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  • Orthogonality

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  • Definition of angle between vectors
  • Definition of orthogonal vectors
  • Definition of parallel vectors
  • Two vectors are orthogonal if and only if the Pythagorean Theorem holds.
  • 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.
  • The null space of a matrix is the orthogonal complement of the column space.
  • 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 matrix A with real entries has orthonormal columns if and only if A inverse equals A 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. 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.

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  • Definition of leading entry in a row of a matrix#23481
  • Definition of least-squares error of a linear system#23947
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  • Definition of left inverse of a matrix#23515
  • Definition of length/norm of a vector#24090
  • Definition of linear combination of vectors#23800
  • Definition of linear dependence relation on a set of vectors#23816
  • Definition of linear equation#23413
  • Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors#23821
  • Definition of linearly independent set of vectors: if a linear combination is 0#23819
  • Definition of linear transformation/homomorphism#24013
  • Definition of LU decomposition#23625
  • Definition of Markov matrix#23595
  • Definition of matrix#23449
  • Definition of matrix diagonalization#23610
  • Definition of matrix equation#23764
  • Definition of matrix in reduced row echelon form#23543
  • Definition of matrix multiplication#23489
  • Definition of matrix multiplication in terms of column vectors#23488
  • Definition of matrix null space (left)#23874
  • Definition of matrix null space (right)#23873
  • Definition of matrix representation of a linear system#23759
  • Definition of matrix representation of a linear transformation#23861
  • Definition of matrix representation of a linear transformation from a vector space to itself#24070
  • Definition of matrix representation of a linear transformation with respect to bases of the spaces#24068
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