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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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301–325 of 715 nodes
  • Definition of sum of subspaces#23968
  • Definition of symmetric matrix#23572
  • Definition of system of linear equations#23416
  • Definition of the 0/trivial subspace#23814
  • Definition of the determinant in terms of the effect of elementary row operations#23726
  • Definition of the imaginary part of a vector in C^n#23790
  • Definition of the least-squares linear fit to 2-dimensional data#23951
  • Definition of the (main) diagonal of a matrix#23455
  • Definition of the real part of a vector in C^n#23789
  • Definition of the standard basis of the m by n matrices#23834
  • Definition of the standard basis of the polynomials of degree at most n#23833
  • Definition of the standard matrix for a linear transformation#23857
  • Definition of the standard/natural basis of R^n (or C^n)#23830
  • Definition of trace of a matrix#23715
  • Definition of transpose of a matrix#23503
  • Definition of trivial linear dependence relation on a set of vectors#23817
  • Definition of trivial solution to a homogeneous linear system of equations#23436
  • Definition of unitary matrix#23588
  • Definition of unit matrix#23554
  • Definition of unit vector#23909
  • Definition of Vandermonde matrix#23594
  • Definition of vector#23779
  • Definition of vector addition#23955
  • Definition of vector-scalar multiplication#23792
  • Definition of vector sum/addition#23786
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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