Let 0 ≠ v ∈ R3.
(a) Show that the cross product Lv[x] = v x x defines a linear transformation on R3.
(b) Find the 3 x 3 matrix representative Av of L, and show that it is skew-symmetric.
(c) Show that every non-zero skew-symmetric 3 x 3 matrix defines such a cross product map.
(d) Show that ker Av is spanned by v.
(e) Justify the fact that the matrix exponentials etAv are rotations around the axis v. Thus, the cross product with a vector serves as the infinitesimal generator of the one-parameter group of rotations around v.