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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 codomain of a linear transformation#23841
  • Definition of coefficient matrix of a linear system#23756
  • Definition of coefficients of a linear equation#23414
  • Definition of cofactor/submatrix of a matrix#23717
  • Definition of column rank of a matrix#23673
  • Definition of column space of a matrix#23875
  • Definition of column vector#23780
  • Definition of complement of a subspace#23975
  • Definition of composition of linear transformations#24023
  • Definition of conjugate of a matrix#23465
  • Definition of conjugate of a vector in C^n#23788
  • Definition of consistent linear system#23431
  • Definition of constant vector of a linear system#23757
  • Definition of coordinates relative to a given basis#23835
  • Definition of coordinate vector/mapping/representation relative to a given basis#23991
  • Definition of cross product#24121
  • Definition of determinant of a matrix as a cofactor expansion across the first row#23718
  • Definition of determinant of a matrix as a product of the diagonal entries in a non-scaled echelon form.#23724
  • Definition of diagonalizable linear transformation#24041
  • Definition of diagonalizable matrix#23611
  • Definition of diagonal matrix#23456
  • Definition of dimension of a vector space (or subspace)#23837
  • Definition of dimension of a vector space (or subspace) being finite or infinite#24009
  • Definition of direct sum of subspaces#23970
  • Definition of distance#23911
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  • Aspen Summer 2016 - Entanglement Matters#1213
  • Math#23168

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