Exercise 15

Let $W_{1},W_{2}$ be subspaces of a vector space.

• Assume that the set $S_{1}$ spans $W_{1}$, and that the set $S_{2}$ spans $W_{2}$. Can $S_{1}\cup S_{2}$ span $W_{1}+W_{2}$? Must it?

• Assume that $S_{1}$ is a linearly independent subset of $W_{1}$ and that $S_{2}$ is a linearly independent subset of $W_{2}$. Can $S_{1}\cup S_{2}$ be a linearly independent subset of $W_{1}+W_{2}$? Must it?