Question

Determine whether the vectors are linearly dependent or independent (1,2,1),(-1,0,1) and (2,-1,4)

Answer 1

$\text{Let $c_1$, $c_2$ and $c_3$ be scalars such that} \\c_1(1,2,1) + c_2(-1,0,1) +c_3(2,-1,4) = 0 \\\text{which can be represented in matrix form as represented as} \\\begin{vmatrix} 1 & -1 & 2\\ 2 & 0 & -1\\ 1 & 1 &4 \end{vmatrix} \begin{vmatrix} c_1\\c_2\\c_3 \end{vmatrix} =\begin{vmatrix} 0\\0\\0 \end{vmatrix} \\\text{Next, we calculate the determinant of the co-efficient matrix} \\1\begin{vmatrix} 0&-1\\ 1&4 \end{vmatrix} +1\begin{vmatrix} 2&-1\\ 1&4 \end{vmatrix} +2\begin{vmatrix} 2&0\\ 1&1 \end{vmatrix} \\=1+9+4=14 \\\text{Since the determinant is not equal to zero, then the solution exists and} \\\text{$c_1=c_2=c_3=0$} \\\text{Therefore the vectors are linearly independent}$