Inner Products
Overview
An inner product is a function that takes two vectors from a
vector space,
and returns a real number.
(More generally, it returns an element of the
field
over which the vector space is defined.)
Definition
An inner product, denoted {% <x|y> %} is a function that takes two vectors and returns a real number.
To be an inner product, the function must satisfy the following properties.
- {% \langle x,y \rangle ^* = \langle y,x \rangle %}
- {% \langle a\vec{x} + b \vec{y}, \vec{z} \rangle = a \langle \vec{x} ,z \rangle + b \langle \vec{y} , z \rangle %}
- {% \langle x,x \rangle > 0 %}
Column Vectors and Inner Product
If {% u, v %} are a
column vectors,
then the standard definition of the inner product on column vectors is
{% \langle u,v \rangle = u^Tv %}
Fourier Decomposition
Let {% e_1,e_2,...,e_n %} be an ortho-normal basis for a vector space V. Then any vector {% \vec{v} \in V %} can be written as
{% v = \langle v,e_1 \rangle e_1 + ... + \langle v, e_n \rangle e_n %}
Because {% e_1,e_2,...,e_n %} is a basis, we can write
{% v = a_1 e_1 + ... + a_n e_n %}
then
{% \langle v,e_1 \rangle = a_1 %}
because {% e_1,e_2,...,e_n %} are orthonormal
Topics
