Linear algebra, row reduced echelon form problem

[AbdelRahmanShady]

So it is normal to find extra information in a proof that is not necessary and removing this extra information , does not affect the rigor and completeness of the proof

 

[Dr.Peterson]

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

Some authors write their proofs to be more helpful to relative beginners, by pointing out facts that the reader could have noticed, but might not have. There is nothing wrong in doing so.

In your other question about this proof, you were troubled that the author omitted some details that you had to fill in; here you are troubled that he included something that you didn't need to be told. Both are normal in proofs, because the author doesn't know the reader.

 

[AbdelRahmanShady]

my question now, is the something included necessary for the proof to be rigorous? is there any step that relies on this fact? or is this extra information that the author noticed has nothing to do with the proof

 

[mmm4444bot]

my question now, is [extra information] included necessary for the proof to be rigorous?

No, Abdel. You've already declared for us the answer to that question (more than once).

Regarding your other questions (posts #40 and #41), we've already answered those for you (more than once).

If you wish to discuss this issue further (or anything else concerning the personal choices made by the book's author in presenting their proof), please contact the author. Thank you.

Moderator Note: The same question has been raised in three threads. Those threads have been combined into a single thread.

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[AbdelRahmanShady]

you misunderstood my question and stopped replys so I can't clear it up. You think I am asking why the author specified this information and that it is obvious to me.
And that is not my question. My question is whether any further step after this information depended on it.

 

[Steven G]

Post your problem to get help.

 

[AbdelRahmanShady]

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

 

[AbdelRahmanShady]

I posted before but other members misunderstood my question and closed my post

 

[Dr.Peterson]

First, here is the entire theorem and proof as found here, for reference:

Theorem REMEF Row-Equivalent Matrix in Echelon Form​

​

Suppose A is a matrix. Then there is a matrix B so that​

1. A and B are row-equivalent.
2. B is in reduced row-echelon form.

​

​

Proof​

Proof of Theorem REMEF​

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 Gauss-Jordan 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 mm 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 increase 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≤i≤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 scalar, 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 mm 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.​

The sentence I believe you are objecting to is in red. I claim that it is worth being aware of, and that one might not think about it, though it is "obvious" when you do think. In particular, I claim that it is in fact used later, in the line in green. It is being shown that at each time through the loop, the "loop precondition" stated at the start (which I put in bold), "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", remains true. This is an essential part of the proof, so it is important to state it. And the fact in red is part of it.

These facts could have been left unstated, trusting that you would see them; but the author wants to say more than the absolute minimum, in part because he is writing to students who don't have a lot of experience with proofs, and can benefit from guidance.

Does that help at all?

 

[AbdelRahmanShady]

Yes, but if you read before the green you would notice that he proved it without needing to state statement red. Statement red doesn’t say that the first j columns are row reduced with r nonzero rows. It simply stated that first j columns in the first r-1 rows are row reduced ignoring r to m other rows. That’s why I feel it is unnecessary not because it is obvious but because it is irrelevant. On the other side I think that statement green is so important.
So is statement red is irrelevant as I am saying or there is any thing else after this statement that depends on it

 

[AbdelRahmanShady]

By the way sorry for being annoying and asking many questions I just want to make sure I fully understand the proof before completing the book

 

[AbdelRahmanShady]

By the way this how the proof could be without statement red. Since the new column j is to the right to each leading one and is below each leading statement 4 of definition RREF is true. And since there are only zeros in column j except row r then statement 3 is true and since it is 1 so statement 2 is true and statement 1 is true since the all zero rows are in r+1 to m rows so I proved statement green without needing to state statement red

 

[mmm4444bot]

Moderator Note: The OP's fourth thread has been merged with the others.

 

[Otis]

There's nothing wrong with how the author chose to present that part of their proof, Abdel. We understand that you would describe the part slightly differently. There's nothing wrong with that, either. It's a free world, brother.

in third parahraph when he said That first j columns in r rows are still row reduced I think he had to say, Since matrix is row reduced in first r rows For the j-1 columns adding new column will not change the leftmost element Leading to definition RREF to still hold, am I right

I think the author is suggesting it is 'obvious' and we simply 'recognize' the fact

I wonder why…I feel it is unnecessary for the proof

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

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.

Unnecessary information in a proof…My new question is why…he mentioned that first j columns in first r-1 rows is still in row reduced echelon form. I feel it was unnecessary

authors can't always accurately judge what a student will be able to follow…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

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?

but what about the other question about the unnecessary information in third paragraph

I got confused because the proof had one unnecessary information. it was in paragraph 3

If you remove this information the flow of reasoning is still rigorous and complete

So it's not wrong then, just extra information…different people would describe the same proof differently.

I saw [no] reason to put extra information in the proof. So I thought I was misunderstanding something. So is it extra information[?]

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.

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

it is normal to find extra information in a proof that is not necessary and removing this extra information does not affect the rigor and completeness of the proof

Some authors write their proofs to be more helpful to relative beginners, by pointing out facts that the reader could have noticed, but might not have. There is nothing wrong in doing so.

my question now, is the something included necessary for the proof to be rigorous?

No, Abdel.

I got confused because the proof had one unnecessary information.

These facts could have been left unstated, trusting that you would see them; but the author wants to say more than the absolute minimum, in part because he is writing to students who don't have a lot of experience with proofs, and can benefit from guidance.

I feel it is unnecessary not because it is obvious but because it is irrelevant.

By the way…I just want to make sure I fully understand the proof

Your ongoing discussion seems to indicate more than that.

By the way this how the proof could be [rewritten by me] without statement red…

By the way, since you're now at the stage where you're rewriting a presentation to suit your personal preferences, I would suggest that you already understand the proof fully.

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[AbdelRahmanShady]

So is there any errors in my own approach.
And sorry for over skepticism as it is the first time for me to self teach math textbook.
So I sometimes confused whether I understand the proofs or I should revise them.
By the way thanks for helping

 

[AbdelRahmanShady]

And one last thing, statement red wasn’t used again in the proof

 

[AbdelRahmanShady]

By the way do you have any tips on self learning linear algebra from a textbook

 

[AbdelRahmanShady]

So is statement red irrelevant and by irrelevant I mean information that no other step after it or before it depend on it and doesn’t add something to the proof

 

[AbdelRahmanShady]

Sorry for being annoying but I still don’t know your opinions is statement red irrelevant and by irrelevant I mean information that no other step after it or before it depend on it and doesn’t add something to the proof

 

[Dr.Peterson]

Sorry for being annoying but I still don’t know your opinions is statement red irrelevant and by irrelevant I mean information that no other step after it or before it depend on it and doesn’t add something to the proof

I've told you my opinion: the red statement is not irrelevant. It could perhaps be omitted, but it is of use, by pointing out where we stand at that point. And if no one else feels like contributing, then perhaps they feel, as I do, that no more answer is really needed. I think you need to just let it go.

I compare proofs to persuasive essays. People may disagree on what points are essential, and what is the best way to express them; and there can be stylistic differences, particularly depending on the intended audience. To criticize an author's proof because you think a line should be omitted is to imply that you know better. Yes, that can be annoying.