1. ## Inner Product Axioms

Hello,

I would appreciate your help on the following problem.

A rule being used to define the mathematics of a vector space is
$
<\vec{u},\vec{v}>=u_{1}v_{2}+u_{2}v_{1}
$

Determine and state which axiom fails.

I tried the four axioms (symmetry, additive, homogeneity, and positive definite), but none seems to fail. Am I missing anything?

2. Originally Posted by hasanbalkan
Hello,

I would appreciate your help on the following problem.

A rule being used to define the mathematics of a vector space is
$
<\vec{u},\vec{v}>=u_{1}v_{2}+u_{2}v_{1}
$

Determine and state which axiom fails.

I tried the four axioms (symmetry, additive, homogeneity, and positive definite), but none seems to fail. Am I missing anything?

it's not positive definite! for example consider $u=v=(1,-1).$

3. Also notice that the inner product of <1, 0> with itself is <1, 0>.<1, 0>= 1(0)+ 0(1)= 0.