Inverse of Linear Operator is Linear Operator

Theorem

Let $X$ be a vector space.

Let $A : X \to X$ be an invertible (in the sense of a mapping) linear transformation with inverse mapping $A^{-1} : X \to X$.

Then $A^{-1}$ is a linear operator.

Proof

Applying Inverse of Linear Transformation is Linear Transformation in the case $U = V = X$ we have:

$A^{-1}$ is a linear transformation.

Since $A^{-1}$ is a linear transformation $X \to X$, we have:

$A^{-1}$ is a linear operator.

$\blacksquare$