Inner Products

What and why

Inner products are operations that take 2 vectors $\vec{x}, \vec{y}$ in the vector space $V$ as inputs and return a scalar. We denote the inner product using angle brackets: $\langle \vec{x}, \vec{y} \rangle$

$\langle \vec{x}, \vec{y}\rangle: V \times V \rightarrow \mathbb{R}$

Inner products can be used to compare two vectors in the vector space and are helpful in finding the projection of a vector onto the a vector subspace.

Quick overview of inner products: