1. The problem statement, all variables and given/known data So there's a linear transformation T: ℝ3 → ℝ4, standard matrix A that satisfies det(A e1) = 5, det (A e2) = 4, det (A e3) = 5 and det (A e4) = 5 If S is the unit sphere, find the 3-dimensional volume of T(S). 2. Relevant equations Volume of sphere = 4/3 * pi * r^3 Volume based on determinants = det(ATA) 3. The attempt at a solution I know the determinant of a matrix can be seen as the scaling factor for the volume change of a transformation. So the answer will be 4/3 * pi * (something). The (something) will probably be some kind of combination of the determinants above (5,4,5,5), but I have no clue how to find its value. I have spent hours on this question, it's driving me crazy. Help!