Consider the following matrix: \begin{equation} \boldsymbol{A} = \begin{bmatrix} 1 & 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \end{bmatrix}, \end{equation} where the entries are from GF(2). Checking with MATLAB, the rank of matrix $\boldsymbol{A}$ turns out to be equal to 3. However, it is clear from simple observation that the addition of first and third columns is equal to the last column. Hence the number of linearly independent columns in $\boldsymbol{A}$ is equal to 2, so the rank should be equal to 2.
What is the reason for this discrepancy?
You have found one column that can be expressed as a linear combination of the others. Hence, the $7$ columns are not linearly independent and the conclusion you can make, is that the rank is at most $6$. However, since column rank and row rank always agree, it can be at most $3$ anyway, so you didn't gain anything.
Now note that the first three columns are in fact linearly independent, so the rank is equal to $3$.