Suppose $T : \mathbb{R}^n \to \mathbb{R}^n$ is a linear map and let $U \subset \mathbb{R}^n$ be a $d$-dimensional subspace where $0 < d < n$ and $\ker T = U$. I was wondering how to make sense of the sentence
The determinant of $T$ restricted to $U^\perp$
I figure that this means that you form the map $S: \mathbb{R}^n \to \mathbb{R}^n$ where $$S(x) =\begin{cases} x \text{ if } x\in U\\ T(x) \text{ otherwise} \end{cases}$$ and then the determinat of $T$ restricted to $U^\perp$ is given by $\det S$. Is this the correct interpretation?