18.06
a0 a1
3 a2 a3
2000 a4 5 a5 3 a6
a7 1.
(a) a8 ATA a9a11a10a11a12a11a13a11a14a11a15a17a16a19a18a11a20a11a21a11a14a11a15a11a22a11a23 64 a24 4 a25 0 a26
(b) a27a11a28a11a22a11a29 ATA a30a17a31a19a32a11a14a11a15a11a22a11a30a11a33a11a34a11a35a11a23 8 a25 2 a26
(c) a27a11a28a11a22a11a36a11a37a11a29a11a23
A =
" 1p
2
1p
21
p2 1p2
# "
8 0 0
0 2 0
# 2
64
1p
2
1p
2 01
p2 1p2 0
0 0 1
3
75
a7 2.
(a) a38a11a26 2 a39a11a40a11a41a11a42a17a43a19a29a11a44a11a45a11a30a11a24a47a46a11a48a11a49a11a50a17a51a19a52a11a53a11a54a11a30a11a14a11a15a17a16a19a18a11a24a47a55a11a56 (1;0)
a25 (1;1) a26
(b) a44a11a26a57a8a11a53a11a54a58a42a17a43 Q a59 A a44a17a60a19a61a11a23 A = Q QT a26a57a62a11a63a11a12a11a13a11a42a17a43a19a29a11a44a11a45
a30a11a23 (Q QT)T = Q QT a26
(c) a38a11a26a57a55a11a64a11a65 (a) a66a17a67a19a26
a7 3,d > 0
a24a57a68a11a49a11a62a11a63a11a30a11a14a11a15a11a22a11a26a57a69a11a70a11a71a11a49a11a30a11a14a11a15a11a22a11a72a11a73a11a74a11a32a11a24a57a75a11a76a11a77a11a78 A a29
a53a11a79a11a30a11a26a19a80a11a81a11a82a11a83a11a64a11a84a11a29 1;d 4 a25 12 4d a26 A a30a11a14a11a15a11a22a11a52a11a85a11a86a11a72a11a29a11a53a11a30a11a26
a7 4.
(a) = 1 a29a88a87a88a35a88a24 = 1 a44a88a89a88a30a88a90a88a91a88a30a88a14a88a15a92a16a93a18a17a94 N(A I) a30a88a95a88a96a88a97a92a98a93a24
a48a11a29 2 a95a11a30a11a26a100a99a19a101 A a44a11a89a11a74a11a14a11a15a11a22 = 1 a49 2 a13a11a90a11a91a11a30a11a14a11a15a17a16a19a18a11a26
(b)A a49a11a102a11a13a11a90a11a91a11a30a11a14a11a15a17a16a19a18a11a24a100a99a19a101a11a48a11a29a11a85a11a44a17a60a19a61a11a30a11a24a57a41a11a103a11a74a11a104a11a80a11a66a11a105a11a74
=
2
64 1 0 00 1 0
0 0 2
3
75
1
(c) a42a17a43
B =
2
64 1 0 00 1 1
0 0 2
3
75
a25 A a49a11a66a17a67a19a30a11a14a11a15a11a22a11a25a11a66a17a67a19a96a107a106a109a108a11a110a11a90a11a91a11a30a11a14a11a15a17a16a19a18a11a26a100a99a19a101 B a66a11a105a11a74 A a26
(d) a42a17a43
C =
2
64 1 1 00 1 0
0 0 2
3
75
a25 A a49a11a66a17a67a19a30a11a14a11a15a11a22a11a26a57a65 A a66a11a111a17a112a17a113a109a12a11a13a11a14a11a15a17a16a19a18a11a23 rank(C I) = 2 a24a100a99
a101 C a66a11a89a11a74a11a14a11a15a11a22 = 1 a114a11a49a11a12a11a13a11a108a11a110a11a90a11a91a11a30a11a14a11a15a17a16a19a18a11a26
2
a0 a1
3 a2 a3
2000 a4 5 a5 3 a6
a7 1.
(a) a8 ATA a9a11a10a11a12a11a13a11a14a11a15a17a16a19a18a11a20a11a21a11a14a11a15a11a22a11a23 64 a24 4 a25 0 a26
(b) a27a11a28a11a22a11a29 ATA a30a17a31a19a32a11a14a11a15a11a22a11a30a11a33a11a34a11a35a11a23 8 a25 2 a26
(c) a27a11a28a11a22a11a36a11a37a11a29a11a23
A =
" 1p
2
1p
21
p2 1p2
# "
8 0 0
0 2 0
# 2
64
1p
2
1p
2 01
p2 1p2 0
0 0 1
3
75
a7 2.
(a) a38a11a26 2 a39a11a40a11a41a11a42a17a43a19a29a11a44a11a45a11a30a11a24a47a46a11a48a11a49a11a50a17a51a19a52a11a53a11a54a11a30a11a14a11a15a17a16a19a18a11a24a47a55a11a56 (1;0)
a25 (1;1) a26
(b) a44a11a26a57a8a11a53a11a54a58a42a17a43 Q a59 A a44a17a60a19a61a11a23 A = Q QT a26a57a62a11a63a11a12a11a13a11a42a17a43a19a29a11a44a11a45
a30a11a23 (Q QT)T = Q QT a26
(c) a38a11a26a57a55a11a64a11a65 (a) a66a17a67a19a26
a7 3,d > 0
a24a57a68a11a49a11a62a11a63a11a30a11a14a11a15a11a22a11a26a57a69a11a70a11a71a11a49a11a30a11a14a11a15a11a22a11a72a11a73a11a74a11a32a11a24a57a75a11a76a11a77a11a78 A a29
a53a11a79a11a30a11a26a19a80a11a81a11a82a11a83a11a64a11a84a11a29 1;d 4 a25 12 4d a26 A a30a11a14a11a15a11a22a11a52a11a85a11a86a11a72a11a29a11a53a11a30a11a26
a7 4.
(a) = 1 a29a88a87a88a35a88a24 = 1 a44a88a89a88a30a88a90a88a91a88a30a88a14a88a15a92a16a93a18a17a94 N(A I) a30a88a95a88a96a88a97a92a98a93a24
a48a11a29 2 a95a11a30a11a26a100a99a19a101 A a44a11a89a11a74a11a14a11a15a11a22 = 1 a49 2 a13a11a90a11a91a11a30a11a14a11a15a17a16a19a18a11a26
(b)A a49a11a102a11a13a11a90a11a91a11a30a11a14a11a15a17a16a19a18a11a24a100a99a19a101a11a48a11a29a11a85a11a44a17a60a19a61a11a30a11a24a57a41a11a103a11a74a11a104a11a80a11a66a11a105a11a74
=
2
64 1 0 00 1 0
0 0 2
3
75
1
(c) a42a17a43
B =
2
64 1 0 00 1 1
0 0 2
3
75
a25 A a49a11a66a17a67a19a30a11a14a11a15a11a22a11a25a11a66a17a67a19a96a107a106a109a108a11a110a11a90a11a91a11a30a11a14a11a15a17a16a19a18a11a26a100a99a19a101 B a66a11a105a11a74 A a26
(d) a42a17a43
C =
2
64 1 1 00 1 0
0 0 2
3
75
a25 A a49a11a66a17a67a19a30a11a14a11a15a11a22a11a26a57a65 A a66a11a111a17a112a17a113a109a12a11a13a11a14a11a15a17a16a19a18a11a23 rank(C I) = 2 a24a100a99
a101 C a66a11a89a11a74a11a14a11a15a11a22 = 1 a114a11a49a11a12a11a13a11a108a11a110a11a90a11a91a11a30a11a14a11a15a17a16a19a18a11a26
2
