Invertible Linear Transformation

A linear transformation can, in some cases, be inverted. Then, the inverted transformation is called the inverse linear transformation

A linear transformation is said to be invertible if it has an associated inverse linear transformation. For it to be invertible, it must be bijective.

Definition §

Inverse Linear Transformation ^formula

Let $T:U\to V$ be a linear transformation. $T$ is said to be invertible if it has an inverse, $T^{-1}:V\to U$ such that:

(T^{-1} \circ T)(\vec{u}) = T^{-1}(T(\vec{u}))= \vec{u} \quad\quad \vec{u} \in U$$

(T \circ T^{-1})(\vec{v}) = T(T^{-1}(\vec{v})) = \vec{v} \quad\quad\vec{v} \in V

So if we have a vector, $\vec{u}$ from the original domain, $U$, and we transform it to $T(\vec{u})$, we can reverse the transformation if $T$ is invertible, by applying $T^{-1}$.

Invertible Linear Transformations ^theorem

Let $T:U\to V$ be a linear transformation

1. $T$ is invertible iff it is bijective.
2. If $T$ is invertible, then it’s inverse, $T^{-1}$ is also a linear transformation

Invertible Linear Transformations (Matrices) ^theorem

Let $T:U\to V$ be a linear transformation. Let $\mathcal{B}$ and $\mathcal{C}$ be the standard, ordered, bases of $U$ and $V$, respectively. Then:

$$T\text{ is invertible}⟺[T{]}_{\mathcal{C},\mathcal{B}}\text{ is invertible}$$

$$[T^{-1}{]}_{\mathcal{B},\mathcal{C}}=([T{]}_{\mathcal{C},\mathcal{B}}{)}^{-1}$$

I.e. we can take the inverse matrix of the transformation matrix to find the inverse transformation. For this to happen:

$$\text{dim}(U)=\text{dim}(V)$$