Linear Algebra Identity Proof Prove that the following identity holds for all $x,y$ are in $\mathbb{R}^n$:
$x\cdot{}y = \frac{1}{4}(\left\|x+y\right\|^2 - \left\|x-y\right\|^2)$

We have
$$
\begin{align}
\frac14 \left(\left\|x + y\right\| ^2 - \left\|x - y\right\|^2\right) &= \frac14 \left((x + y)\cdot(x+y) - (x - y)\cdot(x-y)\right)\\
& = \frac14(x \cdot x + 2x \cdot y + y \cdot y - (x \cdot x - 2 x \cdot y + y \cdot y))\\
&= \frac14(4x \cdot y)\\
&= x \cdot y
\end{align}$$
and so we are done.
