Answer
See answer below
Work Step by Step
We are given:
$S=\{1-3x^2,2x+5x^2,1-x+3x^2\}$
We obtain:
$C_1(1-3x^2)+C_2(2x+5x^2)+C_3(1-x+3x^2)=0$
$C_1-3x^2C_1+2xC_2+5x^2C_2+C_3-xC_3+3x^2C_3=0$
$(C_1+C_3)+x(2C_2-C_3)+x^2(-3C_1+5C_2+3C_3)=0$
We have:
$C_1+C_3=0$
$2C_2-C_3=0$
$-3C_1+5C_2+3C_3=0$
$A=\begin{bmatrix}
1 & 0 & 1 | 0\\
0 & 2 & -1 | 0 \\
-3 & 5 & 3 | 0
\end{bmatrix}$
$\det A=17\ne 0$
Since $\det A \ne 0 $, set $S$ of vector is linearly dependent and therefore, set $S$ is a basis of $P_2(R)$
