Question on basic linear algebra (new to the subject)

It would be nice if someone could find the history of why we use the letters i and j or m and n for the basics when working with Matrices ( A = [aij]mxn ). I tried looking up the information and I was not successful. I understand what they represent in the context of the matter, but not why they are those specific letters.

Regarding your second question: There may be more than one history - but the one I am familiar with is related to the legacy programming language FORTRAN. With the original FORTRAN, the type of a numeric variable was specified by its first letter. If it was I through N, it was an integer. Otherwise, it was floating point. According to the original documentation, the I to N came from "INteger". So any name starting with i, j, k, l, m, or n was an integer. Moving through 1-dimensional array would minimally be i=1 to n. Through a second dimension wold be j=1 to m. A third would be k=1 to l. If you needed a fourth dimension, you would need to use two-letter variable names.
FORTRAN was defined in 1957. So the common used of i,n; j,m goes back at least that far.

.Scott said:

Regarding your second question: There may be more than one history - but the one I am familiar with is related to the legacy programming language FORTRAN. With the original FORTRAN, the type of a numeric variable was specified by its first letter. If it was I through N, it was an integer. Otherwise, it was floating point. According to the original documentation, the I to N came from "INteger". So any name starting with i, j, k, l, m, or n was an integer. Moving through 1-dimensional array would minimally be i=1 to n. Through a second dimension wold be j=1 to m. A third would be k=1 to l. If you needed a fourth dimension, you would need to use two-letter variable names.
FORTRAN was defined in 1957. So the common used of i,n; j,m goes back at least that far.

Very interesting ! thx alot for sharing your knowledge :)

Regarding your first question, I'm not sure what you mean.
This is what I get for C = (Cij)3x3 where Cij = (-1)ˆi+j * (2ˆi)*j is
C1,1=-2; C1,2=-8; C1,3=-18
C2,1=4; C2,2=16; C2,3=36
C3,1=-8; C3,2=-32; C3,3=-72

But I suspect you meant:
C = (Cij)3x3 where Cij = (-1)ˆ(i+j) * (2ˆi)*j

.Scott said:

Regarding your first question, I'm not sure what you mean.
This is what I get for C = (Cij)3x3 where Cij = (-1)ˆi+j * (2ˆi)*j is
C1,1=-2; C1,2=-8; C1,3=-18
C2,1=4; C2,2=16; C2,3=36
C3,1=-8; C3,2=-32; C3,3=-72

But I suspect you meant:
C = (Cij)3x3 where Cij = (-1)ˆ(i+j) * (2ˆi)*j

I posted my first question to quickly so sorry to have wasted you time on it tho I see you figured it out :) Initially the way it was displayed to me I thought C23 was the only solution to the problem and my brain did not undertstand, but I came to see that it was just 1 answer out of all the matrice and therefore they wanted us to build the whole matrice like you did in your response :) thanks again for your very quick responses its an amazing first experience for me on this forum and im greatful I found this gold mine !

Another possible origin of I,j,k is the development of quaternions out of the fertile ground of complex numbers. If you recall 'i' is used in complex number notation to indicate the imaginary portion of the complex number with ##i = \sqrt{-1}##

When Hamilton extended the notion of complex numbers to create a 3D number, he chose to extend the number with j and k as well. As an example, ##1 + 2i + 3j + 4k## and
defined the math to be ## i^2 = j^2 = k^2 = i j k = -1## and ##i \times j = -j \times i = k##
and ##j \times k = -k \times j = i## and ##k \times i = -i \times k = j## .

https://en.wikipedia.org/wiki/Quaternion

Gibbs later reworked quaternions into vectors by keeping the i,j, and k unit vector notation and dropping the real part. Both of these notions precluded programming languages by a few years.

jedishrfu said:

Another possible origin of I,j,k is the development of quaternions out of the fertile ground of complex numbers. If you recall 'i' is used in complex number notation to indicate the imaginary portion of the complex number with ##i = \sqrt{-1}##

When Hamilton extended the notion of complex numbers to create a 3D number, he chose to extend the number with j and k as well. As an example, ##1 + 2i + 3j + 4k## and
defined the math to be ## i^2 = j^2 = k^2 = i j k = -1## and ##i \times j = -j \times i = k##
and ##j \times k = -k \times j = i## and ##k \times i = -i \times k = j## .

https://en.wikipedia.org/wiki/Quaternion

Gibbs later reworked quaternions into vectors by keeping the i,j, and k unit vector notation and dropping the real part. Both of these notions precluded programming languages by a few years.

First off thanks alot for sharing your knowledge :) I can see with your answer and .Scott's that the history behind these letters is alot more intruiging than expected, but also alot more complex. This link will be useful for my research thank you :)

As far as the m and n usage. Cayley coined the term matrix, so the notation is likely from him with m from matrix and the n the next letter in line.

TheJohn said:

It would be nice if someone could find the history of why we use the letters i and j or m and n for the basics when working with Matrices ( A = [aij]mxn ). I tried looking up the information and I was not successful. I understand what they represent in the context of the matter, but not why they are those specific letters.

Well, ##\mathrm{i}## comes in naturally for an i-ndex. That makes ##\mathrm{j}## the natural next choice. And since two aren't sufficient, we get other pairs like ##(m,n)\, , \,(p,q)\, , \,(k,l)\, , \,(\mu,\nu)## etc.

You can name them as you like, but sticking to conventions makes it easier to read. People expect a matrix ##a_{ij}## and its transpose ##a_{ji}.## You could switch this, or use something completely different, e.g. ##a_{\text{tree}\;\text{bush}}## but that would annoy all your readers since they have to think more about your notation than your text. Conventions are helpful. So ## \mathrm{i} ## is an index and the rest is a consequence of it.

Edit: The ##\mathrm{i}## and ##\mathrm{j}## are also tiny which is good for an index. You don't want to have ##a_{\Sigma \Pi}.##

jedishrfu said:

Another possible origin of i, j,k is the development of quaternions out of the fertile ground of complex numbers. If you recall 'i' is used in complex number notation to indicate the imaginary portion of the complex number with ##i = \sqrt{-1}##.

That doesn't seem very likely to me. The i, j, and k in quaternions are all imaginary numbers. i for index, as fresh_42 suggests, seems closer to the mark.

You can muddle around in Google Books and find linear algebra books from the first half of the 20th century using the convention.