I am given that V is a vector space $M_{3,3}(\Bbb R)$ and that function $\langle A,B\rangle\,=\,tr(A^T\cdot B)$ prove this function is an inner product.
So far I have taken these steps:
From the function = $tr( A^T \cdot B)$. Allowing A and B to be 3*3 square matrices. Then it follows we would get , $(a_{11}, a_{22}, a_{33}), (b_{11},b_{22},b_{33})$. Next we would need to prove that it is an inner product using the four axioms:
For the first axiom we can prove this by writing $\langle A^T,B\rangle = A_1{_1} B_1{_1} + A_2{_2} B_2{_2} + A_3{_3} B_3{_3}$, Which by the commutative property of addition can be rewritten as $\langle B, A^T\rangle$.
For the second axiom we can prove this by the distributive property of addition in much the same way as the first axiom.
I am getting a little confused on this third axiom it states that $\ c\,\langle U,V\rangle = \langle c\,U,V\rangle$. The $c$ is just a scalar that would be distributed to both vectors so how could $c$ only be applied to one vector.
Finally since any $M_{3,3}(\Bbb R)$ includes matrices with negative numbers and is odd the trace will be negative therefore $\langle V,V\rangle \lt 0$, so I would have to conclude that this is not a inner product.