I know this question sounds completely idiotic, but I swear its title is exactly what I want to know.
Linear algebra focuses on inherently linear phenomena, characterized by several variables that show up with a power of at most one. Mathematically speaking, vector spaces were created to abstract this concept in a very elegant way, since their elements can be added together and multiplied by a scalar, but they can't be multiplied together.
But just like linear approximation can be useful, sometimes quadratic approximation can be just as useful and maybe even better. So does there exist an algebraic structure that takes vector spaces one step further and allows you to study combinations like $\alpha\mathbf{a}^2 + \beta\mathbf{b}$, but not $\alpha\mathbf{a}^3 + \beta\mathbf{b}$, just like vector spaces allow you to investigate theorems and truths about expressions of the form $\alpha\mathbf{a} + \beta\mathbf{b}$, but not $\alpha\mathbf{a}^2 + \beta\mathbf{b}$?