# Fundamental theorem of linear algebra

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.

• – lhf Jan 14 '14 at 1:03
• The Fundamental Theorem of Fundamental Theorems $\$ This generation's fundamental theorem is the next generation's triviality. – Bill Dubuque Jan 14 '14 at 1:25
• @BillDubuque The/A theorem is fundamental not for not being trivial. This is a question on didactic of mathematics and of history. It may have also a component of ethnomathematics. I teach students from many different countries. Knowing what schools use this name for one theorem or another is important. If you don't have anything important to say spare me your usual made-up, dismissive, "theorems/jokes". – user119256 Jan 14 '14 at 1:54
• @Karene It is not necessarily a joke. I think the original author (whom I've forgotten) intended it to reflect the striking continual progress of mathematics. That we now consider trivial many results that were once considered deep, highlights just how spectacularly successful mathematics has been, and continues to be. – Bill Dubuque Jan 14 '14 at 2:10
• @Karene I confess surprise that any author would designate that theorem (a linear map is determined on a basis) to be the Fundamental Theorem of Linear Algebra. So much so that it sparked me to peruse the top 100 Google Web and Google Books results to see who might do so. But I did not see that usage anywhere (most designated the Rank-Nullity theorem or Strang's variant, with a few outliers, such as Fredholm's Alternative). Can you please provide some references to that usage. Without such it will be difficult to research the history of that usage of the term. – Bill Dubuque Jan 14 '14 at 4:37

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

Strang, Gilbert (1993), "The fundamental theorem of linear algebra", American Mathematical Monthly 100 (9): 848–855. doi:10.2307/2324660.

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.

• The fundamental theorem of algebra is often cited as such. The fundamental theorem of linear algebra I have often cited it as such. Notice you cite a theorem when highlighting the step is important in the proof. Citing a theorem by name is not so much about the theorem being cite but about its relation to the larger proof you are making. But given the case in which citing the theorem is required, the right name is better to use. And yes, the question is about history. Your answer gives the half of the history of the theorem in Wikipedia. I would like to see also about the other. Thanks. – user119256 Jan 14 '14 at 2:01
• @Karene As I have pointed out, you may cite the names of trivial theorems in homework, but I highly doubt you would cite those in research papers. In the case of the fundamental theorem of algebra, it is most often used in trivial ways (completely factor $f \in \mathbb{C}[x]$ etc.), so even if it is an important step, I won't cite its name if the audience has enough expertise—it is the second nature of everyone. In the case of "the fundamental theorem of linear algebra," since the name is controversial, it seems to me that not citing the name is actually better (to avoid confusion). – 4ae1e1 Jan 14 '14 at 3:19
• Uh, forgot to clarify, the fundamental theorem of algebra is by no means trivial, but lack of proof doesn't prevent people from assuming it, as it falls into the category of "so intuitive." I started to assume it at the age of twelve I guess, but not until I got into college did I learn several proofs of it—Liouville's theorem proof, Rouché's theorem proof, Galois theory proof, etc. None is trivial. But still, the result is far more intuitive than the proofs. By the way, you are more than welcome to explore the history, but as suggested above, please don't cite a controversial name. – 4ae1e1 Jan 14 '14 at 3:27
• It is not in homework where names of theorems are important. In a homework students are in a close enough communication (1-1, student-instructor) that niche language like "Theorem 5.11.4 from Friday's lecture" might be understood. When writing a book, preparing a course, writing a paper is when it is more important. Don't forget there are also history papers, pedagogy papers, paper's like Strang's. And I am sorry, but specially because the name is not widely used it becomes entirely my call if I use it or not, even more if I think that Strang's choice is not convenient. – user119256 Jan 14 '14 at 4:23

If there is a general consensus/tradition about what should be called Fundamental Theorems in Arithmetic, Algebra and Calculus, the situation appears less clear cut in Linear Algebra.

Some textbooks do indeed label "Fundamental Theorem" the fact that a linear transformation is completely determined by its values on a basis, but some don't. This past semester I taught a first year course in Linear Algebra and warned the students about the fact that the terminology is not universal, but somehow did not take a position.

My personal impression is that although this fact is used all the time to prove results about linear transformations (it is indeed the fact behind the possibility to code a linear transformation into a matrix, as we all know), many authors are reluctant to label it "Fundamental Theorem" because in itself is not such a deep result and its proof appears to be just a straightforward exercise in applying the definitions. A situation different from the "Fundamentals" of Arithmetic and Calculus, not to mention Algebra, the latter being in fact a not easy result.

Besides, there's another fundamental fact about vector spaces that--in my opinion--may be a likely "Fundamental Theorem" given the ubiquitness of its application, namely the fact that every set of linearly independent vectors can be extended to a basis.

• Could you cite some of those books and maybe in what country(ies) they are used? – user119256 Jan 14 '14 at 12:58
• I second Karene's request for references. I find it quite surprising that such a result would be deemed the fundamental theorem of linear algebra. As I mentioned in a comment above, I perused the top 100 search results in Google Web and Google Books but I could not find any uses of that nomenclature. Perhaps this nomenclature is used only in non-English languages? – Bill Dubuque Jan 14 '14 at 14:06