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

[BluFoot]

Homework Statement

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).

Homework Equations

Volume of sphere = 4/3 * pi * r^3
Volume based on determinants = det(A^TA)

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!

 

[haruspex]

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

 

[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 A_i, det(A^TA) = Ʃ(det(A_i))².