Linear Algebra - Symmetric matrices and bilinear forms

Question

Let $A \in K^{n,n}$ be symmetric.

Define $<\cdot,\cdot> : K^n \times K^n \rightarrow K$ by $$<x, y> = x^TAy.$$

Prove that $<\cdot,\cdot>$ is a symmetric bilinear form.

Please can anyone help me out here?

Answer 1

Observe that

$$\langle y,x\rangle:=y^tAx\stackrel{\text{given}\;A=A^t}=y^tA^tx=(x^tAy)^t=\langle x,y\rangle^t=\langle x,y\rangle$$