Question

determine if set of vectors are linearly independent v1={4,1,2}, v2={-3,0,1} and v3={1,2,1}

Answer 1

Let us determine the determinant of the matrix consisting of coordinates of vectors:

$d = \begin{vmatrix} 4 & 1 & 2 \\ -3 & 0 & 1 \\ 1 & 2 & 1 \end{vmatrix} = 4\begin{vmatrix} 0 & 1 \\ 2 & 1 \end{vmatrix} - 1 \begin{vmatrix} -3 & 1 \\ 1 & 1 \end{vmatrix} + 2 \begin{vmatrix} -3 & 0 \\ 1 & 2 \end{vmatrix}$ ,

$d = 4\cdot(0\cdot1 -2\cdot1 ) -1\cdot(-3\cdot1 -1\cdot 1)+2\cdot(-3\cdot2 - 1\cdot0), \\ d = -16.$

The determinant is not 0, so the system of vectors is linearly independent.