New to Linear Algebra: Confused on Reduced Row Echelon Form

[sniper_spike]

Hey guys, lurked here a bit, but now I'm in this new math course right. So anyway it seems like a completely new language to me. There's some discussions about reduced row echelon form in the textbook I'm using. I was taught (before going into the course) that the object was to get leading 1s in a diagonal, then 0's in every other spot. The textbook doesn't seem to corroborate this, instead it only puts zeros above the 1s. Then I check wiki and it states

'reduced row echelon form... satisfies the additional condition:

Every leading coefficient is 1 and is the only nonzero entry in its column,[2] as in this example'

but then at the same time it shows a matrix which has nonzeros other than 1 and says its also a reduced row echelon form.

https://en.wikipedia.org/wiki/Row_echelon_form
Could anyone help clear up some of these initial confusions I'm having?

 

[micromass]

Can you post the title and author of your textbook??

Anyway, about the wiki example you mention. You should only look at the leading coefficients. For example, the first row has a leading coefficient in the first column. Every other value in that row should be zero. The second row has a leading coefficient in the second column. Every other value in the second column should be zero. And finally, we have a leading coefficient n the third row at the fourth column. This implies that everything else in the fourth column should be zero. Other columns should necessarily have zeroes.

So in our example, the leading coefficients occur in column 1, 2 and 4. So these are the only columns which should have zeroes. The rest of the columns can be anything.

 

[sniper_spike]

What do you mean a leading coefficient in the third row at the fourth column? not third column?

Also its called Elementary Linear Algebra, Applications Version and its by Howard Anton and Chris Rorrers.

In fact it does state these properties of having all zeroes except for the leading coefficient, then in the list of matrices which are examples of reduced row echelon form there is a matrix with a nonzero.

the matrix is

0 1 -2 0 |1
0 0 0 1 |3
0 0 0 0 |0
0 0 0 0 |0

so what's the -2 doing there?

and also a generic form

1 0 * *
0 1 * *
0 0 0 0
0 0 0 0

once again a generic example is

0 1 * 0 0 0 * * 0 *
0 0 0 1 0 0 * * 0 *
0 0 0 0 1 0 * * 0 *
0 0 0 0 0 1 * * 0 *
0 0 0 0 0 0 0 0 1 *

with the *'s being replaced with real numbers

And somewhere I read that you can have a non zero if there is a zero below it. Doesn't make full sense to me.

 

[micromass]

What do you mean a leading coefficient in the third row at the fourth column? not third column?

Also its called Elementary Linear Algebra, Applications Version and its by Howard Anton and Chris Rorrers.

In fact it does state these properties of having all zeroes except for the leading coefficient, then in the list of matrices which are examples of reduced row echelon form there is a matrix with a nonzero.

the matrix is

0 1 -2 0 |1
0 0 0 1 |3
0 0 0 0 |0
0 0 0 0 |0

The only columns we care about are the columns with a leading coefficient. The columns here with a leading coefficient are column 1 and column 4. So those columns should contain all zeroes except for the leading coefficient.
Column 3 contains a -2, but this is not a leading coefficient since there is a 1 in front of it. So column 3 contains no leading coefficients, so nonzero values are allowed.

 

[sniper_spike]

Makes sense. thanks a lot!