Let V = Mnn be the real vector space of all n x n matrices. If A and B

are in V, we define (A,B) = Tr(B^TA) where Tr stands for trace and T

stands for transpose. Prove that this function is an inner product on V.

I don't understand how I would prove that the function is an inner

product.

According to my book, an inner product must satisfy the following

properties:

a) (u,u) > 0; (u,u) = 0 iff u = 0v.

b) (v,u) = (u,v) for any u,v in V.

c) (u+v,w) = (u,w) + (v,w) for any u,v,w in V.

d) (cu,v) = c(u,v) for any u,v in V and c is a real scalar.

I plugged in u = Tr(BTA).

I get how I can prove the first property.

(u,u) would be > 0 because it is a square. And if u = 0, then (u,u) = 0.

I don't get how I would prove the other properties. What would I plug

in v as? Should A and B be u and v, or is there another way? Thank you

in advance!