a)

Let $\displaystyle T : \Re^n \rightarrow \Re^m $ be a linear map. Define $\displaystyle D= \{ x \in \Re^n \mid T x=y \} $, for

some $\displaystyle y \in \Re^m$, where $\displaystyle y \neq 0 $. Is D a subspace? If not explain.

b)

Let H and V be subspaces of $\displaystyle \Re^n $. Is $\displaystyle H \cap V $ a subspace? Is $\displaystyle H \cup V $ a subspace? Explain your answers.

I don't think D is a subspace...but I don't know why.

What does $\displaystyle H \cap V $ and $\displaystyle H \cup V $ represent? There is nothing in part a of the problem that talks about H and V...

Lastly, what exactly are subspaces?

Step by step process with some explanation would be really nice!

Thank you for your time!