Solutions Manual for A First Course in Linear Model Theory, 2e by Nalini Ravishanker, Zhiyi Chi, Dipak Dey (All Chapters)

Solutions Manual for A First Course in Linear Model Theory, 2e by Nalini Ravishanker, Zhiyi Chi, Dipak Dey (All Chapters) 1.1 |a • b| = | − 9| = 9, while kak kbk = √ 6 √ 22 ∼= 11.489 9. 1.2 To verify the Cauchy–Schwarz inequality, first see that the inequality holds trivially if a and b are zero vectors. We therefore assume that both a and b are nonzero. Let c be the vector c = xa − yb, where x = b ′b, and y = a ′b. Clearly, c ′c ≥ 0. We express c ′c in terms of x and y: c ′ c = (xa − yb) ′ (xa − yb) = x 2a ′a − 2xya ′b + y 2b ′b. Since c ′c ≥ 0, and using the definitions of x and y, we see that (b ′b) 2 (a ′a) − 2(a ′b) 2 (b ′b) + (a ′b) 2 (b ′b) ≥ 0 and dividing by b ′b in the last inequality, we see that (b ′b)(a ′a) − (a ′b) 2 ≥ 0, which verifies the Cauchy–Schwarz inequality. We use the Cauchy–Schwarz inequality to deduce the triangle inequality, which can be written in an equivalent form ka + bk 2 ≤ (kak + kbk) 2 . The expression on the left is ka + bk 2 = (a + b) • (a + b) = a • a + 2a • b + b • b = kak 2 + 2a • b + kbk 2 while the expression on the right is (kak + kbk) 2 = kak 2 + 2kakkbk + kbk 2 . Comparing these two formulas, we see that the triangle inequality holds if and only if a • b ≤ kak kbk. By Cauchy–Schwarz inequality, |a • b| ≤ kakkbk, so the triangle inequality follows as a consequence of the Cauchy–Schwarz inequality. The converse is also true; i.e., if the triangle inequality holds, then a•b ≤ kakkbk holds for a and for −a, from which Cauchy–Schwarz inequality follows. If equality holds, i.e., if a • b = kakkbk, then b = ca, for some scalar c. Hence, a • b = ckak 2 , and kakkbk = |c|kak 2 . For nonnull a, this implies that c = |c|, so that c ≥ 0. If b 6= 0, then b = ca, with c 0.