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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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326–350 of 715 nodes
  • Definition of weights in a linear combination of vectors#23802
  • Description of a basis for the null space of a matrix from the reduced row-echelon form.#23870
  • Description of a spanning set for the null space of a matrix from the reduced row-echelon form.#23869
  • Description of the Gram-Schmidt process#23935
  • Determinants#23714
  • Determinants and operations on matrices#23723
  • Determinants axiomatically#23732
  • Determine if a particular set of vectors in R^3 in linearly independent#23818
  • Determine if a particular set of vectors spans R^3#23807
  • Determine if a particular vector is in the span of a set of vectors#23808
  • Determine if a particular vector is in the span of a set of vectors in R^2#23809
  • Determine if a particular vector is in the span of a set of vectors in R^3#23810
  • Dimension#23836
  • Distinct eigenvalues of a Hermitian matrix have orthogonal eigenvectors.#23712
  • Each vector can be written uniquely as a linear combination of vectors from a given basis.#23987
  • Echelon matrices#23536
  • Eigenspaces#23683
  • Eigenvalues and eigenvectors#23680
  • Eigenvalues and operations on matrices#23688
  • Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal.#23574
  • Eigenvectors with distinct eigenvalues are linearly independent.#23694
  • Elementary matrices#23556
  • Elementary matrices are invertible/nonsingular.#23560
  • Equation operations on a linear system give an equivalent system.#23425
  • Equivalence theorem for injective linear transformations: The columns of the matrix of T are linearly independent.#24086
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