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  • Orthogonality

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  • Definition of angle between vectors
  • Definition of orthogonal vectors
  • Definition of parallel vectors
  • Pythagorean equality characterizes orthogonality over the reals
  • Definition of a vector being orthogonal to a subspace
  • Definition of orthogonal complement of a subspace
  • The orthogonal complement of a subspace is a subspace.
  • The direct sum of a subspace and its orthogonal complement is the whole space.
  • A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace.
  • Null spaces are orthogonal complements of the corresponding row or column spaces
  • Definition of orthogonal set of vectors
  • Definition of orthonormal set of vectors
  • An orthogonal set of nonzero vectors is linearly independent.
  • Definition of orthogonal basis of a (sub)space
  • Definition of orthonormal basis of a (sub)space
  • A square real matrix has orthonormal columns iff its inverse is its transpose
  • A matrix with real entries and orthonormal columns preserves norms.
  • A matrix with real entries and orthonormal columns preserves dot products.
  • Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.
  • A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.
  • Description of the Gram-Schmidt process
  • The Gram-Schmidt process converts a linearly independent set into an orthogonal set.
  • Definition of Gram-Schmidt process
  • Definition of orthogonal subspaces
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  1. Definition of angle between vectors

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  • The Cauchy–Schwarz inequalityClose match
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  • The standard inner product of a vector with itself is 0 only for the 0 vectorRelated
  • The standard inner product of a vector with itself is non-negativeRelated
  • Inner products in coordinate spacesRelated
  • Definition of a vector being orthogonal to a subspaceRelated
  • Pythagorean equality characterizes orthogonality over the realsRelated
  • Inner productsRelated
  • Definition of orthogonal set of vectorsRelated
  • Complex inner products conjugate scalars in the first argumentRelated
  • Definition of unit vectorRelated
  • The triangle inequalityRelated
  • Definition of length/norm of a vectorRelated
  • Definition of distanceRelated
  • Definition of norm/length of a vectorRelated
  • Definition of (orthogonal) projection of one vector onto another vectorRelated
  • An orthogonal set of nonzero vectors is linearly independent.Related

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