# Volume of a sphere under a linear transformation R3->R4.

1. Nov 24, 2012

### BluFoot

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!

2. Nov 25, 2012

### haruspex

Thinking about the ℝ1 → ℝ2 suggests a root-sum-squares relationship. Can't see how to generalise it yet.

3. Nov 26, 2012

### haruspex

Have now proved the following for both ℝ1 → ℝ2 and ℝ2 → ℝ3, but still don't see how to prove it in general.
Denoting the square matrices formed by appending the basis vectors as Ai, det(ATA) = Ʃ(det(Ai))2.