Verify the property u v w u w v w If the inner product on

Verify the property (u + v, w) = (u, w) + (v, w), If the inner product on R^2, defined by (u, v) = 2u_1v_1 + 3u_2v_2.

Solution

The vectors u, v,w have not been defined here, but we presume that u = (u₁,u₂),v = (v₁,v₂) and w = (w₁,w₂). Since <u,v> = 2u₁v₂ +3u₂v₂, we have (u+v,w)= 2(u₁+v₁)w₁+3(u₂+v₂)w₂ = (2u₁w₁+3u₂w₂) +(2v₁w₁+3v₂w₂) = <u,w)> +<v,w>.