1 -1 0 0 0 0 0 0 0 1 0 0 000 000 100 010 00 1 (E1):col2 col₂+col₁ Bker (81) V2 (E2):col3+col3-col₁ 0 0 10 0 00 -1 0 0 0 0 00 1 0 The key is now that the vectors of Q under the zero "columns" of 8¹. Q will actually form a basis for ker(8¹). So if we call v₁ through vs the five columns of 8¹. Q, the four vectors in our case that form a basis for the kernel are: 0 0 1 1 0 1 0 0 0 1 00 0 00 0 0 V4 8¹. Q Q=I·E₂E₁, 0 0, V5 Exercise Check that Bker(8¹) forms a basis for ker(8¹) by first showing these four vectors are linearly independent and then using the Rank-Nullity theorem and an analysis of the pivot(s) of 8¹ above to conclude it must be a basis for "dimension reasons".
1 -1 0 0 0 0 0 0 0 1 0 0 000 000 100 010 00 1 (E1):col2 col₂+col₁ Bker (81) V2 (E2):col3+col3-col₁ 0 0 10 0 00 -1 0 0 0 0 00 1 0 The key is now that the vectors of Q under the zero "columns" of 8¹. Q will actually form a basis for ker(8¹). So if we call v₁ through vs the five columns of 8¹. Q, the four vectors in our case that form a basis for the kernel are: 0 0 1 1 0 1 0 0 0 1 00 0 00 0 0 V4 8¹. Q Q=I·E₂E₁, 0 0, V5 Exercise Check that Bker(8¹) forms a basis for ker(8¹) by first showing these four vectors are linearly independent and then using the Rank-Nullity theorem and an analysis of the pivot(s) of 8¹ above to conclude it must be a basis for "dimension reasons".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:1
1
0
0
0
-1 100
0 0 00
1 0 00
0
1 00
0
0 10
0
0 0
(E1):col2col2+col₁
(E2):col3 cols-col₁
Bker (8¹)
---C
V2
1
0
1 1
0 1
0 0
0 0
00
V3
The key is now that the vectors of Q under the zero "columns" of 8¹. Q will actually form a basis for ker(8¹). So if we call
v₁ through v5 the five columns of 8¹ - Q, the four vectors in our case that form a basis for the kernel are:
0
-1
0
0 0
1
0
0
0 1 0
0 0 1
V4
00
00
1
V5
8¹. Q
Q=I·E₂ E₁)
0
0
Exercise Check that Bker(8¹) forms a basis for ker(8¹) by first showing these four vectors are linearly
independent and then using the Rank-Nullity theorem and an analysis of the pivot(s) of 8¹ above to conclude it must be a
basis for "dimension reasons".
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