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I'm learning about linear algebra and in the course we've defined three "elementary row operations"

$$(1) \text{ Switching any two rows}$$ $$(2) \text{ Non-Zero scaling of any row}$$ $$(3) \text{ Adding a multiple of one row to a different row}$$

However it seems all these operations can be performed by the two simpler operations:

$$(1) \text{ Non-Zero scaling of any row}$$ $$(2) \text{ Adding any two rows}$$

I mean why even have switching rows as an operation in the first place when it can be performed by the addition and scaling of rows? It just seems redundant, why add an extra operation that can already be performed by the other operations?

Edit: Because someone in the comments asked how the swapping of rows could be performed with just non-zero scaling and adding rows. If you wanted to swap, say row $p$ with row $q$ you would:

$$(1) \text{ Add row } q \text{ to row } p$$ $$(2) \text{ Multiply row } p \text{ by } -1$$ $$(3) \text{ Add row } p \text{ to row } q$$ $$(4) \text{ Multiply row } q \text{ by } -1$$ $$(5) \text{ Add row } q \text{ to row } p$$ $$(6) \text{ Multiply row } p \text{ by } -1$$

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  • $\begingroup$ @Itay4 You scale a row, add it to another row, then scale the row by the inverse of the first scalar. $\endgroup$
    – Marra
    Apr 25, 2017 at 18:32
  • $\begingroup$ The point of the third operation is exactly this: to make things shorter. You may, in fact, work with only two operations. $\endgroup$
    – Marra
    Apr 25, 2017 at 18:33
  • $\begingroup$ @Marra I thought the point was to have the row operations be as simple as possible? Also why have adding a scaled multiple of a row instead of just the sum of two rows, when you could scale it and then add them? $\endgroup$
    – nathan56
    Apr 25, 2017 at 18:36
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    $\begingroup$ @Marra: there may be more to this, from an algorithmic viewpoint. Using switching, all information can be stored in the matrix as-you-go. If you want to perform a switch using only (1) and (2), you need to store additional information. $\endgroup$
    – mlc
    Apr 25, 2017 at 18:36
  • $\begingroup$ @nathan56 if you want the theory to be strict (in the sense that it reduces everything to the mininum needed) then you are correct. But then, when using Gauss-Jordan's method do solve linear systems, the third operation is a tool that makes everything simpler. Also, as pointed out by the previous comment, this might be due do algorithmic reasons. $\endgroup$
    – Marra
    Apr 25, 2017 at 18:40

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This is an interesting question, but you didn't go deep enough. For instance, why do we even have addition, if we can represent adding in terms of set theory? The answer, as stated above, is simplicity - at a suitable level of abstraction, things become simpler to denote and to describe, talk about, and work with. Most of the topics in math are somewhat arbitrary decisions to draw a line in the sand and say "here's a new branch, let's develop useful notation and techniques for it".

The key word is useful. We use linear algebra because it has great applications both in pure mathematics and applied, and is worthy of study in and of itself. However, it can definitely be simplified to a considerable degree, if we define "simplification" as "reducing the number of fundamental operations". If, instead, we define "simplification" as "decrease the effort to use the mathematics by introducing shorthand for common operations", then having all of these elementary row operations is useful.

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  • $\begingroup$ I appreciate your answer and up-voted you. However wouldn't reducing the number of row operations make things simpler? The author in my book uses this notion of elementary row operations to define "elementary matrices" that is identity matrices which differ by one elementary row operation - he then proves that every invertible matrix can be written as a product of "elementary matrices". If the goal is to decompose invertable matrices into a product of very simple looking matrices, why not use the simpler definition of row operation corresponding to simpler "elementary matrices"? $\endgroup$
    – nathan56
    Apr 25, 2017 at 20:35
  • $\begingroup$ I mean if we allow row permutation to be an elementary row operation, even though it can be performed as a composition of the other operations why not allow the simultaneous scaling of more then one row to be an elementary row operation? i.e. why not then let all diagonal matrices be elementary matrices, this would coincide with @marra's definition of simplicity wouldn't it? $\endgroup$
    – nathan56
    Apr 25, 2017 at 20:37
  • $\begingroup$ And you're right. Basically, we choose to define certain operations because they are convenient or interesting. One such operation that makes proofs shorter is swapping. Also, you're touching on an interesting point: the operation of swapping can be represented as you showed, but we could define many other ways to represent it. The end result is the same. Thus the operation itself might be defined in terms of its end result, and the implementation may be left to the convenience of the mathematician. $\endgroup$
    – Michael Stachowsky
    Apr 25, 2017 at 21:17

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