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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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  • Abstract vector spaces#23953
  • A consistent system with more variables than equations has infinitely many solutions.#23440
  • Adding a multiple of one row to another row does not change the determinant.#23737
  • Addition#23461
  • A determinant is a multilinear function#23736
  • A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns.#23615
  • Adjoining an element not in the span of a linearly independent set gives another linearly independent set.#23983
  • A homogeneous system has a nontrivial solution if and only if it has a free variable.#23437
  • A homogeneous system with more variables than equations has infinitely many solutions.#23441
  • Algebraic properties of R^n (or C^n)#23777
  • Algorithm for computing an LU decomposition#23626
  • A linear system is equivalent to a matrix equation.#23753
  • A linear system is equivalent to a vector equation.#23752
  • A linear transformation has a representation as an upper triangular matrix.#24076
  • A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.#24075
  • A linear transformation is determined by its action on a basis.#23866
  • A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.#24054
  • A linear transformation is given by a matrix whose columns are the images of the standard basis vectors#23858
  • A linear transformation is given by a matrix with respect to a given basis.#23859
  • A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces#24069
  • A linear transformation is injective on its generalized range space.#24040
  • A linear transformation is invertible if and only if it is injective and surjective#24067
  • A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.#24066
  • A linear transformation is surjective if and only if the columns of its matrix span the codomain.#23885
  • A linear transformation is surjective if and only if the image of a basis is a spanning set#24038
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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