Difference between REF/RREF (Gauss vs Gauss-Jordan)?

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In summary, the conversation is about linear algebra and the process of row reduction. The speaker is seeking clarification on whether the same algorithm should be used every time and if they should focus on getting the first "1" in each row before moving on to the next. They also mention that understanding the acronyms for different forms (REF and RREF) can be helpful.
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leo255
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Hi all,

Just started my Linear Algebra class like a month ago. I have been going through my book, as well as other resources, but I am still confused by this. I have a decent grasp of how to row reduce, and, in general, know that I need to get the 1's in a diagonal line, with 0's everywhere else.

Also, when you row reduce, should you go through the same algorithm/procedure every time? In other words, should you try to get the first "1," and then move to the second and third row's, to try to get "0"s in the first spot (instead of working on a "0" in a different spot?

I'm really new to this subject, so I apologize for the noob question. Thanks!
 
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leo255 said:
Hi all,

Just started my Linear Algebra class like a month ago. I have been going through my book, as well as other resources, but I am still confused by this. I have a decent grasp of how to row reduce, and, in general, know that I need to get the 1's in a diagonal line, with 0's everywhere else.
It helps to understand what the acronyms mean -- otherwise they're just bunches of meaningless letters.
REF - row echelon form. The leading nonzero entry in any row is 1, and there are only 0's below that leading entry.
RREF - reduced row echelon form. Same as REF plus there are only 0's above any leading entry.
leo255 said:
Also, when you row reduce, should you go through the same algorithm/procedure every time?
Pretty much, although it's sometimes helpful to switch one or more pairs of rows. Otherwise, you start at the leftmost column, eliminate the entries above and below it and continue in that fashion.
leo255 said:
In other words, should you try to get the first "1," and then move to the second and third row's, to try to get "0"s in the first spot (instead of working on a "0" in a different spot?

I'm really new to this subject, so I apologize for the noob question. Thanks!
 

FAQ: Difference between REF/RREF (Gauss vs Gauss-Jordan)?

What is the difference between REF and RREF?

REF stands for Row Echelon Form and RREF stands for Reduced Row Echelon Form. Both are methods used to solve systems of linear equations.

REF is a simpler form that follows three rules: all nonzero rows are above any rows of all zeros, each leading entry (also called a pivot) of a row is to the right of the leading entry of the row above it, and all entries in a column below a leading entry are zeros. RREF is a more simplified form that follows the same rules as REF, but also requires that the leading entry in each nonzero row is a 1 and is the only nonzero entry in its column.

What is the difference between Gauss and Gauss-Jordan?

Gauss and Gauss-Jordan are both methods used to solve systems of linear equations.

Gauss is also known as Gaussian Elimination and is the process of reducing a matrix to REF by using elementary row operations. Gauss-Jordan, on the other hand, is the process of reducing a matrix to RREF by using elementary row operations. This results in a more simplified and easier to solve system of equations.

Which method is better, Gauss or Gauss-Jordan?

Both methods have their own advantages and disadvantages, so it ultimately depends on the specific problem being solved. Gauss-Jordan may be preferred for its ability to directly obtain the solution to a system of equations, while Gauss may be preferred for its simplicity and ease of understanding.

Can both methods be used to solve any system of linear equations?

Yes, both Gauss and Gauss-Jordan can be used to solve any system of linear equations. However, some systems may be easier to solve using one method over the other. It is important to understand both methods and choose the one that is most efficient for a given problem.

Are there any situations where one method is more useful than the other?

Yes, there are certain situations where one method may be more useful than the other. For example, Gauss-Jordan may be more useful when dealing with systems of equations that have many solutions, while Gauss may be more useful when dealing with systems of equations that have no solutions. It is important to understand the strengths of each method in order to choose the most appropriate one for a given problem.

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