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Nicholas Kong

What is the reason for making the leading entries all ones in a echelon form?

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Is it to make the process of finding the values of the variables that much easier when you go from echelon form of a matrix to reduced echelon form of a matrix?

 

Subject is Linear Algebra.

Edited by warnexus

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Wikipedia tells me that this requirement to have the leading non-zero term be a 1 is not universal. I suspect that it's just done in order to make the echelon form unique. If you don't specify that, then depending on how exactly you carry out the reduction, you could get different results from someone else also reducing that same matrix.

 

(I'm not 100% sure that it does in fact make the echelon form unique, but it feels like it would to me.)

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It doesn't really matter anyway? Multiplying a row by a factor doesn't really change much. Eigenvectors have the same thing going on, you don't solve

 

Ax = x

 

you solve

 

Ax = kx.

Edited by Paradigm Shifter

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It comes from the definition of row echelon form: what you get when you've finished the first part of Gaussian elimination. The people saying that the leading terms all need to be one are saying that you aren't really done with that first part until you've gotten the coefficients to one.

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It comes from the definition of row echelon form: what you get when you've finished the first part of Gaussian elimination. The people saying that the leading terms all need to be one are saying that you aren't really done with that first part until you've gotten the coefficients to one.

Thanks

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When there is a leading 1, then you could write the equation in the form of whichever variable, respectively, has the 1 in it. For example, if x has the 1 in it, then you can write the equation as: x = Whatever equation. Whereas if the x column had a 2 in it, the respective equation would be 2x = ....

 

So for a solid example: 

 

x y z c
1 0 0 1

4 0 3 7

5 1 3 19

 

This matrix states that: x = 1, 4x +3z = 7, 5x + 1y +3z = 19
but if you look below, Row Echelon Form simplifies this: 

 

Simplifying this into Row Echelon Form it would be:

x y z c

1 0 0 1

0 0 1 1

0 1 0 11

 

which then turns into

x y z c

1 0 0 1

0 1 0 11

0 0 1 1

 

which is equivalent to saying

x = 1, y = 11, z = 1, which turns out the be identical to the original matrix.

Edited by J. Faraday

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