inner product

Overview

Inner products give information about angles between vectors in a vector space.

Definition

Let $V$ be a vector space. An inner product is a map

$$⟨,⟩:V\times V⟶\mathbb{R}$$

that satisfies the following properties:

1. Symmetric. $⟨x,y⟩=⟨y,x⟩$

2. Bilinear

3. Positive definite $⟨x,x⟩>0$ for $x\ne 0$

Note on notation

Since an inner product is bilinear, this can be encoded using the tensor product. In this way, the map would be a linear map

\langle , \rangle: V \otimes V \longrightarrow \mathbb{R}

An inner product using coordinates

Let $\{u_i\}$ be a basis for $V$. Then we can build a matrix $A_{ij}=⟨u_i,u_j⟩$. This would give

$$\begin{array}{rl}⟨v,w⟩& =⟨v^i{u}_i,w^j{u}_j⟩\\ & =A_{ij}{v}^i{w}^j\\ & =v^{T}Aw\end{array}$$

for $A$ a symmetric, positive definite matrix.