# Min number dependent columns in a matrix

by gonzo
Tags: columns, dependent, matrix, number
 Emeritus Sci Advisor PF Gold P: 16,092 Min number dependent columns in a matrix Just to demonstrate more clearly why rank probably won't be too useful... consider the following matrices are all rank 3: $$\left(\begin{array}{cccc} 1 & -1 & 0 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & -1 \\ -1 & 0 & 0 & 1 \end{array}\right)$$ $$\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{array}\right)$$ $$\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right)$$ but the minimum number of rows you need to form a dependent set is 4, 2, and 1 respectively. (4, 1, and 2 if you want a dependent set of columns)