
Linear combinations
Often, a text on differential equations will mention "linear combination" of functions. What exactly does that phrase mean? I've been fuzzy on it all semester. The day before the final is a good time to ask, ehe?
For example, there's a question that asks,
Prove that any twodimensional vector can be written as a linear combination of i + j and i  j.
Just as an example, but by all means please use examples that pop into your own mind.
Thanks.

Say you have a vector field $\displaystyle V$ with scalars in some field $\displaystyle F$.
A vector $\displaystyle x\in V$ can be written as a linear combination of some given vectors $\displaystyle v_1,...,v_r\in V$ if there exists scalars $\displaystyle a_1,..,a_r\in F$ such that:
$\displaystyle x=a_1v_1+...+a_rv_r$.
For your example, it suffices to check that $\displaystyle v_1=(1,1)$ and $\displaystyle v_2=(1,1)$ form a basis for $\displaystyle \mathbb{R}^2$ (i guess this is your vector space). Since $\displaystyle \text{det}(v_1\: v_2)\neq 0$, $\displaystyle \{v_1,v_2\}$ form a basis for $\displaystyle \mathbb{R}^2$. So every vector can be written as a linear combination of these vectors.

A "linear combination" of things that we can add or multiply by numbers is an expression in which we have, in fact, multiplied by numbers and added!
The "linear" part comes in when we have things, like functions, that can also be multiplied. Things like $\displaystyle f^2$ and $\displaystyle fg$ are not "linear".