Let gamma = {t^2 - t + 1, t + 1, t^2 + 1} and beta = {t^2 + t + 4, 4t^2 - 3t + 2, 2t^2 + 3} be ordered bases for P_2(R).Find the change of coordinate matrix Q

Let ${\displaystyle t^2+t+4={\gamma }_1(t^2-t+1)+{\gamma }_2(t=1)+{\gamma }_3(t^2+1)}$
${\displaystyle \Rightarrow {\gamma }_1+{\gamma }_3=1,-{\gamma }_1+{\gamma }_2=1,{\gamma }_1+{\gamma }_2+{\gamma }_3=4}$
${\displaystyle \Rightarrow 1+{\gamma }_2=4}$
${\displaystyle \Rightarrow {\gamma }_2=3}$
${\displaystyle -{\gamma }_1+{\gamma }_2=1\Rightarrow -{\gamma }_1+3=1\Rightarrow {\gamma }_1=2}$
${\displaystyle {\gamma }_1+{\gamma }_3=1\Rightarrow 2+{\gamma }_3=1\Rightarrow {\gamma }_3=-1}$ Let ${\displaystyle 4t^2-3r+2={\gamma }_1(t^2-t+1)+{\gamma }_2(t+1)+{\gamma }_3(t^2+1)}$
${\displaystyle \Rightarrow {\gamma }_1+{\gamma }_3=4,-{\gamma }_1+{\gamma }_2=-3,{\gamma }_1+{\gamma }_2+{\gamma }_3=2}$
${\displaystyle \Rightarrow 4+{\gamma }_2=2}$
${\displaystyle \Rightarrow {\gamma }_2=-2}$
${\displaystyle -{\gamma }_1+{\gamma }_2=-3\Rightarrow -{\gamma }_1-2=-3\Rightarrow {\gamma }_1=1}$
${\displaystyle {\gamma }_1+{\gamma }_3=4\Rightarrow 1+{\gamma }_3=4\Rightarrow {\gamma }_3=3}$ Let ${\displaystyle 2t^2+3={\gamma }_1(t^2-t+1)+{\gamma }_2(t+1)+{\gamma }_3(t^2+1)}$
${\displaystyle \Rightarrow {\gamma }_1+{\gamma }_3=2,-{\gamma }_1+{\gamma }_2=0,{\gamma }_1+{\gamma }_2+{\gamma }_3=3}$
${\displaystyle \Rightarrow 2+{\gamma }_2=3}$
${\displaystyle \Rightarrow {\gamma }_2=1}$
${\displaystyle -{\gamma }_1+{\gamma }_2=0\Rightarrow -{\gamma }_1+1=0Ri>\leftrightarrow ow{\gamma }_1=1}$
${\displaystyle {\gamma }_1+{\gamma }_3=2\Rightarrow 1+{\gamma }_3=2\Rightarrow {\gamma }_3=1}$

Not exactly what you’re looking for?

This is helpful 84