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

• WWCY
In summary, exchanging two equations in a linear system doesn't affect the solutions, replacing an equation by a multiple of itself doesn't affect the solution, and adding a constant-multiple of any equation to any other equation doesn't affect the solution set.
WWCY

## Homework Statement

Show that elementary row operations don't affect solutions sets in linear systems

-

## 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"?

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

Do you have any experience of formal proofs?

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

Do you have any experience of formal proofs?

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!

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?

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?

WWCY said:
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?

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

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

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

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!

WWCY said:

## Homework Statement

Show that elementary row operations don't affect solutions sets in linear systems

-

## 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"?

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?

Math_QED said:
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?
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?

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?

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.

WWCY
Alright, thanks a lot guys!

## 1. What are elementary row operations?

Elementary row operations are a set of three operations (interchanging two rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another row) that can be performed on a matrix in order to simplify it and solve linear equations.

## 2. How do elementary row operations affect solution sets?

Elementary row operations do not change the solution set of a system of linear equations. This means that if the original system of equations had a solution, the same solution will still be valid after performing elementary row operations.

## 3. Are there any limitations to using elementary row operations to solve linear equations?

Yes, elementary row operations can only be used to solve linear equations with a finite number of variables. They cannot be used for nonlinear equations or equations with an infinite number of solutions.

## 4. Can elementary row operations be used to find multiple solutions to a system of equations?

No, elementary row operations can only be used to find a single solution to a system of equations. If a system has multiple solutions, additional methods such as parametric equations or substitution must be used to find all possible solutions.

## 5. Why are elementary row operations important in linear algebra?

Elementary row operations are important in linear algebra because they allow us to manipulate matrices and solve linear equations in a systematic and efficient manner. They also help us to understand the properties and relationships of matrices and systems of equations.

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