## Differential Equations and Linear Algebra (4th Edition)

If $ap_1(x)+bp_2(x)+cp_3(x)=a(1-3x^2)+b(2x+x^2)+5c=(5c+a)+(2b)x+(b-3a)x^2=0$, then $2b=0$, thus $b=0$, thus since $b-3a=0$, $a=0$, thus since $5c+a=0$, $c=0$, thus $a=b=0$, thus they are linearly independent.