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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 row equivalent matrices#23479
  • Definition of row operations on a matrix#23475
  • Definition of row reduce a matrix#23545
  • Definition of row space of a matrix#23880
  • Definition of scalar#23778
  • Definition of scalar multiple of a linear transformation#24017
  • Definition of Schur triangulation#23633
  • Definition of similarity transform#23637
  • Definition of similar matrices#23635
  • Definition of singular matrix (not nonsingular)#23643
  • Definition of singular value decomposition (SVD)#23623
  • Definition of size of a matrix#23451
  • Definition of size of a vector#23782
  • Definition of skew-symmetric matrix#23579
  • Definition of solution set of a system of linear equations#23418
  • Definition of solution to a linear equation#23415
  • Definition of solution to a system of linear equations#23417
  • Definition of solution vector of a linear system#23758
  • Definition of spanning/generating set for a space or subspace#23978
  • Definition of span of a set of vectors#23805
  • Definition of square matrix#23454
  • Definition of subspace#23812
  • Definition of subspace spanned by a set of a set of vectors#23813
  • Definition of sum of linear transformations#24015
  • Definition of sum of matrices#23462
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