# Reason for existence of 'swapping' elementary matrix operation

In my Hoffman & Kunze book on Linear Algebra, we define 3 elementary row operations, the first of which is the swapping of two rows.

I'm wondering why we need to even have such an elementary operation since it can be achieved through elementary operations of the second and third kind. Is it simply for convenience & simplicity, or is there some deeper reason? It seems that all the linear algebra books that i have checked always include all 3 operations, though one of them is redundant.

I was under the impression that when we try and list the core properties for an object in mathematics the goal is to always minimize the number of properties per definition.

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Its much easier to say, and write, "interchange rows blah and blah", than to say precisely the same thing with a great many more words and needless obfuscation. So, yes, it's for simplicity and convenience. – David Mitra Jan 26 '13 at 21:14
Good point as Parsimony is desirable. It is my understanding that it is mentioned because its effect when calculating/axiomatically introducing determinants, its effect - while replicable using other operations - is distinct (sign change). Also, it is handy to define the important permutation matrices more clearly than using the other types. – gnometorule Jan 26 '13 at 21:17
Avoiding redundancy in definitions is a goal, but it competes with the goals of making them useful and readily comprehensible. In my view the latter two are more important. In this case identifying row interchange as an elementary row operation is unquestionably useful. – Brian M. Scott Jan 26 '13 at 21:18
Newton's first law of motion is a special case of Newton's second law of motion, but virtually every Physics textbook lists three Newton's laws of motions despite the redundancy. Since there is no monolithic mathematical god who enacts the definition of elementary row operations, I think it is the consensus of the mathematical community to value convenience in interchanging rows over parsimony in the number of types of elementary row operations. – user1551 Jan 26 '13 at 21:58

## 2 Answers

"I was under the impression that when we try and list the core properties for an object in mathematics the goal is to always minimize the number of properties per definition."

It is an amusing parlor game to get the axioms for, say, a vector space down from the usual ten to a single axiom. See here for a single axiom for the propositional calculus (logic without quantifiers); here for a single axiom for group theory. These are interesting exercises, but it is not meant for anyone to use the resulting (highly complicated and non-intuitive) axiom as the way to go in developing the theory. So I am afraid your impression is mistaken.

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Besides the clarity reason, the traditional Gauss-Jordan algorithm is split into a forward phase and a backward phase. In the forward phase, the only elementary row operations of the form 'add a multiple of Row $i$ to Row $j$' that can be used are those in which $i < j$, while the restriction for the backward phase is that $i > j$. So let us distinguish these two cases as two types of 'add a multiple of one row to another' moves.

As it turns out, there is theoretical value (such as the 'LPU' decomposition, a special case of the Bruhat decomposition) in distinguishing the two types and only using one at a time. Since achieving an interchange of rows requires use of both types of the 'add a multiple of one row to another' move, it cannot be achieved when you don't allow mixing of the two types (i.e. you can only begin the backward phase after you are completely finished with the forward phase).

This can be considered an illustration of why giving axioms and definitions in the most conceptually clear way is often more valuable than doing so in the most logically efficient way: conceptual clarity (plus some ingenuity and lots of perseverance!) can help one progress further in developing a mathematical field, while logical efficiency is often of little or no value in doing so.

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