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

**Theorem:** An  	imes $n \times n$ matrix $A$ is nonsingular if and only if Theiff the columns of $A$ form a basis for $\mathbb{R}^n$ (or $\mathbb{C}^n$). Every vector $b$ can be uniquely expressed as $Ax$, which means $A$ is invertible.

# Parents

* Nonsingular matrices and equivalences