You concisely illustrate my point. You misuse words so that others that have learned the material cannot understand what you're writing.
Calling an m-tuple a coefficient and then calling the column span the "coefficient span" does not show understanding of the material.
>I consider it a more fundamental, more useful concept than your "rank".
The dimension of the span is the rank. It's exactly why rank is important. The nullity is the dimension of the kernel, and rank + nullity = n.
Saying rank is not important is like saying dimension is not important. The dimension of a vector space is the first and most important invariant that describes the space. Rank is that dimension for the image of a linear transform. It's absolutely fundamental. It's why when describing some linear algebra thing, one usually starts with something like "Let V be a n-dimensional real vector space" or similar. We rarely (bordering on never) write "Let V be a vector space spanned by the following vectors", and we never write "the coefficient span of the transform".
>The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm
No, it is a fundamental property of the linear transformation. Gaussian elimination is but one of many ways to compute it. And no matter what choices you use in your elimination, you will always get the same rank. This is super important - that Gaussian elimination gives knowledge of the linear transformation that is independent of choice of basis or of steps performed. Changing of either vector space changes the numbers in the matrix, and different Gaussian elimination steps may have differing intermediate steps, but the rank is the rank is the rank.
It is also true that the row span = the column span = the rank, which is also not immediately obvious. These are theorems proven in basic linear algebra, and fundamental to claiming to understand basic linear algebra.
That rank shows up in Gaussian elimination is not some artifact or unique thing to Gaussian elimination. Since rank shows up everywhere, when it also shows up in Gaussian elimination shows that Gaussian elimination is doing something fundamental - it is but one of many, many ways that rank pops up over and over in linear algebra.
Rank (and nullity) are absolutely fundamental.
And no one calls that m-tuple a coefficient.
None of this is in the book above. Zero.
A final nice point to illustrate, here [1] is the index to Strang's Introduction to Linear Algebra. Rank occurs more than any just about every other entry in the index.
Here's Serge Lang's book [2]. After the word dimension, rank occurs the most in the index.
Book after book shows that after the concept of dimension, rank is probably the most important concept in linear algebra.
I could go on and on. You cannot claim to understand linear algebra without know how pervasive and useful rank is.
> You misuse words so that others that have learned the material cannot understand what you're writing.
> Calling an m-tuple a coefficient and then calling the column span the "coefficient span" does not show understanding of the material.
So… I'm not wrong; I just don't know the words? Sounds like I actually do understand the mathematical concepts. (Maybe I'd find the words easier if mathematicians named stuff sensibly… and yes, I know I'm using English to write that sentence. I'm a hypocrite, but that's no excuse for the naming conventions in abstract algebra and category theory.)
> No, it is a fundamental property of the linear transformation. Gaussian elimination is but one of many ways to compute it.
So why does every textbook, every lecture, and Wikipedia talk incessantly about Gaussian elimination (an algorithm for inverting matrices!) when talking about rank? Rank's only useful as a property of the vector spaces, so why treat it like a separate concept?
> It is also true that the row span = the column span = the rank, which is also not immediately obvious.
The size of the row / column span is the rank, surely? Unless I'm misunderstanding the terminology.
And that's because we have several different concepts to describe the same property, for no reason that I can see. That relationship was immediately obvious to me as soon as I worked out what “rank” was, because I'd done my own investigations into linear algebra before I ever got taught it. (Investigations that I wouldn't've thought to do had I not known about the concepts of linear functions, multi-variate functions and inverses, so I'm not claiming to have independently invented linear alegbra.) There's no way they could be different, because it's about the linear independence (which regions of n- or m-dimensional space are reachable by a linear combination of the… you don't want me to say “coefficients”).
Likewise, I don't think we should be using “row” and “column” to describe linear mappings. “We sometimes write numbers in a grid to represent the function” isn't fundamental.
> And no one calls that m-tuple a coefficient.
But it is a coefficient, if you write the transformation the way I wrote it. Why is that wronger than talking about the “rows” and “columns” of a function? Other than convention, of course.
> None of this is in the book above. Zero.
To be fair, the book does purport to teach statistics; it uses, but does not claim to teach, basic linear algebra. A solid grasp of linear algebra is a prerequisite to understanding the book, so if you understand the book you probably understand linear algebra.
>Maybe I'd find the words easier if mathematicians named stuff sensibly
Column space and row space are completely sensible.
>And that's because we have several different concepts to describe the same property, for no reason that I can see.
You do not see. If you think column space and row space are the same thing, then that's completely wrong. They have the same dimension, which is a theorem, but they are not the same space.
>But it is a coefficient,
So is everything, which is why calling this a coefficient, when there is a better word, is useless.
If you have columns m1, m2, m3, m4, and form the linear combination a1m1 + a2m2 +... Then the ai are also coefficients. And they're much more like what people call coefficients since they're scalars. If you want to call the mi the coefficients, what are you calling the ai? Numbers? Integers? Crawdads?
The mi are vectors, they are column vectors, linear combinations of them form a subspace, and the things multiplied by them to form the subspace are called coefficients.
So yes, you can call them coefficients, but you may as well call them numbers, or pointy-things, or anything else you make up, and no one will be able to talk with you, since you insist on doing things in a manner that makes your work unintellgible.
>Likewise, I don't think we should be using “row” and “column” to describe linear mappings. “We sometimes write numbers in a grid to represent the function” isn't fundamental.
... and you're off the deep end again. I'm glad you invented close but not correct linear algebra, that you missed so many important relations, that you use words in the manner you believe they should be, and on and on.
Of course, your methods clearly must be better than centuries of mathematicians - you should publish a book and clear it up for everyone.
>So why does every textbook
It's baffling to me how hard you push at simply learning. Pick up one of those textbooks I mentioned, and look at every page indexed to rank, and look at how it's used.
That you continue to debate this point is astounding and willful ignorance. I've given you why it's important. I've shown that in textbooks it indexed second only to dimension. I've given at least 5 places it shows up. I've explained how it's the dimension of extremely important items.
It becomes more and more important the deeper you go into math, and that is probably the biggest reason it is so important here. The concept of rank is the tip of an iceberg going through everything above linear algebra: Hilbert spaces, operator theory, exact sequences, homology, cohomology, topology, and on and on and on.
I'm done. You don't care to learn. You want to keep claiming your made up vocabulary and sloppy terms that miss important issues are superior. You're far too stubborn to educate. Go do it yourself.
> You do not see. If you think column space and row space are the same thing, then that's completely wrong.
You were the one who wrote column space = row space…?
> If you want to call the mi the coefficients, what are you calling the ai?
I'm calling them the coefficients because they're “part of the function” and don't change. They're the thing you'd naturally call a coefficient. Coefficient isn't as broad a term as you seem to think it is; In f(x) = x² + x + 3, the coefficients are 3, 1 and 1; not 1, x and x².
> You should publish a book and clear it up for everyone.
As soon as I have any actually original work, I plan to. But linear algebra isn't a specialism of mine, so I doubt I ever will.
And no, I don't think the terminology I've used here is an improvement over the status quo; I'm not a complete imbecile. I just don't get why terminology is considered more important than understanding in mathematics education, and why it's practically impossible to use different words for things even when you do have an improvement.)
> That you continue to debate this point is astounding and willful ignorance. I've given you why it's important. I've shown that in textbooks it indexed second only to dimension. I've given at least 5 places it shows up. I've explained how it's the dimension of extremely important items.
If you think any of those are counterarguments, you were never addressing what I was trying to say.
> You want to keep claiming your made up vocabulary and sloppy terms that miss important issues are superior.
Where did I claim this? I believe I said I was “not wrong” (a weaker label than “correct”), and I identified some problems I have with the existing terminology, but I don't think I ever said my (inconsistent, ad-hoc) terminology was better.
Calling an m-tuple a coefficient and then calling the column span the "coefficient span" does not show understanding of the material.
>I consider it a more fundamental, more useful concept than your "rank".
The dimension of the span is the rank. It's exactly why rank is important. The nullity is the dimension of the kernel, and rank + nullity = n.
Saying rank is not important is like saying dimension is not important. The dimension of a vector space is the first and most important invariant that describes the space. Rank is that dimension for the image of a linear transform. It's absolutely fundamental. It's why when describing some linear algebra thing, one usually starts with something like "Let V be a n-dimensional real vector space" or similar. We rarely (bordering on never) write "Let V be a vector space spanned by the following vectors", and we never write "the coefficient span of the transform".
>The concept of “rank” drops out of the (misnamed) “Gaussian Elimination” algorithm
No, it is a fundamental property of the linear transformation. Gaussian elimination is but one of many ways to compute it. And no matter what choices you use in your elimination, you will always get the same rank. This is super important - that Gaussian elimination gives knowledge of the linear transformation that is independent of choice of basis or of steps performed. Changing of either vector space changes the numbers in the matrix, and different Gaussian elimination steps may have differing intermediate steps, but the rank is the rank is the rank.
It is also true that the row span = the column span = the rank, which is also not immediately obvious. These are theorems proven in basic linear algebra, and fundamental to claiming to understand basic linear algebra.
That rank shows up in Gaussian elimination is not some artifact or unique thing to Gaussian elimination. Since rank shows up everywhere, when it also shows up in Gaussian elimination shows that Gaussian elimination is doing something fundamental - it is but one of many, many ways that rank pops up over and over in linear algebra.
Rank (and nullity) are absolutely fundamental.
And no one calls that m-tuple a coefficient.
None of this is in the book above. Zero.
A final nice point to illustrate, here [1] is the index to Strang's Introduction to Linear Algebra. Rank occurs more than any just about every other entry in the index.
Here's Serge Lang's book [2]. After the word dimension, rank occurs the most in the index.
Book after book shows that after the concept of dimension, rank is probably the most important concept in linear algebra.
I could go on and on. You cannot claim to understand linear algebra without know how pervasive and useful rank is.
[1] https://math.mit.edu/~gs/linearalgebra/linearalgebra5_Index....
[2] http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_a...