Notation for changing rows in a matrix

In summary: So this is not the same as ## R2 \leftrightarrow R3, R1 \leftrightarrow R3 ##.In summary, the conversation discusses the use of notation for elementary row operations in a linear algebra course. The speaker mentions that they interpret R1 <-> R2 to mean swap row 1 and row 2, but then questions if this notation is correct. They suggest using right-pointing arrows, R1 -> R2, to indicate that row 1 goes to row 2, but acknowledges that this may be ambiguous. The speaker prefers two approaches for notation: using a single R1 <-> R2 for swapping two rows, or using R1 -> R2' and R2 -> R1'
  • #1
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Homework Statement
Please see below
Relevant Equations
Please see below
For this,
1686280663310.png

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!
 
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  • #2
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'
 
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  • #3
FactChecker said:
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!
 
  • #4
CORRECTION: There is a level of standardization of the notation that I was not aware of. This post is wrong.
ChiralSuperfields said:
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'.
 
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  • #5
FactChecker said:
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!
 
  • #6
FactChecker said:
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 ##).

FactChecker said:
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 ##.
 
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  • #7
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.
 
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  • #8
ChiralSuperfields said:
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.
FactChecker said:
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##
 
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  • #9
pbuk said:
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.
pbuk said:
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. :-)
 
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Related to Notation for changing rows in a matrix

What is the notation for swapping two rows in a matrix?

The notation for swapping two rows in a matrix is often represented using a permutation matrix or by indicating the row swap directly in the matrix notation. For example, if you want to swap row i and row j, you might see it written as Ri ↔ Rj.

How do you denote the operation of adding a multiple of one row to another?

The operation of adding a multiple of one row to another is typically denoted as Ri → Ri + kRj, where you are adding k times row j to row i.

What is the notation for multiplying a row by a scalar?

Multiplying a row by a scalar is usually denoted as Ri → kRi, where k is the scalar you are multiplying row i by.

How is the row reduction process typically notated?

The row reduction process is typically notated using a sequence of elementary row operations. Each step may be indicated with the specific operation being performed, such as Ri ↔ Rj for a row swap, Ri → kRi for row scaling, and Ri → Ri + kRj for row addition.

What is the purpose of using specific notation for row operations in matrices?

The purpose of using specific notation for row operations in matrices is to clearly and concisely communicate the steps being taken to manipulate the matrix. This helps in understanding the transformations being applied, especially in methods like Gaussian elimination or finding the inverse of a matrix.

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