TITLE: Is linear algebra laying the foundation for something important?
QUESTION [73 upvotes]: I'm majoring in mathematics and currently enrolled in Linear Algebra. It's very different, but I like it (I think). My question is this: What doors does this course open? (I saw a post about Linear Algebra being the foundation for Applied Mathematics -- but I like doing math for the sake of math, not so much the applications.) Is this a stand-alone class, or will the new things I'm learning come into play later on?

REPLY [4 votes]: Linear algebra is often (as seems to be the case here by your instructor) taught in a disjoint way from calculus and then eventually the threads are gathered together and merged, but usually hastily with what seems to be tangential asides, leaving most students to miss the point.
The truth of the matter is it turns out that to really understand the functions you study in calculus, you need to understand spaces of functions.  You already get a hint of that when you talk about differentiation and integration as being "inverse" operators.  The natural structure to put on these spaces is that of a vector space (it's not coincidence that high-schoolers learn about the "algebra of functions").  
Linear algebra can be considered the study of vector spaces and natural operations upon them.  Using the tools of linear algebra, such as eigenvalues, you can gain an understanding of linear operators on these function spaces.  This will actually help you understand differential equations, which themselves are the chief motivation for studying calculus in the first place (historically Newton et al studied calculus to solve differential equations in physics).