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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.