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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 matrix A represents the identity map with respect to some pair of bases.

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Definition of distance between vectors#24091
Definition of domain of a linear transformation#23840
Definition of echelon form of a linear system#23422
Definition of (echelon matrix/matrix in (row) echelon form)#23539
Definition of eigenspace of a linear transformation#24050
Definition of eigenspace of a matrix#23684
Definition of eigenvalue/characteristic value of a linear transformation#24042
Definition of eigenvalue of a matrix#23681
Definition of eigenvector/characteristic vector of a linear transformation#24043
Definition of eigenvector of a matrix#23682
Definition of elementary matrix#23557
Definition of entry/component of a vector#23783
Definition of equality of matrices#23459
Definition of equality of vectors#23785
Definition of equation operations on a linear system#23424
Definition of equivalent matrices#23604
Definition of equivalent systems of linear equations#23426
Definition of extended reduced row echelon form of a matrix#23550
Definition of free/independent variable in a linear system#23443
Definition of generalized inverse of a matrix#23534
Definition of generalized kernel/null space of linear transformation#24060
Definition of generalized range space of a linear transformation#24039
Definition of geometric multiplicity of an eigenvalue#23697
Definition of Gram-Schmidt process#24093
Definition of Hermitian/self-adjoint matrix#23597

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