Answer to Question #222315 in Linear Algebra for dante

Question

Express the polynomial 1+3x+4x² as a linear combination of the polynomials

1+2x+3x², -1+x+x² and 2+x+x²

Answer 1

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

$\begin{cases} a-b+2c = 1 \\ 2a+b+c = 3 \\ 3a+b + c = 4 \end{cases}$

subtraction the second equation from the third gives: a =1

addition the first equation and the secong gives 3a + 3c = 4, and from this c = 1/3

from the fist equation: b = a +2c - 1= 2/3

so 1+3x+4x²= (1+2x+3x²) + 2/3 (-1+x+x²) + 1/3 (2+x+x²)