What exactly do we mean when say "linear" combination? I've noticed that the term gets abused alot. For instance, suppose I have
$c_1 x_1 + c_2 x_2 = f(x)$...(1)
Eqtn (1) is such what we say "a linear combination of $x_1$ and $x_2$"
In ODE, sometimes when we want to solve a homogeneous 2nd order ODE like $y'' + y' + y = 0$, we find the characteristic eqtn and solve for the roots and put it into whatever form necessary. But in all casses, the solution takes form of $c_1y_1 + c_2y_2 = y(t)$.
The thing is that $y_1$ and $y_2$ itself doesn't even have linear terms, so does it make sense to say $c_1y_1^2 +c_2y_2^2 = f(t)$ is a "quadratic" combination of y_1 and y_2?

It's a linear combination in the vector space of continuous (or differentiable or whatever) functions. $y_1$ and $y_2$ are vectors (that is, elements of the vector space in question) and $c_1$ and $c_2$ are scalars (elements of the field for the vector space, in this case $\mathbb{R}$). In linear algebra it does not matter what kind of elements vector spaces consists of (so these might be tuples in the case of $\mathbb{R}^n$, or linear operators, or just continuous functions, or something entirely different), but that vector spaces satisfy the axioms which are algebraic laws.
