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I asked this question and got the answer

The Gauss method turns the matrix to upper triangular, then you solve the triangular system

but if "then you solve the triangular system" means that you solve it by using elementary row operations (a.k.a. elementary row operations are performed to substitute), the two methods seem identical: In Gauss-Jordan you use backward elimination (performs EROs to sort of "substitute" away variables from equations) and in Gauss you use substitution to sort of substitute away variables from equations. Since EROs are used to substitute away these variables, the two seems completely identical to me. Can anyone help with this confusing? They must be different but I do not get how if substitution requires elementary row operations to be performed in any case.

I though this part could be a question on its own.

Rodrigo de Azevedo
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That Guy
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2 Answers2

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(a little late but hopefully still of use to anyone else)

In Gaussian elimination with back substitution we first bring the system of equations to row echelon form via elimination. The pivots, the leading coefficients, are used to make zeros below. We then use substitution to solve the system: starting from the lowest equation, we plug (substitute) the solutions to our variables into the equations above.

In Gauss-Jordan elimination we bring the system to row reduced echelon form via elimination. The pivots are used to make zeros below and above. Once in row reduced echelon form, we have our solution.

Please see https://en.wikipedia.org/wiki/Row_echelon_form for pictures of these echelon forms.

To summarize, in Gaussian elimination with back substitution we eliminate downward, and substitute upward; in Gauss-Jordan elimination we eliminate up and down.

rfmvoers
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Backward substitution is a procedure of solving a system of linear algebraic equations $Ux=y$, where $U$ is an upper triangular matrix whose diagonal elements are not equal to zero. By backward elimination, I think what is meant is Gaussian Elimination, the process of performing row operations to make an upper triangular matrix.

So if you have a system $Ax=b$, in order to solve the system, you may perform Guassian elimination (the process of applying row operations to obtain an upper triangular matrix) to obtain $Ux=b'$. Now that you have an upper triangular matrix, you can perform backward substitution (that is start with $u_{n,n}x_{n}=b_n'$, then use that solution to solve the equations for $x_{n-1}$ then $x_{n-2}$ and so on).

The very short of it is: Backs-sub is a solver for a system with an upper triangular matrix, Gaussian Elimination (perhaps also called backward elimination) is a method to obtain a system with an upper traingular matrix.

Check out the wikipedia articles on each for more info: https://algowiki-project.org/en/Backward_substitution and https://en.wikipedia.org/wiki/Gaussian_elimination .

Mark
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  • But again - the substitution in the gaussian(gauss) method could be performed by performing elementary row operations(maybe without explictly stating it). The definition of backward elimination is exactly that - use EROs to substitute away variables from other equations in the linear system - so the two methods still seem completely identical - they both in some sense use elementary row operations to remove variables from equations? – That Guy Apr 21 '20 at 20:28
  • It seems like it is more about the goal than the process. GE ends when you get an upper triangular matrix. back sub is for going from upper triangular to a solution. – Mark Apr 23 '20 at 00:51