I am reading a book on partial differential equations. One of the exercise question in the book is:
Show that the functions $(c_1 + c_2 sin^2x + c_3 cos^2x)$ form a vector space. Find a basis of it. What is its dimension?
I don't know how important this question is for understanding PDE's, but I don't understand the question, nor do I know how to solve it. I know about standard vector spaces on the real numbers. Moreover, I've read about what abstract vector spaces are, but I think my abstract algebra is not developed enough to fully comprehend vector spaces over non-scalar fields.
So my question is: How can a "set of functions" form a vector space? how is that a meaningful statement? And how does this particular set of functions form a vector space?