Linear algebra, row reduced echelon form problem

I am trying to read a book called a first course in linear algebra. The problem that I was reading the Gaussian elimination proof and found some gaps but I am skeptical that I found a gap in proof of an elite mathematician also during the proof I found one unnecessary information in the proof so I feel I am missing something[this is the proof I am talking about. The gap is in paragraph 2 when the author says that the j rows are still row reduced without justifying. The unnecessary part is when in third paragraph the author said j columns in first r rows is still row reduced. Am I missing something][1] [1]:

and the proof was this :

Proof suppose that A has m rows and n columns. We will describe a process for converting A into B via row operations.This procedure is known as Guass-Jordon elimination. Tracing through this procedure will be easier if you recognize that I refers to a row that is being converted, j refers to a column that is being converted, and r keeps track of the number of nonzero rows.Here we go.

  1. Set j =0 and r = 0.
  2. Increase j by 1. If j now equals n+1 then stop.
  3. Examine the entries of A in column j located in rows r+1 through m.if all of these entries are zero then go to step 2.
  4. Choose a row from rows r +1 through m with a nonzero entry in column j. Let I denote the index for this row
  5. increase r by 1.
  6. use the first row operation to swap rows i and r.
  7. use the second row operation to convert the entry in row r and column j to a 1.
  8. use the third row operation with row r to convert every other entry of column j to zero. 9.Go to step 2.
The result of this procedure is that the matrix A is converted to a matrix in reduced row-echelon form, which we will refer to as B. We need to now prove this claim by showing that the converted matrix has the requisite properties of Definition RREF. First, the matrix is only converted through row operations (Steps 6,7,8), so A and B are row-equivalent(Definition REM). It is a bit more work to be certain that B is in reduced row-echelon form. We claim that as we begin Step 2, the first j columns of the matrix are in reduced row-echelon form with r nonzero rows. Certainly this is true at the start when j = 0, since the matrix has no columns and so vacuously meets the conditions of Definition RREF with r = 0 nonzero rows.

In Step 2 we j by 1 and begin to work with the next column. There are two possible outcomes for step 3. Suppose that every entry of column j in rows r+1 through m is zero. Then with no changes we recognize that the first j columns of the matrix has its first r rows still in reduced-row echelon form, with the final m-r rows still all zero.

Suppose instead that the entry in row i of column j is nonzero. Notice that since r+1 is less or equal i less than or equal m, we know the first j-1 entries of this row are all zero. Now, in step 5 we increase r by 1, and then embark on building a new nonzero row. In step 6 we swap row r and row i. In the first j columns, the first r-1 rows remain in reduced row-echelon form after the swap. In step 7 we multiply row r by a nonzero scaler, creating a 1 in the entry in column j of row i, and not changing any other rows. This new leading 1 is the first nonzero entry in its row, and is located to the right of all the leading 1's in the preceding r-1 rows. With Step 8 we insure that every entry in the column with this new leading 1 is now zero, as required for reduced row-echelon form. Also, rows r+1 through m are now all zeros in the first j columns, so we now only have one new nonzero row, consistent with our increase of r by one. Furthermore, since the first j-1 entries of row r are zero, the employment of the third row operation does not destroy any of the necessary features of rows 1 through r-1 and rows r + 1 through m, in columns 1 through j-1.

so at this stage, the first j columns of the matrix are in reduced row-echelon form. When Step 2 finally increases j to n+1, then the procedure is completed and the full n columns of the matrix are in reduced row-echelon form, with the value of r correctly recording the number of nonzero rows
 
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Hello for more clarification my problem is that I feel that the proof is not fully rigorous there is a gap in the reasoning in paragraph 2 in the image when the author said we recognize the first j columns are still row reduced without justification. The other problem was that I found in paragraph 3 when author specified that the first j columns are row reduced in first r-1 rows is just unnecessary. It isn’t required for the proof to be complete. So am I right
 
AbdelRahmanShady said:
Why isn’t my question answered
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I can't say for sure, Abdel. If I were to speculate, I would say that it's likely no one currently has enough personal interest to partially peer review a proof that has already been published in a book.

Also, you'd posted the latest information during only the past 48+ hours. Have you read the guidelines? They ask you to "please have patience". This site is staffed by volunteers. There are no paid employees standing by waiting to provide service on demand.

You've asked questions in two threads which seem similar. You've received replys. Is the specific question that you're thinking about now a restatement of something you've already asked?

Thank you for your understanding, and please be prepared for the possibility that this thread might not receive any more replies.

Moderator Note: The two threads have been merged

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Unnecessary information in a proof​

My new question is why in third paragraph he mentioned that first j columns in first r-1 rows is still in row reduced echelon form.
I feel it was unnecessary for the proof
 
AbdelRahmanShady said:
Hello for more clarification my problem is that I feel that the proof is not fully rigorous there is a gap in the reasoning in paragraph 2 in the image when the author said we recognize the first j columns are still row reduced without justification. The other problem was that I found in paragraph 3 when author specified that the first j columns are row reduced in first r-1 rows is just unnecessary. It isn’t required for the proof to be complete. So am I right
Click to expand...
Please explain why you think the conclusion is not justified. We need to understand your thinking before we can correct it. And if there really is an error in the proof, you have to convince us.

To do this, you can first restate the paragraph in question, in your own words. Then, try to show how what he says is true might not be true.

In particular, what does "with no changes" mean? (That is the justification he gives!)

In general, we write proofs expecting the reader to be able to fill in little gaps; that's why proofs require careful reading. Sometimes a reader needs help; authors can't always accurately judge what a student will be able to follow. But when you ask for help in reading it, you have to make it clear how you are interpreting things, because others may not see it the way you do.
 
the paragraph I am talking about is second paragraph while he was seeing the two different cases, first when rows r to m+1 then he said with no changes we recognize that the first j columns are in Row-Echelon form. While this sound intuitive I feel he needed to prove that the new matrix meet definition of row reduced echelon form
 
I see now that you had already asked this question in another thread, and were answered, and seemed to be satisfied. Why did you ask it again? Please stick with the original thread for that question; maybe put the second question, which I suppose wasn't answered, in a separate thread.
 
but what about the other question about the unnecessary information in third paragraph when he said that j columns in first r-1 rows are in row reduced echelon form.
 
Then ask about that question, not the one that was already answered. And either do it in the original thread where there is already discussion, or else in a new thread focused just on that. Double-posting really bothers me.
 
I was reading a proof of Gaussian elimination, but I got confused because the proof had one unnecessary information. it was in paragraph 3 the part when the author specify that the first j columns in the first r-1 rows are in row reduced echelon form. The proof is the attached image
 

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AbdelRahmanShady said:
the author specify that the first j columns in the first r-1 rows are in row reduced echelon form
Click to expand...
Hello Abdel. The statement above does not appear in the image. Please post the exact statement that you claim is unnecessary (i.e., type it word-for-word, as it appears in the image), and then explain why you believe the inclusion of that particular statement causes confusion. Thanks!

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In line 4 of paragraph 3, the author specifies that in the first j columns, the first r-1 rows remain row reduced
 
So it's not wrong then, just extra information -- like some of the other expository comments. I suppose different people would describe the same proof differently.

How were you confused?

[imath]\;[/imath]
 
I saw that the information but I saw I reason to put extra information in the proof. So I thought I was misunderstanding something.
So is it extra information or am I wrong
 
My point is this: When describing a proof, it's not necessarily wrong to include written statements that certain readers don't need to see. Some authors use more words, other authors use less words, when describing the same process.

You'd thought that you had misunderstood something, but then you figured it out, so there's currently no issue.

[imath]\;[/imath]
 
So you mean some author add extra information in a proof even if it is is unnecessary for the proof. And even if not including it doesn’t decrease the rigour of the proof, am I right
 
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