Linear algebra please show me the steps 14Check if the follo

Linear algebra, please show me the steps

14-Check if the following functions f x +2x-1. g(X x+3 and hx) x3 3x+ 2 is a basis for the vector space P

Solution

Let Af(x)+Bg(x)+Ch(x)=0

then A(x² +2x-1)+B(x+3)+C(x²+3x+2) =0.....................................(1)

Comparing coefficients ,

                A +C=0.......................                                                     (2)

           2A+B+3C=0.............................................................................(3)

         -A+3B+2C=0.............................................................................(4)

From (2) and (3)

                -A+B=0.................................................................................(5)

From (3) and (4)

               -3A+3B=0.............................................................................(6)

                or

                  -A+B=0.............................................................................(7)

So A =1,B=1 and C =-1 is a non-trivial (not all zero) solution for (1)

So f(x), g(x) and h(x) are NOT linearly independent.

So they cannot form a basis for P₂