Question

Let u=(u1, u2, u3) and v=(v1, v2, v3) . Determine whether u1v1-u2v2+u3v3 is inner products on R3. For those that are not, list the axioms that do not hold.

Answer 1

I believe that one of the rules for something to be an inner product is that:\[\langle x,x\rangle =0\Longrightarrow x=0\] if the inner product of something with itself is 0, then that object must be the 0 vector. Maybe there is a non-zero vector that doesnt follow this using the definition of inner product that you have there...