Exercise 1.6.32

Answers

1.: Let $W_{1}$ be the $xy$ -plane with $m = 2$ and $W_{2}$ be the $x$ -axis with $n = 1$ . Then we have $W_{1} \cap W_{2} = W_{2}$ has dimension $1$ .
2.: Let $W_{1}$ be the $xy$ -plane with $m = 2$ and $W_{2}$ be the $z$ -axis with $n = 1$ . Then we have $W_{1} + W_{2} = ℝ^{3}$ has dimension $3 = 2 + 1$ .
3.: Let $W_{1}$ be the $xy$ -plane with $m = 2$ and $W_{2}$ be the $xz$ -axis with $n = 2$ . Then we have $W_{1} \cap W_{2}$ is the $x$ -axis with dimension $1$ and $W_{1} + W_{2}$ is $ℝ^{3}$ with dimension $3 \neq 2 + 2$ .

jephianlin

2011-06-27 00:00