1806
a0 a1
2 a2 a3
1 a4 36 a5a7a6
(a)
q 4 = v (qT1 v)q1 (qT2 v)q2 (qT3 v)q3
q4 = q
4
kq 4k
(b)Q a8a10a9a12a11a10a13a15a14 0 a4 Q a8a10a16a10a17a10a18a10a19a10a17a10a20a10a21a22a6a24a23 QT a8a10a9a12a11a10a13a15a14 1 a25a10a8a10a26a12a27
q4 a8a29a28a29a30a29a31a29a32a33a4a35a34a37a36 q4
a38a29a39a29a40
q1; q2
a41
q3 a6a43a42
QTQ = I a8a10a9a12a11a10a13a15a44a10a14 0 a42 QQT a8a10a9a12a11a10a13a15a44a10a14 1 a25a10a8a10a23a46a45a10a26a12a27 q4 a8a10a28
a30a29a31a29a32a29a42
(c)QTQx = QTb =) x = QTb a23a47a34a37a48
x =
2
64 q
T1 (q1 + 2q2 + 3q3 + 4q4)
qT2 (q1 + 2q2 + 3q3 + 4q4)
qT3 (q1 + 2q2 + 3q3 + 4q4)
3
75 =
2
64 12
3
3
75
a49a29a50 p = Qx = q
1 + 2q2 + 3q3 a42
1
2 a4 24 a5a7a6 (a)
2
64 1 21 3
1 4
3
75x =
2
64 35
K
3
75
a51 b
a52 A a8a29a18a53a11a29a13a29a54a37a55a29a23a46a56a29a57 Ax=b a58a29a59a29a19a29a17a29a60a29a42a46a48a29a55 K=7 a42
(b) a51 b a52 AT a8a10a9a12a11a10a13a10a54a15a55a10a23 x = 0 a14a10a56a10a57a10a8a10a61a10a62a10a63a10a64a10a60a10a23a46a65a10a66
2
64 35
K
3
75
a52 AT a8a29a9a53a11a29a13a29a54a37a23a46a67 K = 8 a68 K = 21=4 a42a46a69a29a14a29a70a29a71a29a72a29a8a29a42
2
3 a4 40 a5a7a6 (a) a73a29a18a29a74 = a75a29a73a29a18a29a74 +a14 a8a29a76a29a30a29a77a29a78a29a74a29a42
(b) a27 P2 = P a79a80a23 (det(P))2 = det(P) a23a81a34a82a48a80a23 det(P) = 0 a68 det(P) = 1 a42
(c) a83a29a84a53a85a37a17a29a73a29a8a29a76a29a30a29a77a29a78a29a74a29a86a29a87a29a23
det(C) = ( b)( b)(a2 b2) + ( a)(a)(a2 b2) = (a2 b2)2:
3