Exercise A10

Chapter 10 Exercise A
1. Solution: If $T$ is invertible, then there exists $S\in\ca L(V)$ such that $TS=ST=I$. Then it fol...
Chapter 9 Exercise A
1. Solution: $V_\mathbb{C}$ is clearly closed under addition. We can write each complex number in th...
Chapter 8 Exercise A
1. Solution: Since $$ T^2(w, z) = T(z, 0) = (0, 0), $$ it follows that $G(0, T) = V$. Therefore ever...
Chapter 7 Exercise A
1. Solution: By definition, we have \[\begin{align*}\langle (z_1,\cdots,z_n),T^*(w_1,\cdots,w_n)\ran...
Chapter 6 Exercise A
2. Solution: It does not satisfy definiteness. For the function takes $(0,1,0)$, $(0,1,0)$ to $0$, b...
Chapter 5 Exercise A
1. Solution: (a) For any $u\in U$, then $Tu=0\in U$ since $U\subset \m{null} T$, hence $U$ is invari...
Chapter 4 Exercise
1. Empty 2. Solution: False. Consider $1=(z^m+1)+(-z^m)\notin \{0\}\cup\{p\in\ca P(\mb F):\deg p=m\}...
Chapter 3 Exercise A
1. Solution: If $T$ is linear, then \[(0,0)=T(0,0,0)=(b,0)\]by 3.11, hence $b=0$. We also have \[T(1...
Chapter 2 Exercise A
1. Solution: We just need to show that $v_1$, $v_2$, $v_3$, $v_4$ can be expressed as linear combina...
Chapter 1 Exercise A
1.Solution: Because $(a+bi)(a-bi)=a^2+b^2$, one has\[\frac{1}{a+bi}=\frac{a-bi}{a^2+b^2}.\]Hence\[c=...