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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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176–200 of 715 nodes
  • Definition of Hessenberg form#23619
  • Definition of homogeneous linear system of equations#23433
  • Definition of how the action of a linear transformation on a basis extends to the whole space#23867
  • Definition of identity linear transformation#24014
  • Definition of identity matrix#23457
  • Definition of ill-conditioned linear system#23446
  • Definition of image (of a point) under a linear transformation#23842
  • Definition of inconsistent linear system#23432
  • Definition of independent subspaces#23972
  • Definition of index of nilpotency#23584
  • Definition of inner/dot product on C^n#23896
  • Definition of inner/dot product on R^n#23895
  • Definition of inner product#24088
  • Definition of inner product space#24089
  • Definition of intersection of subspaces#23966
  • Definition of invariant subspace of a linear transformation.#24052
  • Definition of inverse of a linear transformation#23849
  • Definition of inverse of a matrix#23517
  • Definition of invertible linear transformation#23848
  • Definition of invertible matrix#23518
  • Definition of invertible/nonsingular linear transformation#24021
  • Definition of isomorphic/isomorphism between vector spaces#24001
  • Definition of Jordan form#23620
  • Definition of kernel/null space of linear transformation#24058
  • Definition of kernel of linear transformation#23847
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  • Aspen Summer 2016 - Entanglement Matters#1213
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