1. The problem statement, all variables and given/known data Let u = (u1, u2, u3) and v = (v1, v2, v3) Determine if it's an inner product on R3. If it's not, list the axiom that do not hold. 2. Relevant equations the 4 axioms to determine if it's an inner product are (all letters representing vectors) 1. <u,v> = <v,u> 2. <u+v, w> = <u, w> + <v,w> 3. <ku, v> = k<u,v> 4. <v,v> ≥ 0 and < v,v> = 0 if and only if v = 0 3. The attempt at a solution So <u, v> is defined as u1v1 + u3v3 I'll skip the ones that did work and show axiom 4 which did not hold but I'm confused as to why this doesn't hold. I have a guess but have to make sure that I'm thinking correctly. Axiom 4 does not hold: <v, v > = v1v1 + v3v3 = v12 + v32 ≥ 0 and <0, 0> = (0)(0) + (0)(0) = 0 now to check the other way: if <v, v > = 0 implies that since v12 = 0 => v1 = 0 v32 = 0 => v3 = 0 then it goes to say it's not an inner product on R3. Am I correct to say it's not an inner product on R3 because there are only 2 components for axiom 4? and not 3? (i.e. no v2 showing anywhere) Thank you for any help. Will be much appreciated.