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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 0 matrix#23458
  • Definition of 0/trivial subspace#23964
  • Definition of 0 vector#23784
  • Definition of adjoint (conjugate transpose)#23506
  • Definition of adjugate/classical adjoint of a matrix#23748
  • Definition of (algebraic) multiplicity of an eigenvalue#23696
  • Definition of a lower triangular matrix#23563
  • Definition of angle between vectors#23915
  • Definition of an upper triangular matrix#23562
  • Definition of applying a polynomial to a linear transformation#24048
  • Definition of applying a polynomial to a square matrix#23700
  • Definition of augmented matrix (of a linear system)#23755
  • Definition of automorphism of a vector space#24007
  • Definition of a vector being orthogonal to a subspace#23919
  • Definition of band matrix#23593
  • Definition of basic/dependent/leading variable in a linear system#23442
  • Definition of basis of a vector space (or subspace)#23829
  • Definition of block diagonal matrix#23570
  • Definition of block/partitioned matrix#23568
  • Definition of change of coordinates matrix between two bases#23995
  • Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)#23832
  • Definition of characteristic equation of a matrix#23702
  • Definition of characteristic polynomial of a linear transformation#24044
  • Definition of characteristic polynomial of a matrix#23701
  • Definition of Cholesky decomposition#23629
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