Define an action of Sym(3) on R 3 as follows: for Sym(3) and v = (x1, x2, x3) R 3 , set v = (x(1), x(2), x(3)) (a) (10 points) Show that the subspace...

(a) (10 points) Show that the subspace V = {(x1, x2, x3) : x1 + x2 + x3 = 0} has the following property: if v ∈ V , then σ ∗ v is also in V . (b) (10 points) Find a line (that is a one-dimensional subspace) which has this same property?