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Linear algebra question

  • #1

Homework Statement


Quote from my textbook:

"The linear system Ax=b is consistent if and only if the number of nonzero rows of the augmented matrix [U| c], equals the number of nonzero rows in U."

[U,c] is the rref of [A,b]


Homework Equations





The Attempt at a Solution



I understand "only if" but not "if". "Only if" is true since the elementary row operations do not affect the solutions to the system and it is clear that a linear combination of zeros cannot equal a nonzero number. Can someone help me with "if"?
 

Answers and Replies

  • #2
rock.freak667
Homework Helper
6,230
31
number of nonzero rows in U is the rank of U,r(U)
the no. of nonzero rows of [U|c] is the rank of that matrix,r(U|c)

For a system of eq'ns to be constisent

[tex]r(U)=r(U|c) \leq n[/tex]

n= no . of rows
 
  • #3
Thanks for responding to my question!

Unfortunately, I am still confused. It seems like that just restated my question using the word "rank".


For a system of eq'ns to be constisent

[tex]r(U)=r(U|c) \leq n[/tex]
I think the quote I gave came from a proof of this statement actually...
 
  • #4
rock.freak667
Homework Helper
6,230
31
well saying "if and only if", like restricting the statement so that the statement will be true when that condition is satisfied.
 
  • #5
I figured it out. Thanks for your help.
 

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