Chapter 3.4, Problem 74E

Consider the regular tetrahedron in the accompanying sketch whose center is at the origin. Let ${\stackrel{\to }{v}}_0,{\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3$ be the position vectors of the four vertices of the tetrahedron:
${\stackrel{\to }{v}}_0={\stackrel{\to }{OP}}_0,\mathrm{...},{\stackrel{\to }{v}}_3={\stackrel{\to }{OP}}_3$ .
a. Find the sum ${\stackrel{\to }{v}}_0+{\stackrel{\to }{v}}_1+{\stackrel{\to }{v}}_2+{\stackrel{\to }{v}}_3$ .
b. Find the coordinate vector of ${\stackrel{\to }{v}}_0$ with respect to the basis ${\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3$ .
c. Let T he the linear transformation with $T\left({\stackrel{\to }{v}}_0\right)={\stackrel{\to }{v}}_3,T\left({\stackrel{\to }{v}}_3\right)={\stackrel{\to }{v}}_1$ , and $T\left({\stackrel{\to }{v}}_1\right)={\stackrel{\to }{v}}_0$ . What is $T\left({\stackrel{\to }{v}}_2\right)$ ? Describe the transformation T geometrically (as a reflection, rotation. projection, or whatever). Find the matrix B of T with respect to the basis ${\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,{\stackrel{\to }{v}}_3$ . What is $B^3$ ? Explain.