Is the zero Matrix a vector space?

  • #1

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...
 

Answers and Replies

  • #2
14,880
12,412
Does ##V=\{\vec{0}\}## fulfill the axioms of a vector space?

I don't understand what you wrote. ##M(a,b)=\begin{bmatrix}a & -a-b \\ 0& a\end{bmatrix}## is probably meant to be ##V_M=\{M=M(a,b)\,\vert \,a,b \in \mathbb{R}\}##. So how can ##\begin{bmatrix}1&0\\0&0\end{bmatrix}## be in ##V_M## or in ##V_N\,?## In ##V_M## there are only matrices with a diagonal with two identical entries.

Also the choice of the scalar field plays a role in the third case ##V_{M\cap N}##. You should mention it, because there are two different solutions, depending on the kind of field. I assumed you meant the reals, but this isn't clear.
 

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