Exercise 1.5.1

Answers

1.: No. For example, take $S = {(1, 0), (2, 0), (0, 1)}$ and then $(0, 1)$ is not a linear combination of the other two.
2.: Yes. It’s because $1 \vec{0} = \vec{0}$ .
3.: No. It’s independent by the remark after Definition of linearly independent.
4.: No. For example, we have $S = {(1, 0), (2, 0), (0, 1)}$ but ${(1, 0), (0, 1)}$ is linearly independent.
5.: Yes. This is the contrapositive statement of Theorem 1.6.
6.: Yes. This is the definition.

jephianlin

2011-06-27 00:00