Question

Proof whether the following operations are inner product operations:

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

Answer 1

1. Positive Definite Property:

$$\langle x,x\rangle=2x_1x_1-x_1x_2-x_2x_1+2x_2x_2$$

$$=(x_1^2+x_2^2)+(x_1-x_2)^2\geq0$$

The value equals zero if and only if both summands are zero, i.e., when $x_1=x_2=0$

For any $x\in V, \langle x,x\rangle\geq0;$ and $\langle x,x\rangle=0$ if and only if $x=0.$

2. Symmetric Property

$$\langle x,y\rangle=2x_1y_1-x_1y_2-x_2y_1+2x_2y_2$$

$$=2y_1x_1-y_1x_2-y_2x_1+2y_2x_2=\langle y, x\rangle$$

3. Linearity

$$\langle ax+by,z\rangle$$

$$=2(ax_1+by_1)z_1-(ax_1+by_1)z_2-(ax_2+by_2)z_1$$

$$+2(ax_2+by_2)z_2$$

$$=a(2x_1z_1-x_1z_2-x_2z_1+x_2z_2)$$

$$+b(2y_1z_1-y_1z_2-y_2z_1+y_2z_2)$$

$$=a\langle x,z\rangle+b\langle y,z\rangle$$

Then $\langle x,y\rangle$ is an inner product on $V.$