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 $(1)$.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965): $\S 28$: Theorem $28.7$