# Understanding the definition of row echelon form from Golan.

The definition is given below:

But I do not understand what is $$s(i)$$ and how to know it, could anyone give me a numerical example to explain the definition,please?

• $s\left(i\right)$ is just a dummy variable. He doesn't write $s$ because there is one such integer for each $i$, and you cannot call them all $s$; you have to put the $i$ somewhere to be able to distinguish them from one another. Most people would call them $s_i$ instead, but it's just a matter of notation. – darij grinberg Sep 27 at 21:53
• This may help to clarify the definition:math.stackexchange.com/questions/3355553/… – NoChance Sep 28 at 12:38

It means that for each row $$i$$, all elements before a certain column $$j=s(i)$$ are $$0$$, and that the function $$i\longmapsto j=s(i)$$ is (strictly) increasing, i.e. the number of $$0$$s at the beginning of a row is increasing.

This condition is not satisfied in the counterexample they give, as $$s(1)=1, s(2)=3, s(3)=3, s(4)=5$$.

• But how is $s(i) \leq n+1?$ should not it be strictly less than $n+1$? – Mathstupid Sep 28 at 5:47
• If $s(i)=n+1$, it means that you have a row of $0$s (so all further rows will be $0$ rows). – Bernard Sep 28 at 9:08
• But $n$ is the number of columns not rows. – Mathstupid Sep 28 at 23:52
• Oops! I thought this was a square matrix. However, $\mathcal M_{k\times n}$, in the usual conventions, denotes the set of matrices with $k$ rows and $n$ columns. – Bernard Sep 29 at 0:03
• So what does it mean $s(i) = n +1$ in this case? – Mathstupid Sep 29 at 0:20

$$s(i)$$ is the position (column number) of the first nonzero entry in row $$i$$. In the last matrix in your example $$s(2) = s(3) = 3$$.

If row $$i$$ is all $$0$$ then $$s(i)=n+1$$ even though there are only $$n$$ columns.

• But how $s(i) \leq n+1$? is not it should be strictly less than $n+1$? – Mathstupid Sep 28 at 5:30