# Equivalence theorem for nonsingular matrices: the columns of A span R^n (or C^n).

**Theorem:** An  	imes $n \times n$ matrix $A$ is nonsingular if and only if Theiff the columns of $A$ span $\mathbb{R}^n$ (or $\mathbb{C}^n$). The column space equals the entire space, meaningso $Ax=b$ is solvable for every $b$. For $n$ columns in $n$ dimensions, spanning implies linear independence, hence nonsingularity.

# Parents

* Nonsingular matrices and equivalences