Does the determinant of a matrix made up of column vectors being non-zero imply that the vectors are independent? Let's say we have 3 vectors and we make up a matrix where we depict the vectors as the columns of the matrix. If we calculate the determinant of the matrix and we get a non-zero number, does that mean that the vectors are linearly independent? 

There is a much simpler interpretation of the determinant of a $3\times 3$ matrix. Say we have vectors $x,y,z\in\mathbb{R}^3$. Then the determinant of the matrix $M=(x,y,z)$ with coloumn as the given vectors is called the Scalar Triple product. Let's say the determinant is $D$, then $|D|$ can be interpreted as the volume of the paralellopiped whose adjascent sides are given by $x,y,z$.
Now, suppose $|D|=0$, then we can deduce that $x,y,z$ lie on the same plane, and therefore they are linearly dependent. Conversely, if $x,y,z$ are linearly dependent, they lie on the same plane and hence we can see that $D=0$.
To the best of my knowledge, this interpretation is valid only when the vectors are $3$ dimensional.
