# Category Archives: Chapter 5

## Chapter 5 Exercise C

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 bijective and surjective. Hence $V=\m{null} T \oplus\m{range} T$. If $T$ … Continue reading

## Chapter 5 Exercise B

1. Suppose $T\in\ca L(V)$ and there exists a positive integer $n$ such that $T^n=0$. (a) Prove that $I-T$ is invertible and that \[(I-T)^{-1}=I+T+\cdots+T^{n-1}.\] (b) Explain how you would guess the formula above. Solution: (a) Note that \[ (I-T)(I+T+\cdots+T^{n-1})=I-T^n=I \]and \[ … Continue reading

## 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 invariant under $T$. (b) For any $u\in U$, then $Tu\in\m{range} T \subset U$, hence $U$ is invariant under $T$. 2. Solution: See Linear … Continue reading