>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.