I was wondering if it is possible to determine the number of linearly dependent columns of a matrix by inspection. Consider the following matrix $$H=\begin{pmatrix} 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 \end{pmatrix}.$$ $H$ has three leading columns. From my understanding, this means there are three linearly independent columns (this can also be seen as neither of the first three columns can be expressed as a linear combination of one another).
Can we determine the number of linearly dependent columns of $H$ in a similar way?
Question.
Let $C$ be the binary linear code with generator matrix $$G=\begin{pmatrix} 1 & 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{pmatrix}.$$ What is the minimal distance $d(C)$?
There exists a theorem which states that the minimum distance for linear codes, $d(C)$, is given by $d(c)$=min{$r:G$ has $r$ linearly dependent columns}.