\begin{bmatrix} 3 & 4 & -1\\ -2 & 3 & 1\\ -9 & 5 & 4 \end{bmatrix}
I tried to solve the matrix above using row reduction, I did the following steps.
As soon as I found the last matrix, I realized that there are two parallel planes and that one row would consist of all 0's. The question that I was trying to answer was are these column vectors linearly independent. The mark-scheme says this is indeed the case, however I am quite confused due to the parallel planes I found. Because it geometrically means one plane is following the direction of the other, meaning it is dependent, does this dependency suppose to indicate that the matrix's rows are independent.
One perspective I can agree with the mark-scheme is that one of the columns have no pivots. That is when the matrix is in the following form. \begin{bmatrix} -2 & -14 & 0\\ 0& 0& 0\\ 0& 17& 1 \end{bmatrix} So -2 and 17 are pivots in column 1 and column 2, but 3rd column has no pivots, therefore that column is suppose to be independent.
My ideas are contradicting each other since they are not very clear in my mind, I would be very happy if someone can clarify things to me.