In each case, find a change of variables that will diagonalize the quadratic form q. Determine the index and rank of q.
(a) q = x21 + 2x1x2 + x22
(b) q = x21 + 4x1x2 + x22
(c) q = x21 +x22 +x23 - 4(x1x2 + x1x3 + x2x3)
(d) q = 7x21 + x22 + x23 + 8x1x2 + 8x1x3 -16x2x3
(e) q = 2(x21 + x22 + x23 - x1x2 +x1x3 - x2x3)
(f) q = 5x21 +8x22 +5x23 -4(x1x2 +2x1x3 +x2x3)
(g) q = x21 - x23 - 4x1x2 + 4x2x3
(h) q = x21 +x23 -2x1x2 +2x2x3