Space dimension of a $2 \times 2$ matrix

Question

I've been searching but I can't find the solution of matrix, only vectors.

What's the dimension space of $2 \times 2$ matrices? Find a base to this space. Then, do the same to $n \times n$ matrices.

The problem is that I can't find without the matrix.

Answer 1

The vector space of $2 \times 2$ matrices under addition over a field $\mathbb{F}$ is 4 dimensional. It's $$\operatorname{span}\left\{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} ,\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \right\}.$$ These are clearly independent under addition. Can you generalize?

Answer 2

Let $e_1=(1,0,0,...,0)^T$,$ e_2= (0,1,0,0,...,0)^T$, etc, then the matrices $e_i e_j^T$ with $i,j = 1,...,n$ form a basis for the $n \times n$ matrices.

It is straightforward to check that $A = \sum_{ij} [A]_{ij} e_i e_j^T$ and that $A = 0$ iff $[A]_{ij} = 0$ for all $i,j$.