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:
 * $$(1) \quad \phi$$ is an isomorphism.


 * $$(2) \quad \phi$$ is a monomorphism.


 * $$(3) \quad \phi$$ is an epimorphism.


 * $$(4) \quad \phi \left({B}\right)$$ is a basis of $$H$$ for every basis $$B$$ of $$G$$.


 * $$(5) \quad \phi \left({B}\right)$$ is a basis of $$H$$ for some basis $$B$$ of $$G$$.

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 image 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 $$(1)$$.