For a linear transformation $T:\mathbb{R}^n\to\mathbb{R}^n$ represented by a square matrix $A$, several apparently different properties describe the same situation.

The following conditions are equivalent:

• $A$ is invertible;
• $\ker(T)={\mathbf{0}}$;
• $\operatorname{im}(T)=\mathbb{R}^n$;
• $\operatorname{rank}(A)=n$;
• $\det(A)\ne 0$;
• for every $\mathbf{b}$, the system $A\mathbf{x}=\mathbf{b}$ has exactly one solution.

Geometrically, invertibility means that no input direction is collapsed and every output direction remains reachable. Algebraically, that same fact appears as full rank, a trivial kernel, nonzero determinant and unique solvability.

Visual synthesis: inverse matrices, column space and null space