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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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  • Theorem describing matrix multiplication#23490
  • Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrix#23551
  • Theorem describing spaces associated to the block matrices of the extended reduced row echelon form of a matrix#23552
  • Theorem describing the determinants of elementary matrices.#23745
  • Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrix#23553
  • The orthogonal complement of a subspace is a subspace.#23921
  • The (orthogonal) projection of a vector onto a subspace is the point in the subspace closest to the vector.#23943
  • The permutation expansion for determinants#23727
  • The pivot columns of a matrix are a basis for the column space.#23877
  • The preimage of a vector is a translation of the kernel of the linear transformation#24031
  • The product of square matrices is nonsingular if and only if each factor is nonsingular.#23524
  • The product of upper/lower triangular matrices is upper/lower triangular.#23564
  • The projection of a vector which is in a subspace is the vector itself.#23942
  • The QR decomposition of a nonsingular matrix exists.#23632
  • The range/image of a linear transformation is a subspace.#24056
  • The range of a linear transformation is a subspace#24036
  • The range spaces of powers of a linear transformation form a descending chain#24063
  • The rank of a matrix equals number of pivots in a reduced row echelon form.#23676
  • The rank of a matrix equals the rank of the linear transformation it represents.#23677
  • The rank plus the nullity of a linear transformation equals the dimension of the domain.#24064
  • The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.#23986
  • The row space and the column space of a matrix have the same dimension.#23678
  • The row space of a matrix is a vector space#24108
  • The set containing only 0 is a vector space.#24096
  • The set of all functions on a set is a vector space.#24100
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Container for Linear Algebra#24123
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