I am trying to understand these notes: A Note On The Volume Of A Simplex. Here the author uses an affine transformation $X = AY + B$ where $detA \neq 0$ and $B$ is a colum vector to show that the volume of the simplex generated by the points that make up the $X$ matrix is the same as the one generated by the ones that make the $Y$ matrix.
To do that they first write this equality which I asumme stems from the affine transformation:
$$[X_2 - X_1, \dots, X_{n+1} - X_1] = A [Y_2 - Y_1, \dots, Y_{n+1} - X_1]$$
They then aply a determinant to the eqution and arrive to this conclusion:
\begin{equation} det \begin{bmatrix} X_1, \dots, X_{n+1} \\ 1, \dots, 1 \end{bmatrix} = det A \begin{bmatrix} Y_1, \dots, Y_{n+1} \\ 1, \dots, 1 \end{bmatrix} \end{equation}
So my question is where did the $B$ go? At first I thought since its a fixed vector it could've been assumed to be the vector zero and been discarded but further on its mentionned that $A$ and $B$ are specific matrix so that $Y$'s points look like this: $Y_1 = (y_0, 0, \dots, 0)$, $Y_r = (0, y_{r2}, \dots, y_{rn})$ So now I'm not so sure the $B$ simnply vanished.
If anyone could explain this to me or/and direct me to any aditional reading that could help me understand what happenned here better I would really apreaciate it.
A simplex in $n$ dimensions has $n+1$ vertices. To describe its shape and orientation but not its position, you can subtract the coordinates of one vertex from all the others. That's what's happening in your equation: $X_1$ is subtracted from all the other $X_i$:
$$[X_2 - X_1, \dots, X_{n+1} - X_1] = A [Y_2 - Y_1, \dots, Y_{n+1} - Y_1]$$
Now the offset $B$ occurs on each side of each of these differences, so it cancels out.
$$X_2-X_1=(AY_2+B)-(AY_1+B)=AY_2+B-AY_1-B= AY_2-AY_1=A(Y_2-Y_1)$$
The same happens for all the other differences.