When I studied linear algebra we (our books, our professors) used to call Fundamental theorem of linear algebra the theorem that says:
Fundamental theorem of linear algebra: A linear transformation is determined by its values at a basis.
However in other sources there are other results from linear algebra that are called this way, or other similar superlative names. For example, in Wikipedia they give this name to the relation between kernels and ranges of the linear transformation and its adjoint/transpose. In some basic books I have seen it be called Big theorem to certain versions of this theorem on Wikipedia.
Personally it is my opinion that the name has been misused in the theorem in Wikipedia. For example, the theorem in Wikipedia is an easy exercise using what I am used to call the Fundamental theorem of linear algebra, but maybe not the other way around. Pretty much everything you can say about a linear transformation either passes or follows after using what I am used to call the Fundamental theorem of linear algebra.
Question(s):
. What usages of the name "Fundamental theorem of linear algebra" are more common (perhaps by country/regions)?
It seems to be the use of this name for the theorem in Wikipedia has its roots (origin?) in the paper by Gilbert Strang. I would imagine then examples of regions in which this name is used would be USA, and perhaps Canada.
. What motivates the naming of the theorem in Wikipedia? In particular, can it replace the role of what I am used to call "Fundamental theorem of linear algebra"? More in particular, can it prove it? (Strictly speaking this last question in point 3 doesn't make sense. Within a theory any theorem is a consequence of any other theorem. But we can reasonably understand what this means).
The most complete answer will be accepted.
I'll try to answer the third question: the naming of the theorem in Wikipedia, as the References and External links sections suggest, totally follows from
But you know, MAA is not a journal for frontier research. In particular, the above article is a (very informal) expository article; it doesn't even contain a clearly stated "theorem"—only some vague discussions are presented.
Now I'll state my personal opinion toward "the fundamental theorem." Fundamental theorems are surely important, but most often they are so easy to prove/so intuitive that after you've learned the subjects fairly well, they become your second nature—you never think about you are actually using some "theorem," and you never cite their names. (Did you ever cite The Fundamental Theorem of Calculus? Or Algebra? Unless you are trying to prove them or doing homework about their rather immediate implications, readers of your presentation might take it as an insult to their intelligence.) Therefore, since you're never going to cite their names, you don't need to discern which is which, unless you are a historian of mathematics.