Exercise 1.6.1

Answers

1.: No. The empty set is its basis.
2.: Yes. This is the result of the Replacement Theorem.
3.: No. For example, the set of all polynomials has no finite basis.
4.: No. $ℝ^{2}$ has ${(1, 0), (1, 1)}$ and ${(1, 0), (0, 1)}$ as bases.
5.: Yes. This is the Corollary after Replacement Theorem.
6.: No. It’s $n + 1$ .
7.: No. It’s $m \times n$ .
8.: Yes. This is the Replacement Theorem.
9.: No. For $S = 1, 2$ , a subset of $ℝ$ , then $5 = 1 \times 1 + 2 \times 2 = 3 \times 1 + 1 \times 2$ .
10.: Yes. This is Theorem 1.11.
11.: Yes. It’s ${0}$ and $V$ respectively.
12.: Yes. This is the Corollary 2 after Replacement Theorem.

jephianlin

2011-06-27 00:00