# Is the zero Matrix a vector space?

## Homework Statement

So I have these two Matrices:
M = \begin{pmatrix}
a & -a-b \\
0 & a \\
\end{pmatrix}
and
N =
\begin{pmatrix}
c & 0 \\
d & -c \\
\end{pmatrix}

Where a,b,c,d ∈ ℝ

Find a base for M, N, M +N and M ∩ N.

## Homework Equations

I know the 8 axioms about the vector spaces.

## The Attempt at a Solution

I chose these fours matrices as a base for the first three vector spaces.
\begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix}
\begin{pmatrix}
0 & 1 \\
0 & 0 \\
\end{pmatrix}
\begin{pmatrix}
0 & 0 \\
1 & 0 \\
\end{pmatrix}
\begin{pmatrix}
0 & 0 \\
0 & 1 \\
\end{pmatrix}

I got that the only vector space satisfying M ∩ N is
\begin{pmatrix}
0 & 0 \\
0 & 0 \\
\end{pmatrix}

If M ∩ N constitutes a vector space, I can use the same base as the other three. But if it doesnt, I have to explain why.

Not sure how to go about that...