You can determine whether a set of column vectors is dependent by placing them in a matrix and getting the matrix into RREF. If all columns have a leading entry of $1$, then the set of vectors is independent. If a column does not have a leading entry but instead has one or more nonzero entries that are in the same row as a leading entry, then the set of vectors is dependent. For example, it can be shown that the following set of vectors $S$ is dependent:
$S=\left( \begin{bmatrix} 1 \\ 4 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 4 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 7 \\ -4 \\ 4 \end{bmatrix}, \begin{bmatrix} 0 \\ 11 \\ -4 \\ 6 \end{bmatrix} \right)$
This set of vectors can placed in a matrix $A$ and row reduced.
$A = \begin{bmatrix} 1 & 3 & -1 & 0 \\ 4 & 1 & 7 & 11 \\ 0 & 4 & -4 & -4 \\ 2 & 0 & 4 & 6 \end{bmatrix}$
Row reducing shows that the set of vectors is dependent because column $3$ and column $4$ have nonzero entries in the same row as leading entries.
$RREF(A)=\begin{bmatrix} 1 & 0 & 2 & 3 \\ 0 & 1 & -1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$
Furthermore, the entries of these columns correspond to scalars of linear combinations which show that the set of vectors is linearly dependent. For example, using the elements $3$ and $-1$ as scalars, it can be shown that column vector $4$ is a linear combination of column vectors $1$ and $2$.:
$ \begin{bmatrix} 0 \\ 11 \\ -4 \\ 6 \end{bmatrix} = $ $ 3\begin{bmatrix} 1 \\ 4 \\ 0 \\ 2 \end{bmatrix}$ $-\begin{bmatrix} 3 \\ 1 \\ 4 \\ 0 \end{bmatrix}$
Logically, why does the process of row reduction reveal the scalars which prove linear dependence in a set of vectors? I understand that it does work, but not why it should work.
EDIT: I understand that row operations don't change whether a set of column vectors are dependent/independent. My question was as follows: why in RREF do the entries of a column vector correspond to scalars in a linear combination which can prove linear dependence?
I thought of an analogy which would serve as an adequate answer to my question, but I'm not sure if it is accurate or not. You can see how large certain numbers are in terms of other numbers through division. For example, how large is the number $5$ in terms of $4$? The quotient $5/4=1.25$ tells us that $5$ is $1.25$ times as large as $4$. Similarly, through row reduction you can express column vectors in terms of other column vectors. When I row reduce columns $1$ and $2$ so that they have a leading entry of $1$, I express columns $3$ and $4$ in terms of columns $1$ and $2$.
Is this understanding somewhat accurate (it only has to serve as a general intuition)?