Formally Proving the Invariance of Solutions in Gaussian Elimination

In summary, the use of elementary row operations in Gaussian Elimination does not change the set of solutions to a system of linear equations. This can be formally proven by keeping the system of equations in equation form instead of using matrix representation and understanding that the operations are equivalent to permuting equations, multiplying equations by constants, and adding equations together.
  • #1
adartsesirhc
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I've been doing Gaussian Elimination in a Linear Algebra class, but I have a question:

How do I formally prove that elementary row operations do not change the set of solutions to a system of linear equations?

Thanks.
 
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  • #2
Instead of using matrix representation for the system of equation, keep it in equation form.

Then elementary row operations are the same as either permuting two equations (obviously doesn't change the solutions!), multiplying one equations by a constant (doesn't change the solutions as you can probably easily see), and multipling an equation by a constant and adding the resulting equation to another. If you think about it for a second, you'll see why this doesn't change the solutions either.
 

1. What is Gaussian Elimination?

Gaussian Elimination is a method used to solve systems of linear equations. It involves systematically manipulating the equations to eliminate variables and find a unique solution for each variable.

2. How does Gaussian Elimination work?

Gaussian Elimination works by using elementary row operations to transform the system of equations into an upper triangular matrix. This process involves eliminating variables by adding or subtracting rows from each other and also multiplying rows by constants.

3. What are the benefits of using Gaussian Elimination?

Gaussian Elimination is a widely used and efficient method for solving systems of linear equations. It can easily handle large systems and does not require any special knowledge or techniques. It also provides a systematic approach to finding solutions, making it easy to identify any errors in the calculations.

4. Are there any limitations to Gaussian Elimination?

While Gaussian Elimination is a powerful method for solving systems of linear equations, it does have some limitations. It can only be used for systems of equations with the same number of equations and variables. It also cannot be used for systems with infinite solutions or no solutions.

5. Can Gaussian Elimination be applied to other types of equations?

Gaussian Elimination is specifically designed for solving linear equations. However, it can be adapted and extended to solve other types of equations, such as systems of non-linear equations or differential equations. This requires additional techniques and knowledge, but the basic principles of Gaussian Elimination can still be applied.

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