Exercise C7

Chapter 8 Exercise C
1. Solution: Because $$ 4 = \operatorname{dim} \mathbb{C}^4 = \operatorname{dim} G(3, T) + \operator...
Chapter 7 Exercise C
1. Solution: We give a counterexample. Define $T \in \mathcal{L}(\mathcal{R}^2)$ by $$ \begin{aligne...
Chapter 6 Exercise C
1. Solution: Suppose $w \in \{v_1, \dots, v_m\}^\perp$. Let $v = \in \operatorname{span}(v_1, \dots,...
Chapter 5 Exercise C
1. Solution: It is not said $V$ is finite-dimensional, but I will do it by assuming $\dim V<\inft...
Chapter 3 Exercise C
1. Solution: Suppose for some basis $v_1$, $\cdots$, $v_n$ of $V$ and some basis $w_1$, $\cdots$, $w...
Chapter 2 Exercise C
1. Solution: Let $u_1,u_2,\cdots,u_n$ be a basis of $U$. Thus $n=\dim U=\dim V$. Hence $u_1,u_2,\cdo...
Chapter 1 Exercise C
1. Solution: (a) $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}$ is a subspace of $\mathbb F^3$. B...