Question

Find the basis and dimension of the vectors are (1,-3,1) ( 2,-6,2) and (3,-9,3) .

Answer 1

Consider the augmented matrix

$$\begin{bmatrix} 1 & 2 & 3 & 0 \\ -3 & -6 & -9 & 0 \\ 1 & 2 & 3 & 0 \end{bmatrix}$$

$R_2=R_2+3R_1$

$$\begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 2 & 3 & 0 \end{bmatrix}$$

$R_3=R_3-R_1$

$$\begin{bmatrix} 1 & 2 & 3 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$

Three vectors are collinear.

The basis vector is $(1, -3, 1),$ for example. The dimension of basis is 1.