We'll define the following operation for vectors u and v in

:

Where

Which axioms of inner products are satisfied for <,>? Is it an inner product in

?

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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>** (a

*dditivity 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?