# 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*}

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

## Related Questions

### Calculate if a line will pass through a given point?

2  Asked on December 20, 2020 by akash-jain

### I need to find the value of x. Im only given the a degree how would you solve this?

1  Asked on December 20, 2020 by liz

### Find the Eigenvectors $Tleft(left[begin{array}{ll}a & b \ c & dend{array}right]right)=left[begin{array}{ll}d & b \ c & aend{array}right]$

2  Asked on December 20, 2020

### How to show that there exist a m such that $a_0I+a_1T+dots+a_mT^m=0$?

1  Asked on December 20, 2020 by sunit-das

### Integral $intlimits^{infty}_0frac{tan^{-1}t }{(1+t)^{n+1}} dt$

3  Asked on December 20, 2020 by chunky-norris

### Place any number of parentheses into $1div2div3div4div5div6div7div8div9div10$ to get the number $256/63$

2  Asked on December 20, 2020 by user842445

### I want to show that $ker(h)=2Bbb Z_2^2=2(Bbb Z_2 timesBbb Z_2)$

0  Asked on December 20, 2020 by masmath

### Is $frac{1}{x} = 4$ strictly a linear equation (in one variable)?

1  Asked on December 20, 2020 by ispring-quiz

### Doubt in following one step of Proof

1  Asked on December 19, 2020 by latus_rectum

### How to use stars and bars to count how many terms there are in a polynomial expansion?

2  Asked on December 19, 2020 by grompchompz

### Cross Product in Euclidean Space with infinite Dimensions

1  Asked on December 19, 2020 by enock-kabibi

### How to prove that $lim_{xtoinfty}frac{(log_2 x)^3}{x^n}=0$

4  Asked on December 19, 2020 by shiran-shaharabani

### Probability theory (heads and tails)

1  Asked on December 19, 2020 by dshunevich

### Infinite set as a countably infinite union of infinite disjoint subsets

1  Asked on December 19, 2020 by xuuserac

### Can $Tleft( nright) = 4Tleft({nover 2}right) + 2^{nover2}$ be solved using Akra-Bazzi method?

2  Asked on December 19, 2020 by akash-karnatak

### Integral involving distance to the boundary

0  Asked on December 19, 2020 by isak

### In a compact metric space, a sequence with a certain property is convergent

1  Asked on December 19, 2020 by stakmod

### can you help with this exponential decay question?

2  Asked on December 19, 2020 by perfectoid

### Linear combinations of four-dimensional vectors

2  Asked on December 18, 2020 by sammy

### would relation induced subset break cartesian product – homomorphism of convex hull?

0  Asked on December 18, 2020 by peng-yu