## Invertible Linear Transformations (Pt. II)

**Point of post: **This is a literal continuation of this post. Treat it as a physical continuation.

The next order of business is proving a simple but laborious set-theoretic theorem which will make our lives easier when proving that something is an isomorphism. We begin by defining some terminology. Let be sets and . We call a right inverse for if . Similarly, is a left inverse if . With this said, we may state and prove:

**Theorem: ***Let be sets and . Then, is bijective if and only if has a right inverse and a left inverse . Moreover, if this is true then .*

**Proof: **To prove this theorem we first prove the following claims.

**Claim 1: **A function possesses a left inverse if and only if it’s injective.

and

**Claim 2: **A function possesses a right inverse if and only if it’s surjective.

To prove this first claim we first assume that possesses a left inverse and show that this implies that is injective. To see this we merely note that if then . Conversely, suppose that is injective. If is surjective then we can take , otherwisefix . We then define

evidently then .

*Remark: *Note that for a non-surjective function there is a left inverse for each element of .

Now, to prove the second claim we first assume that possesses a right inverse . Then, for we know that so that is the guaranteed element of which maps to . Thus, is surjective.

Conversely, suppose that is surjective. Then, (tacitly employing the Axiom of Choice) we may select an element for each and define

evidently then as desired. From where the two claims follow.

Now that these claims are proven we can clearly state that is bijective if and only if it possesses left and right inverses. Now, to see that if it does possess right and left inverses then we merely note that if then there is some such that and since is injective this implies that which is clearly absurd. Similarly, if then there exists some such that . Note though that since is surjective we have that for some and thus which is also absurd. The conclusion follows.

Now, as an immediate corollary of this theorem we get

**Corollary: ***Let . Then, if and only if there exists such that *

*Remark: *Note that the above corollary is true even for infinite dimensional spaces.

We now branch out in a different direction and show that, in essence, that while the above corollary is useful, for finite dimensional spaces there is a much, much simpler characterization of isomorphisms. Namely:

**Theorem: ***Let be an -dimensional -space. Then, for the following are equivlent*

**Proof: **We prove this in three steps

To see this we fix a basis for . We claim that if is a monomorphism then is a basis for . To see this we first prove that is linearly independent. To see this we assume that

which by linearity implies that

but, by an earlier characterization of monomorphisms we know that and so

but since is a linearly independent set we may conclude that and thus the linear independence of follows. Note then that this implies that

is an -dimensional subspace of . But, by a often stated theorem this implies that from where the claim follows.

Thus, trimming the fluff we see that is a monomorphism implies that , or said differently that is an epimorphism.

Since is already an epimorphism it suffices to show that is a monomorphism. To do this suppose that . Then, there is some such that . Evidently then

Now, we may extend to a basis for . Note then that

where the last equality is clear from . Thus, is spanned by vectors, and thus . But, this implies that contrary to the assumption that is an epimorphism. It follows that and thus by an earlier theorem it follows that is a monomorphism.

This is trivial since an isomorphism must be a monomorphism by definition.

Thus, having proven all three implications the equivalence of the three statements follows.

**Corollary: ***If then if and only if possesses a right or left inverse which is also a linear transformation.*

**References:**

1. Halmos, Paul R. *Finite-dimensional Vector Spaces,*. New York: Springer-Verlag, 1974. Print

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