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1. If V is a vector space over R and ( , ):VxV-->R is an inner product on V, then for v,w in V, is the value of (v,w) independent of my choice of basis for V used to compute (v,w)?

2. If V is an arbitrary n dimensional vector space over R, are there some standard ways to define an inner product on V?

3. If V is an arbitrary n dimensional vector space over R, and v and w are linearly independent vectors in V, is it always possible to define an inner product on V such that (v,w) is not zero?