Download DETERMINING WHETHER A GIVEN VECTOR IS IN THE COLUMN

DETERMINING WHETHER A GIVEN VECTOR IS IN THE
COLUMN SPACE OF A GIVEN MATRIX
MATT INSALL




−3 −2 0 
 1 















Problem: Let A =  0
2 −6 and let ~u =  14 
. Determine whether












6
3
−9
3
~u ∈ Col(A).
Solution:
We row-reduce the matrix [A|~u], using admissible row operations:

 −3 −2 0



[A|~u] = 
2 −6
 0



6
3
3
−R1

1 


R2

2

14 
 ========⇒


 R3 + 2R1
−9
 3




 0



2
1
0 −1
R1 − 2R2 







0
−3
3

−1 



7 




−7
 3 0 6 −15 




R2



========⇒  0 1 −3 7 
.
R3 + R2
0 0
0
0
This last matrix is the augmented matrix for a consistent system. Thus the
vector ~u is in Col(A).
This document was typeset with LaTeX.
c
2003
Matt Insall,
All rights reserved. Students in my classes may print a copy for use in their assignments,
but for other uses, written permission must be obtained from the author.
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