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I was wondering if the following statements about ERO's (elementary row operations) are mathematically correct. Any help and insight is appreciated. Any additional facts to the table is also welcomed.

ERO's performed on a matrix $A$ will preserve

1) The row space of the matrix A

2) The null space of the matrix A

3) Any linear dependence or independence of the columns of $A$

ERO's performed on a matrix $A$ will NOT preserve

1) The column space of the matrix $A$

2) Any linear dependence or independence of the rows of $A$

note that the preservation of linear dependence / independence refers to preserving not just if the set of vectors is linearly dependent / independent - but also the exact relationships between the vectors themselves (e.g. vector #1 is exactly 2x that of vector #3)

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What you have now is correct (see the edit I made).

I would also add: ERO's preserve the dimension of the columns space (and nullspace of $A^T$), even if they don't preserve the spaces precisely. Also, "the linear dependencies of the columns of $A$" is precisely the information conveyed by the nullspace of $A$. That is: for any $x_1,\dots,x_n$ such that $x_1 \mathbf a_1 + \cdots + x_n \mathbf a_n = 0$, we have $$ A \pmatrix{x_1\\ \vdots \\ x_n} = \pmatrix{0 \\ \vdots \\ 0} $$ Similarly, the nullspace of $A^T$ conveys the linear dependencies of the rows of $A$.

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  • $\begingroup$ Ive seen that you changed the original Any linear independence of the columns of A. But I just did it on paper. Consider the following i.imgur.com/rHTsRXI.jpg $\endgroup$ – AlanSTACK May 31 '16 at 8:36
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    $\begingroup$ It is not at all clear to me which row operations you used to go from the first matrix to the second matrix (it should be impossible to do so). In any case, you had the exact same statement in both categories, and you know that can't be right. $\endgroup$ – Omnomnomnom May 31 '16 at 8:39
  • $\begingroup$ The first statement was linear dependence the second statement was linear independence. The 2 ERO's I used was simultaneously adding the rows to each other $\endgroup$ – AlanSTACK May 31 '16 at 8:40
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    $\begingroup$ That is not an elementary row operation. You can replace one of the rows with the sum of two rows, but you can't do that to both rows simultaneously. At least, that is not what is usually meant by "elementary row-operation". $\endgroup$ – Omnomnomnom May 31 '16 at 8:41
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    $\begingroup$ I mean the second thing. That second thing is exactly the information that is conveyed by the nullspaces of $A$ and of $A^T$, as I try to clarify in my answer. $\endgroup$ – Omnomnomnom May 31 '16 at 8:47

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