If a Region W in ℝ3 has volume $$\sqrt{\pi}$$compute the volume of $$f(W) = \{f(\vec v): \vec v \in W\}$$
where $$f:R^{3}\to R^{3}$$ is given by $$f(x_{1},x_{2},x_{3}) = (x_{1}-x_{2},3x_{1}-x_{3},2x_{2}-x_{3})$$
I know how to find the volume of an area by using the determinant, but not sure how to start with this one.
Use simply the change of variables theorem: $$\text{vol}(f(W)) = \int_{f(W)}1 = \int_W 1|\det f| = \cdots$$ (as $f$ is linear, $\forall x: Df(x) = f$)
It is a fact proven in geometric measure theory that a linear (or an affine) map multiplies the volumina of all measurable sets with the same factor. In your case it is therefore sufficient to compute the volume of $f(I)$, where $I$ is the unit cube spanned by the three standard basis vectors of ${\mathbb R}^3$.
