Notation for changing rows in a matrix

[member 731016]

Homework Statement

Please see below

Relevant Equations

Please see below

For this,

What was wrong with the notation I used for showing that I has swapped the rows? The marker put a purple ?

Any help greatly appreciated!

Many thanks!

 

[FactChecker]

I would interpret R1 <-> R2 to mean swap row 1 and row 2. But then you put R2 <-> R1, so that would swap them back where they started.
The note that your teacher put there is a little troublesome.
I would use right-pointing arrows, R1 -> R2 to indicate that row 1 goes to row 2. But then does it overwrite row 2? It's a little ambiguous.
I would prefer one of these two approaches:
1) A single R1 <-> R2.
CORRECTION: There is a level of standardization of the notation that I was not aware of. This option #2 is wrong.
2) R1 -> R2' and R2 -> R1'

 

[member 731016]

I would interpret R1 <-> R2 to mean swap row 1 and row 2. But then you put R2 <-> R1, so that would swap them back where they started.
The note that your teacher put there is a little troublesome.
I would use right-pointing arrows, R1 -> R2 to indicate that row 1 goes to row 2. But then does it overwrite row 2? It's a little ambiguous.
I would prefer one of these two approaches:
1) A single R1 <-> R2.
2) R1 -> R2' and R2 -> R1'

Thank you for you reply @FactChecker!

Your approaches are very helpful. The course textbook uses the first which I think I will use.

Many thanks!

 

[FactChecker]

CORRECTION: There is a level of standardization of the notation that I was not aware of. This post is wrong.

The course textbook uses the first which I think I will use.

The second approach would be more useful when several rows are moved, but not necessarily swapped.
If you were going to move row 1 to row 2, row 2 to row 3, and row 3 to row 1, you could put:
R1 -> R2' and R2 -> R2' and R3 -> R1'.

 

[member 731016]

The second approach would be more useful when several rows are moved, but not necessarily swapped.
If you were going to move row 1 to row 2, row 2 to row 3, and row 3 to row 1, you could put:
R1 -> R2' and R2 -> R2' and R3 -> R1'.

Thank you for your reply @FactChecker!

 

[pbuk]

I would use right-pointing arrows, R1 -> R2 to indicate that row 1 goes to row 2. But then does it overwrite row 2? It's a little ambiguous.
I would prefer one of these two approaches:
1) A single R1 <-> R2.
2) R1 -> R2' and R2 -> R1'

Why are you inventing your own notation? There is a universally recognised notation for elementary row operations, in this case R1 <-> R2 (or rather $R1 \leftrightarrow R2$).

The second approach would be more useful when several rows are moved, but not necessarily swapped.
If you were going to move row 1 to row 2, row 2 to row 3, and row 3 to row 1, you could put:
R1 -> R2' and R2 -> R2' and R3 -> R1'.

No, do NOT do this, rotating three rows is not an elementary row operation. To achieve this you should write $R1 \leftrightarrow R2, R1 \leftrightarrow R3$.

 

[fresh_42]

I read it as a wrong positioning. I think it should have been
$$
A\stackrel{R_1\leftrightarrow R_2}{\sim} B
$$
and not
$$
\left. A\sim B \right| R_1\leftrightarrow R_2\, , \,R_2\leftrightarrow R_1
$$
which is too late, since you already did it at $\sim$, and ambiguous since you seem to do and re-do it.

 

[Mark44]

What was wrong with the notation I used for showing that I has swapped the rows? The marker put a purple ?

As already mentioned, I believe the note was that your notation appeared after the 2nd matrix, not between the 1st and 2nd matrices.

The note that your teacher put there is a little troublesome.
I would use right-pointing arrows, R1 -> R2 to indicate that row 1 goes to row 2.

As already mentioned by @pbuk, the standard notation for swapping rows m and n is $R_m \leftrightarrow R_n$. There are only three row operations:

1. Exchanging (swapping) two rows: $R_m \leftrightarrow R_n$
2. Replacing a row by a nonzero multiple of itself: $R_m \leftarrow kR_m$
3. Replacing a row by the sum of another row and itself: $R_m \leftarrow R_m + R_n$

 

[FactChecker]

Why are you inventing your own notation? There is a universally recognised notation for elementary row operations, in this case R1 <-> R2 (or rather $R1 \leftrightarrow R2$).

Ok, I'll buy that. I didn't know that the notation was so standardized.

No, do NOT do this, rotating three rows is not an elementary row operation. To achieve this you should write $R1 \leftrightarrow R2, R1 \leftrightarrow R3$.

This is interesting. Although it makes sense, I didn't realize that there was so much standardization here. So the notation: $R1 \leftrightarrow R2, R1 \leftrightarrow R3$ is not commutative. The second $R1$ is the original row $R2$.
This type of standardization is significant when a mathematical notation of row operations is developed. It is intellectually satisfying. :-)