Left Inverse Implies Right Inverse for Linear Operators

In linear algebra, a fundamental property of linear operators is that the existence of a left inverse implies the existence of a right inverse. This statement holds true for linear operators on a vector space. Let's delve into the proof.

Theorem: If an operator T ∈ L(V) has a left inverse, then T has a right inverse.

Proof:

Let V be a vector space, and let T be a linear operator on V (T ∈ L(V)). Suppose T has a left inverse, denoted by S, such that:

S o T = I_V

where I_V is the identity map on V. Our goal is to demonstrate the existence of an operator R such that:

T o R = I_V

This R would then be the right inverse of T.

Starting Point: We know S o T = I_V. Let's compose both sides of this equation with T on the right:

(S o T) o T = I_V o T
Associativity: Using the associative property of composition:

S o (T o T) = T
Defining R: Now, define an operator R as R = T o S.
Manipulating T o R: Let's examine the expression T o R:

T o R = T o (T o S) = (T o T) o S
Substitution: Substitute (T o T) with S o (T o T) from step 2:

T o R = (S o (T o T)) o S
Further Manipulation: Applying associativity again:

T o R = S o ((T o T) o S)
Another Substitution: Substitute (T o T) once more with S o (T o T):

T o R = S o (S o (T o T))
Rearranging and Simplifying: Using associativity to rearrange and simplify:

T o R = (S o S) o (T o T) = I_V o (T o T) = T o T
Conclusion: We have shown that T o R = T o T. Since operators in a vector space have unique inverses, we can conclude that R = T is the right inverse of T.

Therefore, we have proven that if a linear operator T on a vector space V has a left inverse, it must also have a right inverse.

Left Inverse Implies Right Inverse for Linear Operators