text{null }T=U$. --- **Problem 20:** Suppose $W$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T$ is injective if and only if there exists $S \in \mathcal{L}(W, V)$ such that $ST$ is the identity map on $V$. Let $W$ be finite-dimensional and $T\in \mathcal{L}(V,W)$. $(\Rightarrow)$ Suppose $T$ is injective. Then, by [[3B2 - Injective and Surjective]], $V$ must be finite dimensional with $\dim V \leq{} \dim W$. Suppose that $v_{1},\ldots,v_{n}$ is the basis for $V$. Then, since $T$ is injective, null $T=\{ 0 \}$. Then, $Tv=T(a_{1}v_{1}+\ldots+a_{n}v_{n})$ is the range for $T$. Thus, $(Tv_{1},\ldots, Tv_{n})=w_{1},\ldots,w_{n}$ is a basis for range $T$. Suppose $S\in \mathcal{L}(W,V)$ such that $Sw_{i}=v_{i}$ for $i=1,\ldots,n$. Consider an arbitrary $v\in V.$ $ \begin{align} STv & =ST(a_{1}v_{1}+\ldots a_{n}v_{n}) \\ & =S(a_{1}w_{1}+\ldots+a_{n}w_{n}) \\ & =a_{1}v_{1}+ \ldots+ a_{n}v_{n} \\ & = v \end{align} $ Thus $ST$ is the identity map on V. $(\Leftarrow)$ Suppose $ST$ is the identity map on $V$. Then, for any arbitrary $v\in V$, $STv=Sw=v$. --- **Problem 21:** Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T$ is surjective if and only if there exists $S \in \mathcal{L}(W, V)$ such that $TS$ is the identity map on $W$. --- **Problem 22:** Suppose $U$ and $V$ are finite-dimensional vector spaces and $S \in \mathcal{L}(V, W)$ and $T \in \mathcal{L}(U, V)$. Prove that $\dim \operatorname{null} ST \leq \dim \operatorname{null} S + \dim \operatorname{null} T.$ --- **Problem 23:** Suppose $U$ and $V$ are finite-dimensional vector spaces and $S \in \mathcal{L}(V, W)$ and $T \in \mathcal{L}(U, V)$. Prove that $\dim \operatorname{range} ST \leq \min{\dim \operatorname{range} S, \dim \operatorname{range} T}.$ --- **Problem 24:** Suppose $W$ is finite-dimensional and $T_1, T_2 \in \mathcal{L}(V, W)$. Prove that $\operatorname{null} T_1 \subset \operatorname{null} T_2$ if and only if there exists $S \in \mathcal{L}(W, W)$ such that $T_2 = ST_1$. --- **Problem 25:** Suppose $V$ is finite-dimensional and $T_1, T_2 \in \mathcal{L}(V, W)$. Prove that $\operatorname{range} T_1 \subset \operatorname{range} T_2$ if and only if there exists $S \in \mathcal{L}(V, V)$ such that $T_1 = T_2 S$. --- **Problem 26:** Suppose $D \in \mathcal{L}(\mathcal{P}(\mathbf{R}), \mathcal{P}(\mathbf{R}))$ is such that $\deg Dp = (\deg p) - 1$ for every nonconstant polynomial $p \in \mathcal{P}(\mathbf{R})$. Prove that $D$ is surjective. _[The notation $D$ is used above to remind you of the differentiation map that sends a polynomial $p$ to $p'$. Without knowing the formula for the derivative of a polynomial (except that it reduces the degree by 1), you can use the exercise above to show that for every polynomial $q \in \mathcal{P}(\mathbf{R})$, there exists a polynomial $p \in \mathcal{P}(\mathbf{R})$ such that $p' = q$.]_ --- **Problem 27:** Suppose $p \in \mathcal{P}(\mathbf{R})$. Prove that there exists a polynomial $q \in \mathcal{P}(\mathbf{R})$ such that $5q'' + 3q' = p$. _[This exercise can be done without linear algebra, but it's more fun to do it using linear algebra.]_ --- **Problem 28:** Suppose $T \in \mathcal{L}(V, W)$, and $w_1, \ldots, w_m$ is a basis of $\operatorname{range} T$. Prove that there exist $\varphi_1, \ldots, \varphi_m \in \mathcal{L}(V, \mathbf{F})$ such that $Tv = \varphi_1(v) w_1 + \cdots + \varphi_m(v) w_m$ for every $v \in V$. --- **Problem 29:** Suppose $\varphi \in \mathcal{L}(V, \mathbf{F})$. Suppose $u \in V$ is not in $\operatorname{null} \varphi$. Prove that $V = \operatorname{null} \varphi \oplus {au : a \in \mathbf{F}}.$ --- **Problem 30:** Suppose $\varphi_1$ and $\varphi_2$ are linear maps from $V$ to $\mathbf{F}$ that have the same null space. Show that there exists a constant $c \in \mathbf{F}$ such that $\varphi_1 = c\varphi_2$. --- **Problem 31:** Give an example of two linear maps $T_1$ and $T_2$ from $\mathbf{R}^5$ to $\mathbf{R}^2$ that have the same null space but are such that $T_1$ is not a scalar multiple of $T_2$.