Linear Transformation of Vector Space Equivalent Statements

Theorem
Let $$G$$ and $$H$$ be $n$-dimensional vector spaces.

Let $$\phi: G \to H$$ be a linear transformation.

Then these statements are equivalent:
 * 1) $$\phi$$ is an isomorphism.
 * 2) $$\phi$$ is a monomorphism.
 * 3) $$\phi$$ is an epimorphism.
 * 4) For every basis $$B$$ of $$G$$, $$\phi \left({B}\right)$$ is a basis of $$H$$.
 * 5) For some basis $$B$$ of $$G$$, $$\phi \left({B}\right)$$ is a basis of $$H$$.

Proof

 * (1) implies (2) by definition.


 * (2) implies (4) by Linear Transformation of Vector Space Monomorphism and Results concerning Generators and Bases of Vector Spaces.


 * (4) implies (5) by basic logic.


 * Suppose $$\phi \left({B}\right)$$ is a basis of $$H$$. Then the range of $$\phi$$ is a subspace of $$H$$ generating $$H$$ and hence is $$H$$ itself. Thus (5) implies (3).


 * Finally, (3) implies that $$\phi$$ is injective.

If $$\phi$$ is surjective, the dimension of its kernel is $$0$$ by Sum of Nullity and Rank of Linear Transformation.

Hence $$\phi$$ is an isomorphism and therefore (3) implies (1).