Orthogonality Orthogonality Net protein utilization Category="Abstract algebra"Category="Linear algebra"In Mathematics, orthogonal is synonymous with Perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. It means at right angles. It comes from the Greek "ortho", meaning "right", and "gonia", meaning "angle". Two streets that cross each other at a right angle are orthogonal to each other.

Formally, two vectors x and y in an Inner product space V are orthogonal if their inner product 〈 x, y 〉 is zero. This situation is denoted x ⊥ y.

Two subspaces A and B of V are called orthogonal subspaces if each vector in A is orthogonal to each vector B. Note however that this does not correspond with the geometric concept of perpendicular planes. The largest subspace that is orthogonal to a given subspace is its Orthogonal complement.

A Linear transformation T : V → V is called an orthogonal linear transformation if it preserves the inner product. That is, for all pairs of vectors x and y in the inner product space V,

〈 Tx, Ty 〉 = 〈 x, y 〉.
This means that T preserves the angle between x and y, and that the lengths of Tx and x are equal.

The word normal is sometimes also used in place of orthogonal. However, normal can also refer to vectors of unit length. In particular, orthonormal refers to a collection of vectors that are both orthogonal and of unit length. So the orthogonal usage of the term normal is often avoided.

In Euclidean vector spaces

For example, in a 2- or 3-dimensional Euclidean space, two vectors are orthogonal if their Dot product is zero, i.e., they make an angle of 90° or π/2 radians. Hence orthogonality of vectors is a generalization of the concept of Perpendicular. Also a line through the origin is orthogonal to a plane through the origin if they are perpendicular. Note however that there is no correspondence with regard to perpendicular planes.

In 4D the orthogonal complement of a line is a Hyperplane and vice versa, and that of a plane is a plane.

Several vectors are called pairwise orthogonal if any two of them are orthogonal, and a set of such vectors is called an orthogonal set. They are said to be orthonormal if they are all unit vectors. Non-zero pairwise orthogonal vectors are always linearly independent.

Orthogonal functions

We commonly use the following inner product to say that two functions f and g are orthogonal:
〈 f, g 〉 =  ∫b
 
a		 f (x)g (x)w (x) dx = 0.
Here we introduce a nonnegative Weight function w (x), and we write
〈 f, gw 〉 = 〈 f, g 〉 w .
We write the norms with respect to this inner product and the weight function as