# Show: Elementary row operations don't affect solution sets

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1. Jan 13, 2017

### WWCY

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
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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. Jan 13, 2017

### PeroK

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

Do you have any experience of formal proofs?

3. Jan 13, 2017

### WWCY

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!

4. Jan 13, 2017

### PeroK

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?

5. Jan 13, 2017

### WWCY

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?

6. Jan 13, 2017

### PeroK

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

7. Jan 13, 2017

### WWCY

How could I use this to prove the statement? Do I use a general augmented matrix from a1n to amn to do this?

8. Jan 13, 2017

### PeroK

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!

9. Jan 13, 2017

### Math_QED

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?

10. Jan 13, 2017

### Staff: Mentor

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?

11. Jan 15, 2017

### WWCY

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?

12. Jan 15, 2017

### PeroK

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.

13. Jan 15, 2017

### WWCY

Alright, thanks a lot guys!