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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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Equivalence theorem for nonsingular matrices: the equation Ax=0 has only the trivial solution.
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Nonsingular matrices and equivalences