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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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  • Definition of orthogonal subspaces#24094
  • Definition of orthogonal vectors#23916
  • Definition of orthonormal basis of a (sub)space#23929
  • Definition of orthonormal set of vectors#23926
  • Definition of parallel vectors#23917
  • Definition of permutation matrix#23555
  • Definition of pivot#23476
  • Definition of pivot column#23477
  • Definition of pivot position#23478
  • Definition of positive-definite matrix#23713
  • Definition of pre-image (of a point) under a linear transformation#23843
  • Definition of pre-image of linear transformation#24019
  • Definition of QR decomposition#23631
  • Definition of quadratic form#24122
  • Definition of range of a linear transformation#24020
  • Definition of range of linear transformation#23846
  • Definition of rank factorization of a matrix#23628
  • Definition of rank of a linear transformation#23883
  • Definition of rank of a matrix#23674
  • Definition of rational form#23621
  • Definition of reduced LU decomposition#23627
  • Definition of reduced row echelon form of a matrix#23544
  • Definition of right inverse of a matrix#23516
  • Definition of R^n (or C^n)#23781
  • Definition of (row) echelon form of a matrix#23542
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