Orthogonality

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In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek ὀρθός (orthos), meaning "straight", and γωνία (gonia), meaning "angle".

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Definitions

  • Two vectors x and y in an inner product space V are orthogonal if their inner product \langle x, y \rangle is zero. This situation is denoted x \perp y.
  • Two vector subspaces A and B of an inner product space V are called orthogonal subspaces if each vector in A is orthogonal to each vector in B. The largest subspace that is orthogonal to a given subspace is its orthogonal complement.
  • A linear transformation T : V \rightarrow 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,

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