Inverse of Linear Operator is Linear Operator
Jump to navigation
Jump to search
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.
$\blacksquare$