Orthogonal subspaces
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u,v) = 0. When the bilinear form corresponds to a pseudo-Euclidean space, there are non-perpendicular vectors that are hyperbolic-orthogonal. In the case of function spaces, families of orthogonal functions are used to form a basis.
The word comes from the Greek ὀρθός (orthos), meaning "upright", and γωνία (gonia), meaning "angle". The ancient Greek ὀρθογώνιον orthogōnion (< ὀρθός orthos 'upright' + γωνία gōnia 'angle') and classical Latin orthogonium originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle.
A set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set. Nonzero pairwise orthogonal vectors are always linearly independent.
In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface. For example, the y-axis is normal to the curve y = x2 at the origin. However, normal may also refer to the magnitude of a vector. In particular, a set is called orthonormal (orthogonal plus normal) if it is an orthogonal set of unit vectors. As a result, use of the term normal to mean "orthogonal" is often avoided. The word "normal" also has a different meaning in probability and statistics.
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