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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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  • Equivalence theorem for nonsingular matrices: the matrix A is a change-of-basis matrix.#23670
  • Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.#23671
  • Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.#23656
  • Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.#23668
  • Equivalence theorem for nonsingular matrices: the null space of the matrix A is {0}.#23667
  • Equivalence theorem for nonsingular matrices: there is a pivot position in every row of A.#23655
  • Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).#23652
  • Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.#23650
  • Equivalence theorem for nonsingular matrices: the rows of A span R^n (or C^n).#23648
  • Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.#23660
  • Equivalence theorems for injective transformations#24077
  • Equivalent matrices represent the same linear transformation with resect to appropriate bases.#23605
  • Every basis for a vector space contains the same number of elements#24008
  • Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.#24006
  • Every matrix has an eigenvalue over the complex numbers.#23687
  • Every matrix is row-equivalent to a matrix in reduced row echelon form.#23548
  • Every matrix is row-equivalent to only one matrix in reduced row echelon form.#23549
  • Every nilpotent matrix is similar to one with 1 on subdiagonal blocks and all other entries 0.#23582
  • Every square matrix is conjugate#23566
  • Every square matrix is similar the sum of a diagonal and a nilpotent matrix.#23585
  • Every square matrix is similar to one in Jordan form.#23639
  • Example of a linear transformation on R^2: generic#23888
  • Example of a linear transformation on R^2: projection#23891
  • Example of a linear transformation on R^2: rotation#23890
  • Example of a linear transformation on R^2: shear#23889
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