Question

let F^mxn be the set of all mxn matrices over the field F. is F^mxn is a vector space?

Answer 1

Let V be the vector space 

Let $V={A=(a_{ij})\in M_{m\times n}: a_{ij}\in R}$

Let $X=(x_{ij}),Y=(y_{ij})\in V$ and $\alpha,\beta\in Q$

Now lets us check if vector $\alpha X+\beta Y$ is an element of V.

$\alpha X+\beta Y=\alpha(x_{ij})+\beta (y_{ij})$

        $=(\alpha x_{ij})+(\beta y_{ij})\\[9pt] =(\alpha x_{ij}+\beta y_{ij})$

Since $(\alpha x_{ij})+(\beta y_{ij})\in R \forall i$ , Vector $\alpha X+\beta Y$ is an element of V.

So V is the vector space over the field F. Which contains all the $m\times n$ matrices.