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Linear Algebra

Exercise 2.1.36

Answers

1.: In this case we have $R (T) = V$ and $N (T) = {(a_{1}, 0, 0, \dots)}$ . So naturally we have $V = R (T) + N (T)$ . But $V$ is a direct sum of them since $R (T) \cap N (T) = N (T) \neq {0}$ .
2.: Take $T_{1} = U$ in the Exercise 2.1.21. Thus we have $R (T_{1}) = {(0, a_{1}, a_{2}, \dots)}$ and $N (T_{1}) = {(0, 0, \dots)}$ . So we have $R (T_{1}) \cap N (T_{1}) = {0}$ but $R (T_{1}) + N (T_{1}) = R (T_{1}) \neq V$ .

jephianlin

2011-06-27 00:00

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