(Solved): 1. Let T:R3R3 be the linear transformation determined by the matrix a000b000c ...

1. Let $T:R^3?R^3$ be the linear transformation determined by the matrix $?\begin{array}{lll}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}?$, where $\mathrm{a},\mathrm{b}$, and $\mathrm{c}$ are positive numbers. Let $\mathrm{S}$ be the unit ball, whose bounding surface has the equation $x_1^2+x_2^2+x_3^2=1$. a. Show that $\mathrm{T}(\mathrm{S})$ is bounded by the ellipsoid with the equation $\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}+\frac{x_3^2}{c^2}=1$. b. Use the fact that the volume of the unit ball is $4?/3$ to determine the volume of the region bounded by the ellipsoid in part (a).