a)

Let be a linear map. Define , for

some , where . Is D a subspace? If not explain.

b)

Let H and V be subspaces of . Is a subspace? Is a subspace? Explain your answers.

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

What does and 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!