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Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is an isomorphism.

Created over 8 years ago, updated 10 days ago

Theorem: An $n \times n$ matrix $A$ is nonsingular iff $T(x) = Ax$ is an isomorphism. An isomorphism is a bijective linear transformation, and for finite-dimensional spaces of the same dimension, injectivity, surjectivity, and bijectivity are all equivalent.