## Elementary Linear Algebra 7th Edition

$S$ is linearly independent set of vectors.
Consider the combination $$a(x^2+3x+1)+b(2x^2+x-1)+c(4x)=0, \quad a,b,c\in R.$$ Which yields the following system of equations \begin{align*} a+2b&=0\\ 3a+b+4c&=0\\ a-b&=0. \end{align*} The determinant of the coefficient matrix is given by $$\left| \begin {array}{cccc} 1&2&0\\ 3&1&4\\1&-1&0\end {array} \right|=12$$ Since determinant is non zero, hence there exist a unique solution for the above system; that is, the trivial solution, $$a=0,\quad b=0, \quad c=0.$$ Then, $S$ is linearly independent set of vectors.