Question

The vectors V1 - (1, 1, 2, 4), V2 - (2, - I , -5 , 2), V3 = (1, -1 , -4 , 0) and

V4 = (2, 1, 1 ,6 ) are linearly independent. Is it true ? Justify your answer

Answer 1

Let $A =\begin{pmatrix} 1 & 1 & 2 & 4\\ 2 & -1 & -5 & 2\\ 1 & -1 & -4 & 0\\ 2 & 1 & 1 & 6 \end{pmatrix}$

$\begin{pmatrix} 1 & 1 & 2 & 4\\ 2 & -1 & -5 & 2\\ 1 & -1 & -4 & 0\\ 2 & 1 & 1 & 6 \end{pmatrix}$ ~ $[V_2 -2V_1, V_3 - V_1, V_4 - 2V_1]$ ~$\begin{pmatrix} 1 & 1 & 2 & 4\\ 0 & -3 & -9 & -6\\ 0 & -2 & -6 & -4 \\ 0 & 0 & -1 & -2 \end{pmatrix}$ ~

~$[V_3 - \cfrac{2}{3}V_2]$ ~ $\begin{pmatrix} 1 & 1 & 2 & 4\\ 0 & -3 & -9 & -6\\ 0 & 0&0&0\\ 0&0&-1&-2 \end{pmatrix}$

So we can see $rank \ A = 3 < 4$

Hence the vectors are linearly dependent