This post assumes familiarity with calculus and linear algebra.
I recently took MTH2201: Differential Equations and Linear Algebra at Florida Tech. The academic demographic of the class was as follows:
Needless to say, the course was focused more on problem-solving techniques than actual math. I was not fully aware of this when the course began, then this happened:
Professor: So, you're good with linear independence, right? Guess what, functions can also be linearly independent.
Class: ...
Professor: Yeah, you use this thing called the Wronskian to test for linear independence. Here it is: \[begin{bmatrix}a & b\c & d\end{bmatrix}\]
Two vectors \(\vec{a}\) and \(\vec{b}\) are Linearly independent if and only if \[ \vec{a} = c\vec{b}, c\in \mathbb{R} \] i.e. if the two vectors are multiples of each other.
\[ c_1f_1(x) + c_2f_2(x) + ... + c_nf_n(x) = 0 \]