Linear Algebra Done Right · Problem 9C.17

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · determinants

Problem

Suppose a,b,ca, b, c are positive numbers. Find the volume of the ellipsoid {(x,y,z)∈R3:x2a2+y2b2+z2c2<1}\left\{(x, y, z) \in \mathbf{R}^3 : \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} < 1\right\} by finding a set Ω⊆R3\Omega \subseteq \mathbf{R}^3 whose volume you know and an operator TT on R3\mathbf{R}^3 such that T(Ω)T(\Omega) equals the ellipsoid above.
Intuitive explanation

An ellipsoid is a stretched version of a sphere—the key is to find the right linear transformation that maps a unit ball (whose volume you know) into your ellipsoid. If you scale the xx, yy, and zz coordinates by factors of aa, bb, and cc respectively, the equation x2+y2+z2<1x^2 + y^2 + z^2 < 1 becomes x2a2+y2b2+z2c2<1\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} < 1. The volume changes by the determinant of the transformation matrix—which is exactly the product of the scaling factors.

Intuitive explanation

An ellipsoid is a stretched version of a sphere—the key is to find the right linear transformation that maps a unit ball (whose volume you know) into your ellipsoid. If you scale the xx, yy, and zz coordinates by factors of aa, bb, and cc respectively, the equation x2+y2+z2<1x^2 + y^2 + z^2 < 1 becomes x2a2+y2b2+z2c2<1\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} < 1. The volume changes by the determinant of the transformation matrix—which is exactly the product of the scaling factors.

Related theory

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