I'm having a little trouble with this question of determining whether the operation is an inner product:
Verify that the operation $<x, y> = x_1y_1 − x_1y_2 − x_2y_1 + 3x_2y_2$ where $x= (x_1, x_2)$ and $y = (y_1, y_2)$ is an inner product in $R^2$.
Now I do understand for the above to be an inner product, it has to satisfy 4 axioms:
- $<u,v>$ = $<v,u>$
- $\alpha$$<u,v>$= <$\alpha$ u, v>
- $<u+v,w>$= $< u, w>+<v,w>$
4a. $(u,u)$≥ 0
4b. $(u,u)$ = 0 $\implies$$u$=0
I have managed to proof axiom 1 & 2 but I'm having trouble with proofing axiom 3, 4a and 4b.