Let Z be the set of all vectors fo the form $\displaystyle \begin{bmatrix} 2a - 3b - e \\ 0 \\ a + c +2d \\ -3b + c + d + e \end{bmatrix}$, where $\displaystyle a, b, c, d, e \ \epsilon \ R$. Z is a subspace of $\displaystyle R^4$.

a) Find a set that spans Z.

b) Is the set from part (a) a basis for Z? Please explain.

c) Does Z have any two dimensional subspaces? If so give an example of such a subspace. If not, please explain why not.

Attempted Answers:

For part (a) I got: $\displaystyle \{(2, 0, 1, 0), (-3, 0, 0, -3), (0, 0, 1, 1) \}

$.This is the column space of W. Correct?

Part (b) seems like Yes but I don't know the reason. Part (c) I have no clue.