Understanding the Wronskian

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:

• 5% physics majors (<-- including me!)
• 10% various other disciplines
• 85% engineers

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:

Linear Independence: A Quick Refresher

Two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are Linearly independent if and only if $$\overrightarrow{a}=c\overrightarrow{b},c\in \mathbb{R}$$ i.e. if the two vectors are multiples of each other.

$$c_1{f}_1(x)+c_2{f}_2(x)+...+c_n{f}_n(x)=0$$