# Chapter 4 - Vector Spaces - 4.4 Spanning Sets and Linear Independence - 4.4 Exercises - Page 178: 41

$S$ is linearly independent set of vectors.

#### Work Step by Step

Consider the combination $$a(2-x)+b(2x-x^2)+c(6-5x+x^2)=0, \quad a,b,c\in R.$$ Which yields the following system of equations \begin{align*} 2a-6c&=0\\ -a+2b-5c&=0\\ -b+c&=0. \end{align*} The determinant of the coefficient matrix is given by $$\left| \begin {array}{cccc} 2&0&-6\\ -1&2&-5\\0&-1&1\end {array} \right|=-12$$ 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.

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