Volume of an ellipsoid
Task number: 2608
Let a linear map \(f:\mathbb R^3\to\mathbb R^3\) sends vectors
\(\mathbf a^T=(1{,}3,1),\mathbf b^T=(1{,}0,3),\mathbf c^T=(1{,}1,1)\) onto vectors
\(f(\mathbf a)^T=(3{,}1,0),f(\mathbf b)^T=(1{,}0,2),f(\mathbf c)^T=(4{,}1,5)\).
Determine the volume of the ellipsoid \(f(B_3)\), that arises as the image of a unit ball \(B_3\) (i.e. a ball of the unit radius) under \(f\).
Solution
The linear map satisfies \(f(\mathbf u)=[f]_{KK}\mathbf u\) for the matrix
\([f]_{KK}=\mathbf B\mathbf A^{-1}= \begin{pmatrix} 3 & 1 & 4 \\ 1 & 0 & 1 \\ 0 & 2 & 5 \\ \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 \\ 3 & 0 & 1 \\ 1 & 3 & 1 \\ \end{pmatrix}^{-1} \).
The volumes are scaled by the factor \(|\det([f]_{KK})|\), i.e. \(V(f(B_3))=|\det([f]_{KK})| \cdot \frac{4}{3}\pi=\frac{|\det(\mathbf B)|}{|\det(\mathbf A)|}\cdot \frac{4}{3}\pi=\pi\)
Answer
The volume of the ellipsoid is \(\pi\).
