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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 matrix-scalar multiplication#23471
  • Definition of matrix-vector product#23483
  • Definition of m by n matrix#23452
  • Definition of minimal polynomial of a linear transformation#24045
  • Definition of minimal polynomial of a matrix#23703
  • Definition of multilinear function#23733
  • Definition of nilpotent linear transformation#23860
  • Definition of nilpotent matrix#23581
  • Definition of nonsingular matrix: matrix is invertible#23641
  • Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solution#23642
  • Definition of nontrivial solution to a homogeneous linear system of equations#23435
  • Definition of normal matrix#23600
  • Definition of norm/length of a vector#23908
  • Definition of (not necessarily orthogonal) projection onto a component of a direct sum#23940
  • Definition of nullity of a linear transformation#23884
  • Definition of nullity of a matrix#23675
  • Definition of one-to-one/injective linear transformation#23845
  • Definition of onto/surjective linear transformation#23844
  • Definition of orthogonal basis of a (sub)space#23928
  • Definition of orthogonal complement of a subspace#23920
  • Definition of orthogonally diagonalizable matrix#23577
  • Definition of orthogonal matrix#23586
  • Definition of (orthogonal) projection of one vector onto another vector#23938
  • Definition of (orthogonal) projection onto a subspace#23941
  • Definition of orthogonal set of vectors#23925
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