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.


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


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


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


 * $(4): \quad \map \phi B$ is a basis of $H$ for every basis $B$ of $G$.


 * $(5): \quad \map \phi B$ 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 $\map \phi B$ 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 Rank Plus Nullity Theorem.

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