Why is it useful to write a vector as a finite (or infinite) linear combination of basis vectors? I'm working on a project in an applied mathematics course and a professor asked me a basic question:
What is useful about writing any element of a vector space in terms of a possibly infinite linear combination of basis vectors?
I didn't really have an answer. So I'm wondering what's so useful about it? Why can't we just come up with an alternate expression for any arbitrary element and use that?

Sometimes there is a linear transformation that you want to understand. If you can find a basis such that the linear transformation has a simple effect on the basis vectors (like it simply scales them, for example) then this helps a lot to understand what the linear transformation does to an arbitrary vector (which can be written as a linear combination of the special basis vectors).
