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Show: Elementary row operations don't affect solution sets

  1. Jan 13, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that elementary row operations don't affect solutions sets in linear systems

    2. Relevant equations
    -

    3. The attempt at a solution
    It's pretty easy to come up with a random linear system and perform ERO on them and showing that solutions are not affected, but is there all there is to it? Do I need to show that this works throughout all possible forms of linear systems or will this be known as "proving"?
     
  2. jcsd
  3. Jan 13, 2017 #2

    PeroK

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    Yes, to prove this you need to show it for all linear systems of equations.

    Do you have any experience of formal proofs?
     
  4. Jan 13, 2017 #3
    I have very little experience of proving statements, especially in LA (I just got through 2 lectures). Could you provide me with a guideline? Thanks!
     
  5. Jan 13, 2017 #4

    PeroK

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    The key here is that having the same solutions to two sets of equations can be stated as:

    ##x## is a solution to one system iff it is a solution to the other system.

    Does that make sense to you?
     
  6. Jan 13, 2017 #5
    Not really, sorry. Unless you mean that x is a solution to the system if it is a solution to all equations in that system?
     
  7. Jan 13, 2017 #6

    PeroK

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    Yes, ##x = (x_1, x_2, \dots ,x_n)## is a solution to the system if it is a solution to all equations in the system
     
  8. Jan 13, 2017 #7
    How could I use this to prove the statement? Do I use a general augmented matrix from a1n to amn to do this?
     
  9. Jan 13, 2017 #8

    PeroK

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    Putting things in a matrix only complicates matters. Think of the simplest row operation: exchanging two rows. How could you prove that doesn't change the solutions?

    Hint: try to think about it as simply as possible!
     
  10. Jan 13, 2017 #9

    Math_QED

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    Try to think about the relation between the matrix and the system.

    What happens when I exchange 2 rows in the matrix? What system does the new matrix correspond to? Do the original system and the new system have the same solution?
     
  11. Jan 13, 2017 #10

    Mark44

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    I agree with @PeroK here -- that bringing in matrices adds unnecessary complication. The original problem statement mentions linear systems (of equations), with no mention of matrices. The questions to consider are these:
    • Does interchanging two equations change the solution?
    • Does replacing an equation by a multiple of itself change the solution?
    • Does replacing one equation by adding a multiple of another equation to the first equation change the solution?
     
  12. Jan 15, 2017 #11
    Forgive me if I'm wrong, but are you guys saying that it can be solved as such:

    1. Prove that interchanging 2 equations don't affect the solution set,
    2. Show that constant multiples of an equation don't affect its solution,
    3. Show how adding a constant-multiple of eqn 1 to eqn 2 doesn't affect a solution set because we're actually adding the same values to both sides of eqn 2,
    4. Generalise this result across all systems of linear equations?
     
  13. Jan 15, 2017 #12

    PeroK

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    Yes, but I've tidied this up a bit:

    For any system of linear equations:

    1. Show that interchanging any two equations doesn't affect the solution set,
    2. Show that multiplying any equation by a non-zero constant doesn't t affect the solution set.
    3. Show that adding a constant-multiple of any equation to any other equation doesn't affect the solution set.
     
  14. Jan 15, 2017 #13
    Alright, thanks a lot guys!
     
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