Question

Find wheather the vectors (2 0 0 0) (2 0 0 8) (0 0 0 0) and (2 0 9) are linearly dependent or not

Answer 1

matrix with these vectors as columns:

$A=\begin{pmatrix} 2 & 2&0&2 \\ 0 & 0&0&0 \\ 0 & 0&0&9 \\ 0 & 8&0&x \\ \end{pmatrix}$

$|A|=0$ (in the second row all entries are 0).

So, the vectors are linearly dependent.