Hello,

Let's rephrase the properties with the notations your text uses =)

a) (A,B) > 0; (A,A) = 0 iff A = 0B. << be careful, it's iff, not if

b) (B,A) = (A,B) for any A,B in V.

c) (A+B,C) = (A,C) + (B,C) for any A,B,C in V.

d) (mA,B) = m(A,B) for any A,B in V and m is a real scalar.

A, B and C are matrices.

For example

Use basic properties such as linearity of the trace to prove them

Try to do them, and even if you have a slight doubt, don't hesitate to post it here.