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## 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 [itex]A=a_{ij},B=b_{ij}[/itex] for elements of [itex]M_{mn}(F)[/itex] . Define [itex]A+B=(a_{ij}+b_{ij})[/itex] and for [itex]a\in F[/itex] denote [itex]\alpha A=(\alpha a_{ij})[/itex]. Then,

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

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

I'm kind of stuck in justifying the claims.