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I understand the concepts of the inner product in Rn as well as the vector space of C[a,b] as the integral operator, however i don't understand how to obtain or prove the inner product space of two 2x2 matrices?

Example: consider two matrices u,v which are row 1 [a b] row 2 [c d]

and row 1[e f],row 2[g h]

where u is the 1st matrix and v is the second

for example if the inner product of <u,v>= ae+2bf+3cg+4hd

How do i satisfy the 4 axioms that prove the subspace spanned by these two matrices is infact an inner product?

Thanks.

P.S. i use lay's linear algebra book and there is NOTHING on matrices of inner products, only integral operators, R^n but not matrices and we weren't taught in class how...?