prove that <u,v>=u1v1-2u1v2-2u2v1+5u2v2 is an in inner product on R^2,where u=(u1,u2),v=(v1,v2)

Answer 1

1) It is easy to show that <u, v> = <v, u>

2) <u, u> = u₁² -4u₁u₂ + 5u₂² = (u₁ - 2u₂)² + u₂² >=0

3) Let w = <w₁, w₂>, then for scalers a and b we must show that

<av + bw, v> = a<u,v> + b<w, v>

This is easy to show. Just plug in and compare the two sides. You will se that they are the same.

This proves that the given formula defines and inner product.