Linearly Dependent Columns of a Matrix

I want to make sure I'm understanding what the matrix of a linear transformation says about its null space and range.

It's clear for me with rows (as this is how Gaussian elimination seems to be applied in most examples)

For an $$m \times n$$ matrix: After applying Gaussian elimination to the rows turning it into RRE form, if any row(s) can be row reduced to a row of zeros, then the number non-zero rows is the row rank, (i.e. dimension of the range).

$$n$$, the column count of the matrix, minus this number, is the dimension of the null space.

How does this work for columns?

For example, the matrix: $$\begin{bmatrix} 1 & 2 & 0 & 1 & 0\\ 2 & 4 & 1 & 0 & 0\\ 3 & 6 & 0 & 0 & 1 \\ 4 & 8 & 0 & 0 & 0 \\ \end{bmatrix}$$

Where $$c_i$$ denotes the column, it's clear that $$-2c_1 + c_2 + 0c_3 + 0 c_4 + 0 c_5=0$$. But on first sight, it's not that arithmetically easy to do row reduction. So I would prefer to use any conclusion by looking at columns. In this case, what can we say about this matrix?