3. Suppose T is a n x n matrix such that for any u harmonic on R, v(x) = u(Tx) is also harmonic. Prove that T is a scalar multiple of an orthogonal matrix (i.e. T = kR, where k E R and RR' = R'R=1) (Hint: Since the above holds for any harmonic function u, consider the functions u = Wij(x) = x;!; and u = w;j(x) = x; 2;. Show that these functions are harmonic and then set v(x) = w;j(Tx), which is also harmonic. From here, deduce the relation between the entries of the matrix T to show it is a scalar multiple of an orthogonal matrix) 3. Suppose T is a n x n matrix such that for any u harmonic on R, v(x) = u(Tx) is also harmonic. Prove that T is a scalar multiple of an orthogonal matrix (i.e. T = kR, where k E R and RR' = R'R=1) (Hint: Since the above holds for any harmonic function u, consider the functions u = Wij(x) = x;!; and u = w;j(x) = x; 2;. Show that these functions are harmonic and then set v(x) = w;j(Tx), which is also harmonic. From here, deduce the relation between the entries of the matrix T to show it is a scalar multiple of an orthogonal matrix)