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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 codomain of a linear transformation#23841
  • Definition of coefficient matrix of a linear system#23756
  • Definition of coefficients of a linear equation#23414
  • Definition of cofactor/submatrix of a matrix#23717
  • Definition of column rank of a matrix#23673
  • Definition of column space of a matrix#23875
  • Definition of column vector#23780
  • Definition of complement of a subspace#23975
  • Definition of composition of linear transformations#24023
  • Definition of conjugate of a matrix#23465
  • Definition of conjugate of a vector in C^n#23788
  • Definition of consistent linear system#23431
  • Definition of constant vector of a linear system#23757
  • Definition of coordinates relative to a given basis#23835
  • Definition of coordinate vector/mapping/representation relative to a given basis#23991
  • Definition of cross product#24121
  • Definition of determinant of a matrix as a cofactor expansion across the first row#23718
  • Definition of determinant of a matrix as a product of the diagonal entries in a non-scaled echelon form.#23724
  • Definition of diagonalizable linear transformation#24041
  • Definition of diagonalizable matrix#23611
  • Definition of diagonal matrix#23456
  • Definition of dimension of a vector space (or subspace)#23837
  • Definition of dimension of a vector space (or subspace) being finite or infinite#24009
  • Definition of direct sum of subspaces#23970
  • Definition of distance#23911
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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