Question

Note: In the following, if $\mathbb{F}$ is a set and both and are positive integers, then matrices with entries from $\mathbb{F}$. The problem below has many applications. If is a linear map from complex vector space to itself, and $\lambda$ is an eigenvalue of , then $\lambda$ is a simple eigenvalue of if  $ker((T-\lambda I)^2)\subset ker(T-\lambda I)$.

1. Suppose is a vector space of dimension over field $\mathbb{F}$ where you may assume that $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$, and let be a linear map from to . Show that if and only if .

2. Use the previous result to prove the following. Assume that $\mathbb{F}$ is $\mathbb{R}$ or $\mathbb{C}$, and assume that .

(a). If $M\in \mathbb{F}^{n \times n}$ , and is idempotent, then $\mathbb{F}^n=ker(M)\oplus Range(M)$ .

(b). If $T: V \to V$ is linear, and $\lambda$ is an eigenvalue of , then $V=ker((T-\lambda I)^n)\oplus Range((T-\lambda I)^n)$ .

(c). If $T: V \to V$ is linear, and $\lambda$ is a simple eigenvalue of , then $V=ker((T-\lambda I))\oplus Range((T-\lambda I))$ .

  

mxn V = ker(T) φ Range (T) ker(T)2 C ker (T)

Answer 1

Tis e kur Ст*) C ker. Cr) Comunal.hten une err CT) dlim (V) (»-n,it 서 we w.macham et nt2 swospaum af v

Sinl C 4I