Inverse of a non-squared matrix working on a subspace

Question

Let $A$ be a topological manifold in $\mathbb{R}^n$ of dimension $m < n$. Let $W$ be a linear transformation $\mathbb{R}^n \to \mathbb{R}^m$ (or $m \times n$ matrix). Assume that $W$ is one-to-one on $A$. Therefore, there is an inverse mapping $W^{-1}: W(A) \to A$. How can I compute this mapping?