## 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 $$f$$.

• #### Resolution

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. $$V(f(B_3))=|\det([f]_{KK})| \cdot \frac{4}{3}\pi=\frac{|\det(\mathbf B)|}{|\det(\mathbf A)|}\cdot \frac{4}{3}\pi=\pi$$

• #### Result

The volume of the ellipsoid is $$\pi$$.