Question

QUESTION 1 Find the coordinate vector [p(x)]ℬ of p(x) = 5 − x + 3x 2 with respect to the basis ℬ = {u1, u2, u3 } of P2 where u1 = 1 − x + 3x 2 , u2 = 2 and u3 = 3 + x 2

Answer 1

$In\,\,basis\,\,\left\{ 1,t,t^2 \right\} \\p=\left[ \begin{array}{c} 5\\ -1\\ 3\\\end{array} \right] \\Transition\,\,matrix\\T=\left[ \begin{matrix} 1& 2& 3\\ -1& 0& 0\\ 3& 0& 1\\\end{matrix} \right] \\The\,\,coordinates\\p'=T^{-1}p=\left[ \begin{matrix} 1& 2& 3\\ -1& 0& 0\\ 3& 0& 1\\\end{matrix} \right] ^{-1}\left[ \begin{array}{c} 5\\ -1\\ 3\\\end{array} \right] =\left[ \begin{matrix} 0& -1& 0\\ 0.5& -4& -1.5\\ 0& 3& 1\\\end{matrix} \right] \left[ \begin{array}{c} 5\\ -1\\ 3\\\end{array} \right] =\left[ \begin{array}{c} 1\\ 2\\ 0\\\end{array} \right] \\$

I find the inverse matrix with Gauss method: