We'll define the following operation for vectors u and v in :
Which axioms of inner products are satisfied for <,>? Is it an inner product in ?
I'm not sure how to answer this question but I know that it must satisfy all the axioms in order to be an inner product. These are the properties of inner products:
<u,v>=<v,u> (symmetry property)
<u+v,w>=<u,w>+<v,w> (additivity property)
<ku,v>=k<u,v> (homogeneity property)
<v,v> ≥ 0 and <v,v>=0 if and only if v=0 (positivity property)
I know that matrix multiplication not commutative, so I'm guessing the symmetry property is violated. I think the only condition it satisfies is the homogeneity. But I'm not sure how to go about proving the other ones. Any ideas?