Question

Show whether the first three vectors are linearly independent

V1= 1, 1, -2, -2

V2= 2, -3, 0, 2

V3= -2, 0, 2, 2

V4= 3, -3 -2, 2

Answer 1

$\begin{cases} x_1+2x_2-2x_3=0, (1)\\ x_1-3x_2=0, (2)\\ -2x_1+2x_3=0, (3)\\ -2x_1+2x_2+2x_3=0. (4) \end{cases}$

From (2): $x_2=x_1/3$, from (3): $x_3=x_1,$ to (1):

$x_1+2x_1/3-x_1=0,$

$x_1=0,$

(2): $x_2=0,$

(3): $x_3=0.$

Only $x_1=x_2=x_3=0$ is the solution.

So, by the defenition, vectors are linearly independent.