# Thread: Basis for this set:

1. ## Basis for this set:

$\displaystyle S=\lbrace \begin{bmatrix} 0&0\\0&0 \end{bmatrix} \rbrace$.

I.e. it is a singleton.

I've already showed that it is a subspace of $\displaystyle M_2(R)$ (hopefully you'll agree that it is indeed a vector space).

But the problem is:

How can I find a basis for it?!

A basis must span the set S, but it must also be a linearly independent set!

So if I understand correctly, $\displaystyle \lbrace \begin{bmatrix} 0&0\\0&0 \end{bmatrix} \rbrace$ spans S, but it is not linearly independent (since any singleton set containing just the 0 vector is linearly dependent).

What am I to do?

2. By definition, the span of the empty set is the zero vector.
(kind of makes sense if we understand the concept of the empty sum
evaluating to zero). Thus, your basis is: the empty set.

Best,
ZD

3. ## Null set

The null set is considered the basis for the unique 0 dimensional subspace of a vector space which consists of just the additive identity.

4. Thanks guys! I knew it had to have something to do with that.