What is Orthonormal?

Two vectors x and y are orgonal and each vector has the length of one are said to be orthonormal.
length of x = sqrt(<x, x>) = 1
or simply <x, x> = 1
<y, y> = 1
<x, y> = 0

What is Orthogonal?

Two vectors x and y are described as orthogonal if their inner product is zero:
<x, y> = 0

Inner Product

The inner product is denoted by <x, y> where x and y are vectors.
If x = (1, 2, 3) and y = (6, 5, 4), then
<x, y> = (1)(6) + (2)(5) + (3)(4) = 28.
It is obvious that <x, y> is a scalar.

A Field

A field (F, +, .) is a set F with operations `+` (or addition) and `.` (or multiplication) satisfying the properties including closure for addition (x + y also in F),
commutativity for addition (x + y = y + x),
associativity for addition (x + (y+z) = (x+y) + z),
existence of additive identity (x + 0 = x),
existence of additive inverse (-x),
closure for multiplication (x.y also in F),
commutativity for multiplication (x.y = y.x),
associativity for multiplication (x.(y.z) = (x.y).z),
existence of multilicative identity (x.1 = x),
existence of multiplicative inverse (1/x), and
distributive law (x.(y+z) = (x.y) + (x.z)).

The set of Integers

The set of integers is considered as the set of natural numbers, zero, the negative of natural numbers. This is {..., -3, -2, -1, 0, 1, 2, 3, ...}.
It is denoted by Z.

Natural Numbers

The natural numbers {1, 2, 3, 4, ...} usually denotes by N.

The Euclidean norm (or length)

A convenient measure of the "length" of a vector x is the Euclidean norm:
|x| = norm of x = (x,x)^(1/2)

The product of two n-dimensional vectors

(x,y) = x1 y1 + x2 y2 + ... + xn yn = x y