Nonsingular matrices and transformations

This concerns square matrices and also transformations $T:V\rightarrow W$ where $V$ and $W$ have the same finite dimension.

Theorem. Let $A$ be an $n \times n$ matrix with entries in a field $F$, with the corresponding matrix transformation $\tau _ A: F ^ n \rightarrow F ^ n$. Also let $T:V\rightarrow W$ be a linear transformation between vector spaces over $F$, both of dimension $n$, such that $T$ has matrix $A$ with respect to particular bases of $V$ and $W$. The following conditions are equivalent.

1. det$A \neq 0$.

2. $A$ row-reduces to the $n \times n$ identity matrix.

3. $A$ has rank $n$ (``full rank'', meaning the maximum rank possible).

4. The rows of $A$ are linearly independent.

5. The columns of $A$ are linearly independent.

6. Some system of linear equations with coefficient matrix $A$ has a unique solution.

7. Every system of linear equations with coefficient matrix $A$ has a unique solution.

8. $A$ has a right inverse, i.e., there is an $n \times n$ matrix $B$ with $AB = I$.

9. $A$ has a left inverse, i.e., there is an $n \times n$ matrix $B$ with $BA = I$.

10. $A$ has a two-sided inverse $A^{-1}$.

11. $A$ has nullity 0; in other words, nullspace$~\tau _ A = \{$0$\}$.

12. $Av =$   0$\Rightarrow v =$   0

13. 0 is not an eigenvalue of $A$, i.e., there is no $v \neq 0$ with $Av = 0 v$.

14. $\tau _ A$ is one-to-one.

15. $\tau _ A$ is onto.

16. $\tau _ A$ is an isomorphism of $F^n$ with itself (an ``automorphism'' of $F^n$)

17. Nullspace$(T)= \{$0$\}$.

18. $T$ is one-to-one.

19. $T$ is onto.

20. $T$ is an isomorphism.

21. The matrix of $T$ with respect to any bases of $V$ and $W$ is nonsingular.

Definition. When any (and so all) of these conditions is satisfied, then $A$ is said to be nonsingular. Otherwise A is singular.

(``Singular'' means ``special'' or ``unusual'', not to be confused with ``single''. So a system of linear equations with nonsingular coefficient matrix has a single solution.)