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Linear Algebra

Exercise 5.4.3

Answers

1.: We have $T (0) = 0$ and $T (v) \in V$ any $v \in V$ and for arbitrary linear operator $T$ on $V$ .
2.: If $v \in N (T)$ , we have $T (v) = 0 \in N (T)$ . If $v \in R (T)$ , we have $T (v) \in R (T)$ by the definition of $R (T)$ .
3.: If $v \in E_{λ}$ , we have $T (v) = λv$ . Since $T (λv) = λT (v) = λ^{2} v$ , we know that $T (v) \in E_{λ}$ .

jephianlin

2011-06-27 00:00

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