Exercise 2.1.14

Answers

1.: The sufficiency is due to that if $T (x) = 0$ , ${x}$ can not be independent and hence $x = 0$ . For the necessity, we may assume $\sum a_{i} T (v_{i}) = 0$ . Thus we have $T (\sum a_{i} v_{i}) = 0$ . But since $T$ is one-to-one we have $\sum a_{i} v_{i} = 0$ and hence $a_{i} = 0$ for all proper $i$ .
2.: The sufficiency has been proven in Exercise 2.1.13. But note that $S$ may be an infinite set. And the necessity has been proven in the previous exercise.
3.: Since $T$ is one-to-one, we have $T (β)$ is linear independent by the previous exercise. And since $T$ is onto, we have $R (T) = W$ and hence span $(T (β)) = R (T) = W$ .

jephianlin

2011-06-27 00:00