A lot of the stuff we do to solve differential equations are taken word for word from linear algebra. The concept of linear independence, determinant of the Wronskian used to determine independence, adding a particular solution to the kernel to get the general solution – all this stuff that I'm used to applying on vectors seems to work with functions. Why?
[Math] Why do the concepts of linear algebra apply to differential equations
linear algebraordinary differential equations
Best Answer
Because differentiation is a linear operator, and hence can be represented by a matrix acting on a function.
Specific example for OP
$f''+3f'+2f=0 \implies f=ae^{x}+be^{2x}\equiv (a,b)\cdot(e^{x},e^{2x})\equiv \mathbf{a\cdot f}$
Therefore, the solutions to the differential equations can be represented as vectors in a function space with the basis $(e^{x},e^{2x})$
Second Addition
As you can see by all the answers on this page, solutions to linear ODEs can be represented by vectors in the space of functions. If you want more details, you can check out this nice paper.