Let $V$ be a $k$-dimensional subspace of $\mathbb{R}^n$, with $k < n$. Choose an orthonormal basis $(a_1, \ldots, a_k)$ of $V$ and let $A$ be the $k \times n$ matrix whose rows are $(a_1^T, \ldots , a_k^T)$. Define a linear mapping $L: V \rightarrow \mathbb{R}^k$ by
$ L(x) = Ax. $
From the definition of $A$ it is clear that
$ L^{-1}(y) = A^T y. $
Therefore, $L$ has an inverse while $A$ (being rectangular) has not. This seems odd...
While writing this it came to me that $A$ probably isn't the matrix representing $L$ in any basis, since such should be a $k \times k$ matrix, but I am a bit confused - what then is $A$?