If possible write the polynomial vector qx 1 x 2x2 as a l

If possible, write the polynomial \"vector\' q(x) = 1 + x + 2x^2 as a linear combination of P_1(x) = 1 + 2x + 3x^2, p_2(x) = 1 + 4x + 6x^2 and p_3(x) = 2 - 3x - 5x^2. Setup the necessary equations By Hand, and then solve By Computer by converting these polynomials from the Vector Space P_2(x) into the Vector Space R^3 so that you can use row matrix notation and matrix operations for these polynomial vectors. Give the final linear combination if it exists.

Solution

Let q(x) be written as,

   q(x) = a ( p1(x) ) + b ( p2(x) ) + c ( p3(x) )

1 + x + 2x² = a ( 1+ 2x +3x² ) + b ( 1 + 4x + 6x²) + c ( 2 - 3x - 5x² )

                  = x² (3a +6b - 5c) + x ( 2a + 4b - 3c) + 1 ( a + b + 2c )

    Therefore     3a +6b - 5c = 2 --------eq.1

                         2a + 4b - 3c = 1 --------eq.2

                        a + b + 2c = 1 ------------eq.3

By solving eq.1 and eq.2, we get

                     3 (a+2b) - 5c = 2

                     2 (a+2b) -3c = 1

we get   c = -1, b = -4,  a = 7

The final linear combination,

1 + x + 2x² = 7 ( 1+ 2x +3x² ) - 4 ( 1 + 4x + 6x²) - 1 ( 2 - 3x - 5x² )