# Proving M_mn(F) over F is a vector space

## Homework Statement

Show M_mn(F) (the collection of mxn matrices over F) over a field F is a vector space.

## The Attempt at a Solution

Denote $A=a_{ij},B=b_{ij}$ for elements of $M_{mn}(F)$ . Define $A+B=(a_{ij}+b_{ij})$ and for $a\in F$ denote $\alpha A=(\alpha a_{ij})$. Then,

(a) If $\alpha\in F$ and $A\in M_{mn}(F)$ ,$\alpha A=(\alpha a_{ij})\in M_{mn}(F)$ since $F$ is closed under scalar multiplication? not sure

(b) If $\alpha\in F and A,B\in M_{mn}(F)$ , $\alpha(A+B)=\alpha(a_{ij}+b_{ij})=\alpha a_{ij}+\alpha b_{ij}\in M_{mn}(F)$ because of distributive property of $F$?

I'm kind of stuck in justifying the claims.

## Answers and Replies

Any ideas?

Dick
Science Advisor
Homework Helper
The whole problem is more awkward to state than it actually is. Take the first one. alpha*a_ij isn't in F because F is closed under 'scalar multiplication'. It's because F is closed under multiplication in F. alpha is in F, a_ij is in F. So alpha*a_ij is in F. So alpha*A is in M_nm(F). There's nothing deep going on there.