1. ## Subspaces..

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!

2. Originally Posted by Linnus
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.
no, because if $\displaystyle y \neq 0,$ then $\displaystyle 0 \notin D.$

b)
Let H and V be subspaces of $\displaystyle \Re^n$. Is $\displaystyle H \cap V$ a subspace?
yes, just check the conditions required for a set to be a subspace.

Is $\displaystyle H \cup V$ a subspace? Explain your answers.
it's a subspace iff either $\displaystyle H \subseteq V$ or $\displaystyle V \subseteq H.$ because otherwise choose $\displaystyle h \in H - V, \ v \in V - H.$ then: $\displaystyle h + v \notin H \cup V.$