# True or false: The non-pivot columns of a matrix are always linearly dependent.

True or false: The non-pivot columns of a matrix are always linearly dependent.

This is false, I just don't really understand why. Thanks for any help!!

• Non-pivot columns need not to be linearly dependent as a subset of the matrix columns but non-pivot columns are linear combinations of the pivot columns. – Algebraic Pavel May 13 '15 at 10:05

What can you say about the following matrix? $$\begin{pmatrix}1&0&1&2\\0&1&1&0\end{pmatrix}$$
• @mataxu: Yes, $$c_1 \begin{pmatrix} 1 \\ 1 \end{pmatrix} + c_2 \begin{pmatrix} 2 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$$ has only trivial solution. – Ángel Mario Gallegos May 13 '15 at 5:52
Hint In the matrix $$\begin{pmatrix}1 & 1\end{pmatrix}$$ the second column is the only nonpivot column and it is nonzero.