"Is it possible to find a 3x3 matrix such that the dimension of the space generated by the line-vector is different from the generated by the column-vector?"
I think that there is a more clever way to solve this problem, but my approach was to noticed that, if it exist, the dimensions are 2 for line/columns and 1 for columns/line. Suppose the dimension of the vector line is one, so that we can reduce the matrix to \begin{bmatrix} \mu a & \mu b & \mu c\\ a & b & c\\ 0& 0 & 0 \end{bmatrix}
But we can notice that the columns vector is also LD, so that it is also 1D. The contrary is also true.
Even if this is right, i want to know if there is another way to prove/disprove it, and for considering cases with matrix whose dimensions > 3
