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Why did we call a row operation "elementary"?

Mathematics Asked by Neothehero on December 21, 2020

  1. Why we called the three actions of row operation "elementary"? Is there a thing called "advanced" or "complicated" row operation?
  2. I’ve seen the word "non-elementary" row operation is used to describe things like $R_1-R_2$, which is not written in the conventional $-1R_2 + R_1$. Is this usage correct? In particular, what should $R_1-R_2$ be called?
    begin{align*}
    &text{a) A non-elementary row operation} \
    &text{b) An elementary row operation} \
    &text{c) Just a row operation} \
    &text{d) It is not a row operation}
    end{align*}

One Answer

The sense of "elementary" here is that all the operations that preserve the row-space of a matrix can be be produced by combining various elementary row operations.

Thus these are the elementary steps that can be taken to calculate a (reduced) row echelon form of a matrix, a basic tool for solving several kinds of problems involving the row-space of a matrix and the solutions (if any) of a linear system of equations.

With regard to what $R_1 - R_2$ ought to be called, something is missing from the description. It would indeed be an elementary row operation if this "new row" immediately replaces $R_1$. If you wanted it to replace $R_2$, you would have to perform that row operation by combining two "elementary row operation" steps (first replace $R_2$ with $R_2 - R_1$, then multiply the resulting new second row by non-zero scalar $-1$).

If you wanted to do something completely different with $R_1 - R_2$, then it would possibly either not be a row operation or not a row operation that preserved the row space of the matrix. For example, if you replaced $R_3$ with $R_1 - R_2$ (leaving $R_1,R_2$ as they are), you might well be making the row space of the matrix smaller.

Correct answer by hardmath on December 21, 2020

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