Linear algebra, row reduced echelon form problem

A

AbdelRahmanShady

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I was reading robert beezer book about linear algebra and in a part he proved guassian elimination using a constructive proof
The proof involved adding a new column
and perform some operations on the matrix that dont change its solution set but keep it row reduced ,the problem is during second paragraph he said adding a new column with r+1 to m rows of matrix is zero then the matrix is row reduced
but he didnt specify how the new matrix met the definition of row reduce echolon form,
Am I missing something or is the proof missing
 

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  • row reduced 1.jpg
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  • row reduced 2.jpg
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My question in the second paragraph author states we recognize that matrix is still in row reduced without justifying why
 
When we perform step 2, the first (j-1) collumns are in RREF, with r non-zero rows. There are now two cases:
(i)
1626859011302.png
In which case, the first j columns are in RREF - just check the definition.
1626859785182.png

or (ii)
There is a non-zero element in column j, 'below' the first r rows:
1626859821324.png
and that is what the next two paragraphs are about.
 
Didnt he have to say since j -1 columns are row reduced and the new column has r+1 to m rows zero then any non zero element in the j column in first r rows will not be the left most element so the left most elements in first r rows dont change leading to definition RREF to still hold, why wasnt it necessary to mention all that
 
If explicitly showing that the top-left [imath]r \times j[/imath] matrix is in RREF, then I would go through all of the points of the definition, showing that they apply.
1626900349967.png
I certainly think it is worth highlighting that the fact the elements of column j from rows r+1 to m are all 0, means that we do not increment r so when j is incremented the top-left [imath]r \times (j-1)[/imath] matrix is in RREF and each row from 1 to r has a non-zero element in the first (j-1) columns.
 
I wanted to show explicitely that after adding new column the first j columns are in row reduced form, so am I correct tgat the proof is missing that part
 
And 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 -
1626907483692.png
However often when we try to prove the 'obvious' it turns out not to be true!
If you did want to prove it explicitly, you would need to show that all 4 criteria of the definition are fulfilled, and not just observe that 'the new column will not change the leftmost element' - there is more to the definition than this.
 
I think the proof will be something like this, since the leftmost non zeri entry in first r rows didnt change it so property 2 still holds as the entries are still 1 and also propert 1 of RREF still holds as there are r non zero rows then r+1 to m zero rows property 3 also holds as the columns of the left most entries disnt change and property 4 holds as the column and rows of leftmost non zero entries disnt change and since it was true before adding column j it is true after addinb the column,
Is this a good proof
I was proof the bit in second paragraph
 
AbdelRahmanShady said:
I think the proof will be something like this, since the leftmost non zeri entry in first r rows didnt change it so property 2 still holds as the entries are still 1 and also propert 1 of RREF still holds as there are r non zero rows then r+1 to m zero rows property 3 also holds as the columns of the left most entries disnt change and property 4 holds as the column and rows of leftmost non zero entries disnt change and since it was true before adding column j it is true after addinb the column,
Is this a good proof
I was proof the bit in second paragraph
Click to expand...
That will do it.
 
In one of attachments there is a proof of gaussian elimination.
My problem that in second paragraph he said "we recognize the first j columns are in row reduce echelon form without justifying".
I said that to solve gap in paragraph 2, you just need to see that leftmost leading one in first rows didn't change.
So, since their columns and rows didn't change statement 4 of the row reduced definition still apply, moreover, they are still ones so, statement 2 still apply, and the j-1 columns that contains leading 1s didn't change so the column contains zeros except for the row of leading 1
so statement 3 still apply, finally all zero rows are in rows r+1 through m so statement 1 still hold.
My problem is that I am skeptic if my proof is rigorous enough.
Moreover, I wonder why in paragraph 3 he mentioned that "after swap the j columns in first r rows are still row reduced echelon form"". I feel it is unnecessary for the proof
thanks for help in advance.
 
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