Abstract Algebra/Linear Algebra

In a nutshell, linear algebra is the study of functions that have the property of linearity:

$f(\bar x + \bar y) = f(\bar x) + f(\bar y)$

$f( \alpha \bar x) = \alpha f( \bar x)$

This condition essentially excludes 'badly behaved' functions, for instance $f(x) = \frac{1}{x}$ has a singularity at zero and so is not defined there. Linear algebra is very well understood and therefore any problem that can be formulated in those terms will be much easier to solve, in fact much of modern physics is just that.

Taylor's theorem states that (for a certain class of functions) $f(a + h) = f(a) + h f'(a) + \frac{h^2}{2!} f''(a) + ... + \frac{h^k}{k!} f^{(k)} (a) + ...$ and so for h sufficiently small we get a very good linear approximation, since small h implies that f(a + h) = f(a) + hf'(a) + X where X is small enough to be 'ignored', and hf'(a) will be very close to f(h)