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# Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.

Put content here**Theorem:** An $n \times n$ matrix $A$ is nonsingular iff $T(x) = Ax$ is one-to-one (injective). $T$ is injective iff $\ker(T) = \{0\}$, which means $Ax = 0$ has only the trivial solution.

# Parents

* Nonsingular matrices and equivalences