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

**Theorem:** An  	imes $n \times n$ matrix $A$ is nonsingular if and only if Theiff the rows of $A$ span $\mathbb{R}^n$ (or $\mathbb{C}^n$). This means every vector can be expressed as a linear combination of the rows. Since there are $n$ rows in an $n$-dimensional space, spanning implies the rows arethey form a basis, which is equivalent to nonsingularity.

# Parents

* Nonsingular matrices and equivalences