# Linear Algebra — Why is this theorem significant?

My book, professor, and friends make this theorem look very significant:

If A is an m by n matrix, then the following are either all true or all false:

• Each vector b in R^m is a linear combination of the columns of A.

• For each b in R^m, the equation Ax = b has a solution.

• The columns of A span R^m.

• A has a pivot position in every row.

My question is: isn't that a fairly obvious tautology? I mean, the definition of matrix multiplication simply expands the second equation into a linear combination, so why do people get so excited about this?

Sometimes I get the feeling that, in linear algebra, we're just finding fifty ways to state the same thing in different words, and getting excited even though they all stem from the same definition. :\

(Apologies in advance about the formatting, I'm not sure how the math formatting works here.)

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The equivalence of those two statements establishes a connection between something you want (knowing when you can solve certain equations) and a certain property those equations may have. It is quite not earth shattering---and I suspect the other statements in that theorem were more interesting. In any case, you should not subestimate the value of being able to view the same truth in many, many different forms: I would venture the claim that most of the power and usefulness of linear algebra stems from the fact that it is found all over the place, in many different forms and shapes. – Mariano Suárez-Alvarez Jan 31 '11 at 7:43
Well, you are ignoring one little fact: you arrived quite late to the party! Those definitions are the result of a very long, arduous process of trying to capture what is important to prove certain things that, in specific situations, were proved originally by hand and concretely. That linear algebra has reached a point where basic facts seem to be obvious is not a hint of its underlying triviality but a testament of the amazing work of giants in picking a good formalization! – Mariano Suárez-Alvarez Jan 31 '11 at 7:59
Oh well... Faith will come. – Mariano Suárez-Alvarez Jan 31 '11 at 8:24
@Mehrdad: Yes, this is a fairly trivial theorem (at least the first three items), since the proof just consists of "unraveling the definitions". I wish all students understood this as easily as you do. Good for you, now you can move on to more interesting things! :-) – Hans Lundmark Jan 31 '11 at 8:43
One last comment: the hallmark of a well-developed theory in mathematics is often that the definitions are sufficiently refined so that what ought to be completely obvious is in fact completely obvious. In view of that you may want to take the apparent triviality of that theorem to be the a posteriori justification for the precise phrasing of the definition. – Willie Wong Jan 31 '11 at 11:36