Prove that the $\langle x,y\rangle$ is an inner product.

Question

Let's consider the vector space $\mathbb{R}^2$ with $\langle x,y\rangle=2x_1y_1+x_2y_2$. Prove $\langle \cdot,\cdot \rangle$ is an inner dot product.

I know that these 4 properties have to be satisfied to prove it is an inner dot product:

1. $\langle x+z,y\rangle=\langle x,y\rangle+\langle z,y\rangle$

2.$\langle cx,y\rangle=c\langle x,y\rangle$

3. $\langle y,x\rangle=\overline{\langle x,y\rangle}$

4. If $x\neq 0$, then $\langle x,x\rangle$ is a positive real and $\langle 0,0\rangle=0$

I'm just not sure how to apply the definition to the actual problem.

Answer 1

1. $\langle x+z,y \rangle = 2(x_1+z_1)y_1 + (x_2+z_2)y_2 = 2x_1y_1+x_2y_2 + 2z_1y_1 + z_2y_2 = \langle x,y\rangle + \langle z,y\rangle$
2. $\langle cx,y\rangle = 2cx_1y_1 + cx_2y_2 = c(2x_1y_1 + x_2y_2) = c\langle x,y\rangle$
3. This is trivially true because the norm is symmetric, and we are working on reals.
4. $\langle x,x\rangle = 2x_1^2+x_2^2$, which is $>0$ if and only if $x \neq (0,0)$.