4.4.29. For each of the following matrices A, (i) find a basis for each of the four fundamental subspaces; (ii) verify that the image and cokernel are orthogonal complements; (iii) verify that the coimage and kernel are orthogonal complements: 5 (-) ( 2 ). (1) (2) 1 0 (e) 7 3 7 3 12 3 1 4 1 1 2 5 2 203 (f) 2 (c) 1 -2 -3 1 0 1 2 -1 0 -3 (d) -2 3 0 3 1 5 -4 -2 0 -2 2 3 4 4 -1 2 (g) 1 2 0 -1 1 3 0 -1 2 -3 1 1 1 3 3 2 -2 2 2 -5 2 -3 -3 1 4 -5 -2 9 -4 6 -2 -1 2
4.4.29. For each of the following matrices A, (i) find a basis for each of the four fundamental subspaces; (ii) verify that the image and cokernel are orthogonal complements; (iii) verify that the coimage and kernel are orthogonal complements: 5 (-) ( 2 ). (1) (2) 1 0 (e) 7 3 7 3 12 3 1 4 1 1 2 5 2 203 (f) 2 (c) 1 -2 -3 1 0 1 2 -1 0 -3 (d) -2 3 0 3 1 5 -4 -2 0 -2 2 3 4 4 -1 2 (g) 1 2 0 -1 1 3 0 -1 2 -3 1 1 1 3 3 2 -2 2 2 -5 2 -3 -3 1 4 -5 -2 9 -4 6 -2 -1 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
7.1.39
Transcribed Image Text:4.4.29. For each of the following matrices A, (i) find a basis for each of the four fundamental
subspaces; (ii) verify that the image and cokernel are orthogonal complements; (iii) verify
that the coimage and kernel are orthogonal complements:
5
(-) ( 2 ). (1) (2)
1
0
(e)
7
3
7 3 12
3
1 4
1 1 2
5 2
203
(f)
2
(c)
1
-2
-3
1
0 1
2
-1 0 -3 (d)
-2
3
0
3
1
5
-4 -2
0 -2
2 3
4 4
-1
2
(g)
1 2 0
-1 1 3
0
-1
2
-3
1
1
1
3 3 2
-2
2
2
-5
2
-3
-3
1
4 -5 -2
9
-4
6
-2
-1
2
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