Exercise C

1. Solution: Because $$ 4 = \operatorname{dim} \mathbb{C}^4 = \operatorname{dim} G(3, T) + \operatorname{dim} G(5, T) + \operatorname{dim} G(8, T), $$ it follows that the multiplicities of the eigenvalues of…

1. Solution: We give a counterexample. Define $T \in \mathcal{L}(\mathcal{R}^2)$ by $$ \begin{aligned} Te_1 = e_1\\ Te_2 = -e_2 \end{aligned} $$ where $e_1, e_2$ is the standard basis of $\mathbb{R}^2$.…

1. Solution: Suppose $w \in \{v_1, \dots, v_m\}^\perp$. Let $v = \in \operatorname{span}(v_1, \dots, v_m)$. We have that $$ v = a_1 v_1 + \dots a_m v_m $$ for some…

1. Solution: It is not said $V$ is finite-dimensional, but I will do it by assuming $\dim V<\infty$. If $T$ is invertible, then $\m{null}{T}=0$ and $\m{range} T=V$ since $T$ is…

1. Solution: Suppose for some basis $v_1$, $\cdots$, $v_n$ of $V$ and some basis $w_1$, $\cdots$, $w_m$ of $W$, the matrix of $T$ has at most $\dim \m{range} T-1$ nonzero…

1. Solution: Let $u_1,u_2,\cdots,u_n$ be a basis of $U$. Thus $n=\dim U=\dim V$. Hence $u_1,u_2,\cdots,u_n$ is a linearly independent list of vectors in V with length $\dim V$. By 2.39,…

1. Solution: (a) $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}$ is a subspace of $\mathbb F^3$. By 1.34, to show a subset is a subspace, we just need to check Additive identity, Closed under addition…