Question

Show that the vectors (3, 0, -3), (-1, 1, 2), (2, 1, 1) and (4, 2, -2) are linearly dependent in R3.

Answer 1

Taking into account that $(3, 0, -3)+ (-1, 1, 2)=(2,1,-1)$ and $(4,2,-2)=2(2,1,-1)=2[(3, 0, -3)+ (-1, 1, 2)]=2(3, 0, -3)+ 2(-1, 1, 2)=2(3, 0, -3)+ 2(-1, 1, 2)+0(2,1,1)$

we conclude that the vector $(4,2,-2)=2(3, 0, -3)+ 2(-1, 1, 2)+0(2,1,1)$ is a linear combimations of vectors $(3, 0, -3), (-1, 1, 2)$ and $(2,1,1)$, and therefore  the vectors $(3, 0, -3), (-1, 1, 2), (2, 1, 1)$ and $(4, 2, -2)$ are linearly dependent in $R^3$ .