Proving M_mn(F) over F is a vector space

[autre]

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.

 

[autre]

Any ideas?

 

[Dick]

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.