It is well known that if a linear map $H$ is bijective then we have $$Vol(S)=\text{det}H \, Vol(\mathcal{X})$$
Now I want to know the case when $H$ is a surjective map. How to calculate the volume of the image set? Specifically, when $\mathcal{X}$ is a simplex. Given a linear map $H: \text{R}^m \rightarrow \text{R}^n$, where $m > n$. Consider a simplex $\mathcal{X}=\{(x_1,x_2,\ldots,x_m)|\sum_{i=1}^{m} x_i \leq 1,x_i \geq 0\}$, and the volume of $\mathcal{X}$ is $1/m!$. Is there some formula on the volume of $H\mathcal{X}$?