Problem Suppose $T \in \mathcal{L}(V,W)$ and $v_1, \dots, v_m$ is a list of vectors in $V$ such that $Tv_1, \dots, Tv_m$ is a linearly independent list in $W$. Prove that $v_1, \dots, v_m$ is linearly independent. Solution Let $a_1, \dots, a_m \in \mathbb{F}$ such that $0 = a_1v_1 + \dots + a_mv_m$, then: $$0 \stackrel{T3.11}{=} T(0) = T(a_1v_1 + \dots + a_mv_m) \stackrel{D3.2}{=} a_1 Tv_1 + \dots + a_m Tv_m$$ Then the linear independence of $Tv_1, \dots, Tv_m$ implies $a_1 = \dots = a_m = 0$ and this in turn implies the linear independence of $v_1, \dots, v_m$ by theorem 2....

Problem Suppose $T \in \mathcal{L}(\mathbb{F}^n, \mathbb{F}^m)$. Show that there exist scalars $A_{j,k} \in \mathbb{F}$ for $j = 1, \dots, m$ and $k = 1, \dots, n$ such that $$T(x_1, \dots, x_n) = (A_{1,1} x_1 + \dots + A_{1,n}x_n, \dots, A_{m,1}x_1 + \dots + A_{m,n}x_n)$$ for every $(x_1, \dots, x_n) \in \mathbb{F}^{n}$ Solution Let $\mathbf{e}_{j}$ denote the vector in $\mathbb{F}^n$ with a $1$ at the $j$‘th coordinate and $0$’s everywhere else. Then $\mathbf{e}_1, \dots, \mathbf{e}_n$ is the standard basis of $\mathbb{F}^n$ as per Example 2....

Problem Suppose $b,c \in \mathbb{R}$. Define $T: \mathcal{P}(\mathbb{R}) \to \mathbb{R}$ by $$Tp = \left(3p(4) + 5p^\prime(6) + bp(1)p(2), \int_{-1}^{2}x^3p(x)dx + c\sin p(0)\right)$$ Show that $T$ is linear if and only if $b = c = 0$ Solution $(\rightarrow)$ Assume $T$ is a linear map, then if $p = \pi$ we must have: $$\left(3\pi + b\pi^2, \frac{15}{4}\pi\right) = T\pi = T\left(\frac{\pi}{2} + \frac{\pi}{2}\right) = T\frac{\pi}{2} + T\frac{\pi}{2}$$ $$= 2 \cdot \left(\frac{3}{2}\pi + b\frac{\pi^2}{4}, \frac{15}{8}\pi + c\right) = \left(3\pi + b\frac{\pi^2}{2}, \frac{15}{4}\pi + 2c\right)$$ The second coordinate implies $c = \frac{1}{2}(\frac{15}{4}\pi - \frac{15}{4}\pi) = 0$ and the first coordinate implies $b\frac{\pi^2}{2} = 0$ implying $b = 0$....

Problem Suppose $b,c \in \mathbb{R}$. Define $T: \mathbb{R}^3 \to \mathbb{R}^2$ by $$T(x,y,z) = (2x - 4y + 3z + b, 6x + cxyz)$$ Show that $T$ is linear if and only if $b = c = 0$ Solution $(\rightarrow)$ By theorem 3.11 we have $T0 = 0$, therefore $(0,0) = T(0,0,0) = (b,0)$ implying $b = 0$. Use additivity of $T$ and $b = 0$ as such: $$(1,6+c) = T(1,1,1) = T((1,0,0) + (0,1,1))$$ $$\stackrel{D3....

Problem You might guess, by analogy with the formula for the number of elements in the union of three subsets of a finite set, that if $U_1, U_2, U_3$ are subspaces of a finite-dimensional vector space, then $$\dim(U_1 + U_2 + U_3) = \dim U_1 + \dim U_2 + \dim U_3$$ $$- \dim (U_1 \cap U_2) - \dim(U_1 \cap U_3) - \dim(U_2 \cap U_3)$$ $$+ \dim(U_1 \cap U_2 \cap U_3)$$ Prove this or give a counterexample....