In each case either show that the statement is true or give an example showing that it is false.
(a) Every independent set in Rn is orthogonal.
(b) If {X, Y) is an orthogonal set in Rn, then {X, X + Y) is also orthogonal.
(c) If {X, Y) and {Z, W) are both orthogonal in Rn, then {X, Y, Z, W] is also orthogonal.
(d) If (X1, X2} and [Y1 ,Y2, Y3] are both orthogonal and Xi ∙ Yj = 0 for all i and j, then {X1, X2, Y1, Y2, Y3] is orthogonal.
(e) If {X1 X2,..., Xn] is orthogonal in IR", then Rn = span{X1,X2,...,Xn}.
(f) If X ≠ 0 in Rn, then {X} is an orthogonal set.