Part II Suppose that B is a 4x3 matrix with full column rank. Then the number of pivot columns is and the number of free variables is In this case, the only vector in the nullspace of Bis The general problem Bx d then has at most solution(s). If C is a 2x4 matrix with full row rank, then the number of special solutions for the nullspace problem Cx O is and therefore, if C'x = d has a solution, then it actually has infinitely many.
Part II Suppose that B is a 4x3 matrix with full column rank. Then the number of pivot columns is and the number of free variables is In this case, the only vector in the nullspace of Bis The general problem Bx d then has at most solution(s). If C is a 2x4 matrix with full row rank, then the number of special solutions for the nullspace problem Cx O is and therefore, if C'x = d has a solution, then it actually has infinitely many.
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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Transcribed Image Text:Part II
Suppose that B is a 4x3 matrix with full column rank. Then the
number of pivot columns is
and the number of free
variables is
In this case, the only vector in the nullspace of
B is
The general problem Bx
d then has at most
solution(s).
If C is a 2x4 matrix with full row rank, then the number of special
solutions for the nullspace problem Cx = 0 is
and
therefore, if Cx =
d has a solution, then it actually has infinitely
many.
Transcribed Image Text:Part I
Suppose we wish to solve the system of equations
x + y + z =
- 3
2x + 3y + z =
Зх + 4у + 22 —
12
The augmented matrix for this system is
1
1
1
-3
3
1
-9
3
4
-12
1
1 1
- 3
Reducing to echelon form gives us
0 1
1
- 3
from which
-
0 0 0 0
we can see that column 3 is a free column. If we set the third
variable, z, equal to 0 and then use back substitution, we get y =
-3
and then =
V Therefore the particular
solution to the system Ax = b is xp
-3
Also, considering the nullspace problem Ax
solution xn =
0, we get the special
-2
1
1
The general solution of Ax
x = xp + cxn.
b is the set of all vectors of the form
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