I like to divide math into 3 disjoint sets: Algebra, Analysis (Calculus), Geometry (Topology). *
Now the most useful one is Analysis. The use of partial differencial equation and Fourier series is one of the big uses in applied math. As well as integration.
It is also true that Geometry is used in applied math. For instance, Differencial Geometry and I am sure certain concepts from Topology are used as well.
But how about Algebra? That is my favorite out of the 3 (or 4) and is sad to see that it is possibly the smallest one by far. I cannot possibly image how Field and Galois theory or Cyclotomic Extensions, Abelian Varieties are used. Can someone give examples?
This is true for all except Linear Algebra. I seen it used heavily by JakeD. And I have seen online lectures by Professor Strang from MIT University. I am convinced it is extremely important. There are many strange concepts about matrices: transposes, orthogonality, decompositions .... I never used them nor have any idea what they are since Linear Algebra never interested me**.
So my question is what is so special about this "strange" linear algebra that is used by applied mathematicians?
*)Sometimes 4. We can think of Combinatorics as the other fourth one. In fact, it was a great honor when I saw MathWonk on PhysicsForums make the same classification.
**)Except for Vector Spaces and Inner Product Spaces. Which is used a lot in pure math.