--- **Problem 1:** Suppose $b, c \in \mathbf{R}$. Define $T : \mathbf{R}^3 \to \mathbf{R}^2$ by $T(x, y, z) = (2x - 4y + 3z + b,\ 6x + cxyz).$ Show that $T$ is linear if and only if $b = c = 0$. **($\Rightarrow$):** Suppose $b=c=0$. Then $T(x,y,z)=(2x-4y+3z,6x)$. Let $u=(u_{1},u_{2},u_{3})\in \mathbb{R}^{3}$ and $v=(v_{1},v_{2},v_{3})\in \mathbb{R}^{3}$. Then $ \begin{align} T(u+v) & = (2(u_{1}+v_{1})-4(u_{2}+v_{2})+3(u_{3}+v_{3}),6(u_{1}+v_{1}))\\ & = (2u_{1}-4u_{2}+3u_{3},6u_{1})+(2v_{1}-4v_{2}+3v_{3},6v_{1}) \\ & = Tu + Tv \end{align} $ So it is additive. By inspection, homogeneity $T(\lambda u)=\lambda Tu$, holds. (**$\Leftarrow$):** Suppose $T$ is a linear map. Then $T(0)=0$. $T(0,0,0)=(b,0)$. Thus it must be that $b=0$. Consider $2T(1,1,1)=2(\ldots, 6+c)=(\ldots,12+2c)$ and $T(2,2,2)=(\ldots, 12 + 8c)$. Then $12+c=12+8c\implies c=0$. Thus $c=b=0$. --- **Problem 2:** Suppose $b, c \in \mathbf{R}$. Define $T : \mathcal{P}(\mathbf{R}) \to \mathbf{R}^2$ by $Tp = \left(3p(4) + 5p'(6) + bp(1)p(2),\ \int_{-1}^{2} x^3 p(x) dx + c \sin p(0)\right).$ Show that $T$ is linear if and only if $b = c = 0$. **($\Rightarrow$):** Suppose $b=c=0$. Then, $T(p+q)=(3(p+q)(4)+5(p'+q')(6), \int_{-1}^{2} x^{3}(q+p)(x) \, dx)$. by inspection $(p+q)$ can be separated and we get $Tp+Tq$. Again by inspection, homogeneity $T(\lambda u)=\lambda Tu$, holds. so $T$ is linear. **($\Leftarrow$):** Suppose $T$ is a linear map and let $p(x)=1$ and $\lambda=2$. Then $2Tp=2\left( 3+0+b, \int_{-1}^{2} x^{3} \, dx + c\sin(1)\right)=\left( 6+2b,\frac{15}{2} + c\sin(1)\right)$. And $T(2p)=\left( 6+4b,\frac{15}{2}+c\sin(2) \right)$. Then, $6+2b=6+4b\implies b=0$. And $\frac{15}{2}+c\sin(1)=\frac{15}{2}+c\sin(2)\implies c=0$. Thus $b=c=0$. --- **Problem 3:** Suppose $T \in \mathcal{L}(\mathbf{F}^n, \mathbf{F}^m)$. Show that there exist scalars $A_{j,k} \in \mathbf{F}$ for $j = 1, \ldots, m$ and $k = 1, \ldots, n$ such that $T(x_1, \ldots, x_n) = (A_{1,1}x_1 + \cdots + A_{1,n}x_n,\ \ldots,\ A_{m,1}x_1 + \cdots + A_{m,n}x_n)$ for every $(x_1, \ldots, x_n) \in \mathbf{F}^n$. _The exercise above shows that $T$ has the form promised in the last item of Example 3.4._ --- **Problem 4:** Suppose $T \in \mathcal{L}(V, W)$ and $v_1, \ldots, v_m$ is a list of vectors in $V$ such that $Tv_1, \ldots, Tv_m$ is a linearly independent list in $W$. Prove that $v_1, \ldots, v_m$ is linearly independent. --- **Problem 5:** Prove the assertion in 3.7. --- **Problem 6:** Prove the assertions in 3.9. --- **Problem 7:** Show that every linear map from a 1-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if $\dim V = 1$ and $T \in \mathcal{L}(V, V)$, then there exists $\lambda \in \mathbf{F}$ such that $Tv = \lambda v$ for all $v \in V$. --- **Problem 8:** Give an example of a function $\varphi : \mathbf{R}^2 \to \mathbf{R}$ such that $\varphi(av) = a\varphi(v)$ for all $a \in \mathbf{R}$ and all $v \in \mathbf{R}^2$ but $\varphi$ is not linear. _[The exercise above and the next exercise show that neither homogeneity nor additivity alone is enough to imply that a function is a linear map.]_ --- **Problem 9:** Give an example of a function $\varphi : \mathbf{C} \to \mathbf{C}$ such that $\varphi(w + z) = \varphi(w) + \varphi(z)$ for all $w, z \in \mathbf{C}$ but $\varphi$ is not linear. (Here $\mathbf{C}$ is thought of as a complex vector space.) _[There also exists a function $\varphi : \mathbf{R} \to \mathbf{R}$ such that $\varphi$ satisfies the additivity condition above but $\varphi$ is not linear. However, showing the existence of such a function involves considerably more advanced tools.]_ --- **Problem 10:** Suppose $U$ is a subspace of $V$ with $U \neq V$. Suppose $S \in \mathcal{L}(U, W)$ and $S \neq 0$ (which means that $Su \neq 0$ for some $u \in U$). Define $T : V \to W$ by $Tv = \begin{cases} Sv & \text{if } v \in U, \ 0 & \text{if } v \in V \text{ and } v \notin U. \end{cases}$ Prove that $T$ is not a linear map on $V$. --- **Problem 11:** Suppose $V$ is finite-dimensional. Prove that every linear map on a subspace of $V$ can be extended to a linear map on $V$. In other words, show that if $U$ is a subspace of $V$ and $S \in \mathcal{L}(U, W)$, then there exists $T \in \mathcal{L}(V, W)$ such that $Tu = Su$ for all $u \in U$. --- **Problem 12:** Suppose $V$ is finite-dimensional with $\dim V > 0$, and suppose $W$ is infinite-dimensional. Prove that $\mathcal{L}(V, W)$ is infinite-dimensional. --- **Problem 13:** Suppose $v_1, \ldots, v_m$ is a linearly dependent list of vectors in $V$. Suppose also that $W \neq {0}$. Prove that there exist $w_1, \ldots, w_m \in W$ such that no $T \in \mathcal{L}(V, W)$ satisfies $Tv_k = w_k$ for each $k = 1, \ldots, m$. --- **Problem 14:** Suppose $V$ is finite-dimensional with $\dim V \geq 2$. Prove that there exist $S, T \in \mathcal{L}(V, V)$ such that $ST \neq TS$.