Question

Check whether each of the following subsets of R 3 is linearly independent. i) {(1,2,3),(−1,1,2),(2,1,1)}. ii) {(3,1,2),(−1,−1,−3),(−4,−3,0)

Answer 1

Let us check whether each of the following subsets of $\R^3$ is linearly independent.

i) $\{(1,2,3),(−1,1,2),(2,1,1)\}$

Since $\begin{vmatrix}1 & 2 & 3\\ −1 &1 &2 \\ 2 & 1 & 1\end{vmatrix}=1+8-3-6-2+2=0$, we conclude that the subset $\{(1,2,3),(−1,1,2),(2,1,1)\}$ is linearly dependent.

ii) $\{(3,1,2),(−1,−1,−3),(−4,−3,0)\}$

Taking into account that $\begin{vmatrix}3 &1&2\\−1&−1&−3\\−4&−3&0\end{vmatrix}=0+6+12-8-0-27=-17\ne 0,$ we conclude that the subset $\{(3,1,2),(−1,−1,−3),(−4,−3,0)\}$ is linearly independent.