# Equivalence theorem for nonsingular matrices: the columns of A are linearly independent.

**Theorem:** An  	imes $n \times n$ matrix $A$ is nonsingular if and only if Theiff the columns of $A$ are linearly independent. The only solution to $c_1 \mathbf{a}_1 + \cdots + c_n \mathbf{a}_n = 0$ is all $c_i = 0$. This is equivalent to $Ax=0$ having only the trivial solution.

# Parents

* Nonsingular matrices and equivalences