Finding a non-zero vector in Col A Definiton: The column space of an $m \times  n$ matrix $A$, written as $\operatorname{Col} A$, is the set of all linear combinations of the columns of $A$. If $A = [ a_1 \ldots a_n ]$, then $\operatorname{Col} A = \operatorname{Span}\{a_1,\ldots,a_n\}$.

$$A= \begin{bmatrix} 2 & 4 & -2 & 1 \\ -2 & -5 & 7 & 3 \\ 3 & 7 & -8 & 6
  \end{bmatrix}$$
Find a nonzero vector in $\operatorname{Col} A$. 
Solution: It is easy to find a vector in $\operatorname{Col} A$. Any column of $A$
  will do.

I'm confused on why any column of $A$ works because $A$ in rref form is:
$$\begin{bmatrix} 1 & 0 & 9 & 0 \\ 0 & 1 & -5 & 0 \\ 0 & 0 & 0 & 1
  \end{bmatrix}$$
So wouldn't only columns 1, 2, and 4 work because column 3 is linearly dependent, and thus not a part of the span?

As you stated the column space is the set of all linear combinations of the columns of $A$. So if the matrix is not the zero matrix, you will be able to find some non zero vector in $Col A$. Colum 3 is non zero and in fact your column space is the set of all vectors in 3D because $x\times col1 + y \times col2 + z\times col3$ gives any vector in $\mathbb{R}^3$.
