The Invertible Matrix Theorem

Let $A$ be a square $n \times n$ matrix. Then the following statements are equivalent. That is, for a given $A$, the statements are either all true or all false.

1. $A$ is an invertible matrix.
2. $A$ is row equivalent to the $n \times n$ identity matrix.
3. $A$ has $n$ pivot positions.
4. The equation $A\vec{x} = 0$ has only the trivial solution.
5. The columns of $A$ form a linearly independent set.
6. The linear transformation $\vec{x} \mapsto A\vec{x}$ is one-to-one.
7. The equation $A\vec{x} = \vec{b}$ has at least one solution for each $\vec{b}$ in $\mathbb{R}^n$.
8. The columns of $A$ span $\mathbb{R}^n$.
9. The linear transformation $\vec{x} \mapsto A\vec{x}$ maps $\mathbb{R}^n$ onto $\mathbb{R}^n$.
10. There is an $n \times n$ matrix $C$ such that $CA = I$.
11. There is an $n \times n$ matrix $D$ such that $AD = I$.
12. $A^T$ is an invertible matrix.

Theorem

Let $T : \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation and let $A$ be the standard matrix for $T$. Then $T$ is invertible if and only if $A$ is an invertible matrix. In that case, the linear transformation $S$ given by $S(\vec{x}) = A^{-1} \vec{x}$ is the unique function satisfying (1) and (2). $$S(T(\vec{x})) = \vec{x} \quad \text{for all} \quad \vec{x} \in \mathbb{R}^n \quad \text{(1)}$$ $$T(S(\vec{x})) = \vec{x} \quad \text{for all} \quad \vec{x} \in \mathbb{R}^n \quad \text{(2)}$$