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.
. 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.