Question

(1,2,-1,0),(1,3,1,2), (4,2,1,0), (6,1,0,1) examine whether the following vectors are linearly independent or dependent.

Answer 1

The vectors are linearly independent if and only if the matrix with columns of these vectors has maximum rank. Since the matrix is 4X4, it is if and only if the determinant of this matrix is non-zero. Find

$\left| \begin{matrix} 1& 1& 4& 6\\ 2& 3& 2& 1\\ -1& 1& 1& 0\\ 0& 2& 0& 1\\\end{matrix} \right|=2\left| \begin{matrix} 1& 4& 6\\ 2& 2& 1\\ -1& 1& 0\\\end{matrix} \right|+\left| \begin{matrix} 1& 1& 4\\ 2& 3& 2\\ -1& 1& 1\\\end{matrix} \right|=\\=2\left( -1\left( 4-12 \right) -1\left( 1-12 \right) \right) +1\left( 3-2 \right) -1\left( 2+2 \right) +4\left( 2+3 \right) =55\ne 0$

Since the determinant is non-zero, the vectors are linearly independent.