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  • Nonsingular matrices and equivalences

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  • Definition of nonsingular matrix: matrix is invertible
  • Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solution
  • Definition of singular matrix (not nonsingular)
  • Proof of several equivalences for nonsingular matrix
  • Equivalence theorem for nonsingular matrices: the equation Ax=b has a unique solution for all b.
  • Equivalence theorem for nonsingular matrices: the equation Ax=b has a solution for all b.
  • Equivalence theorem for nonsingular matrices: the equation Ax=0 has only the trivial solution.
  • Equivalence theorem for nonsingular matrices: the rows of A span R^n (or C^n).
  • Equivalence theorem for nonsingular matrices: the columns of A span R^n (or C^n).
  • Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.
  • Equivalence theorem for nonsingular matrices: the columns of A are linearly independent.
  • Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).
  • Equivalence theorem for nonsingular matrices: the columns of A are a basis for R^n (or C^n).
  • Equivalence theorem for nonsingular matrices: the dimension of the column space of A is n.
  • Equivalence theorem for nonsingular matrices: there is a pivot position in every row of A.
  • Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.
  • Equivalence theorem for nonsingular matrices: the matrix A has an inverse.
  • Equivalence theorem for nonsingular matrices: the matrix A has a left inverse.
  • Equivalence theorem for nonsingular matrices: the matrix A has a right inverse.
  • Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.
  • Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.
  • Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is onto/surjective.
  • Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax has an inverse.
  • Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is an isomorphism.
  • Equivalence theorem for nonsingular matrices: the determinant of A is nonzero.
  • Equivalence theorem for nonsingular matrices: the matrix A has rank n.
  • Equivalence theorem for nonsingular matrices: the null space of the matrix A is {0}.
  • Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.
  • Equivalence theorem for nonsingular matrices: the matrix A does not have 0 as an eigenvalue.
  • Equivalence theorem for nonsingular matrices: the matrix A is a change-of-basis matrix.
  • Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.
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  1. Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.

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  • A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.#23824
  • A subset of a linearly independent set is linearly independent.#23980
  • A vector can be written uniquely as a linear combination of vectors from independent subspaces.#23973
  • A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.#23934
  • 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.#23923
  • Axioms of a vector space#23798
  • Bases#23828
  • Basic properties#23957
  • Basic properties of linear transformations#23863
  • Basic terminology#23412
  • Basic terminology and notation#23448
  • Block matrices#23567
  • Canonical forms of matrices#23608
  • Change of coordinates matrices are invertible#23998
  • Characteristic and minimal polynomials#23699
  • C^n is a vector space.#24098
  • Cofactors#23716
  • Composition#24022
  • Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.#23999
  • Conjugation#23464
  • Container for Linear Algebra#24123
  • Coordinates#23989
  • Coordinate vector spaces#23776
  • Cramer's rule#23722
  • Definition and terminology#23954
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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