Definition and Properties of an Inner Product

Section 5.2 Definition and Properties of an Inner Product

An inner product \(\left\langle \vec{u}\vert \vec{v}\right\rangle\) is a generalization of the dot product with the following properties:

\begin{align} \left\langle \vec{u}\vert \vec{v}\right\rangle \amp = \left\langle \vec{v}\vert \vec{u}\right\rangle^*\tag{5.2.1}\\ \left\langle \vec{u}\vert \lambda\vec{v}+\mu\vec{w}\right\rangle \amp=\lambda\left\langle \vec{u}\vert \vec{v}\right\rangle +\mu\left\langle \vec{u}\vert \vec{w}\right\rangle\tag{5.2.2} \end{align}

Like the dot product, the inner product is always a rule that takes two vectors as inputs and the output is a scalar (often a complex number).

The existence of an inner product is NOT an essential feature of a vector space. A vector space can have many different inner products (or none).

In analogy with inner products, we call the square root of the inner product of a vector with itself \(\braket{v}{v}\) the norm or the length of the vector. Similarly, if the inner product of two vectors is zero \(\braket{v}{w}=0\) we say that the vectors are orthogonal even when these statements have no obvious geometric meaning.

***Add links to sections about dot products for matrices/columns (dotprod) and how to do inner products in multiple reps. (innerprod). ***