Question

Proof whether the following operations are inner product operations:

⟨x, y⟩ = 2x₁y₁ − x₁y₂ − x₂y₁ + 2x₂y₂, x=(x₁, x₂), y=(y₁, y₂)

Answer 1

 $⟨x, y⟩$ satisfies the following properties:

Linearity:

$(ax+by,z)=a(x,z)+b(y,z)$

$(ax+by,z)= 2(ax_1+by_1)z_1 − (ax_1+by_1)z_2 − (ax_2+by_2)z_1 + 2(ax_2+by_2)z_2=$

$=a( 2x_1z_1 − x_1z_2 − x_2z_1 + 2x_2z_2)+b( 2y_1z_1 − y_1z_2 − y_2z_1 + 2y_2z_2)$

$a(x,z)+b(y,z)=a( 2x_1z_1 − x_1z_2 − x_2z_1 + 2x_2z_2)+b( 2y_1z_1 − y_1z_2 − y_2z_1 + 2y_2z_2)$

Symmetric Property:

$(x,y)=(y,x)$

Positive Definite Property:

$⟨x, x⟩ = 2x_1x_1 − x_1x_2 − x_2x_1 + 2x_2x_2=(x_1-x_2)^2+x_1^2+x_2^2\ge0$

so,

$⟨x, y⟩$ is inner product