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  • Orthogonality

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  • Definition of angle between vectors
  • Definition of orthogonal vectors
  • Definition of parallel vectors
  • Pythagorean equality characterizes orthogonality over the reals
  • 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.
  • Null spaces are orthogonal complements of the corresponding row or column spaces
  • 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 square real matrix has orthonormal columns iff its inverse is its 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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551–575 of 718 nodes
  • Spans#23804
  • Subspaces#23811
  • Subspaces associated to a linear transformation#24055
  • Subspaces associated to a matrix#23872
  • Switching two rows multiplies the determinant by -1.#23738
  • Symmetric matrices#23571
  • Symmetric matrices are square.#23573
  • Terminology#23754
  • The 0 scalar multiplied by any vector equals the 0 vector.#23960
  • The 0 vector is unique.#23958
  • The 0 vector multiplied by any scalar equals the 0 vector.#23961
  • The additive inverse of a vector equals the vector multiplied by -1.#23963
  • The additive inverse of a vector is called the negative of the vector.#23956
  • The additive inverse of a vector is unique.#23959
  • The adjoint of a matrix-scalar product is the product of the adjoint and the conjugate.#23508
  • The adjoint of a product of matrices is the product of the adjoints in reverse order.#23513
  • The adjoint of a sum is the sum of the adjoints.#23507
  • The Cauchy-Schwarz inequality#24092
  • The Cauchy–Schwarz inequality#23913
  • The Cayley-Hamilton theorem for a linear transformation#24046
  • The Cayley-Hamilton theorem for a matrix.#23707
  • The change of coordinates matrix between two bases exists and is unique#23996
  • The characteristic polynomial applied to the matrix gives the 0 matrix.#23706
  • The column space of a matrix is a vector space#24109
  • The column space of an m-by-n matrix is a subspace of R^m (or C^m)#23876
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