Let A and C be the following ordered bases of P3(t): A=(1,1+t,1+t+t2,1+t+t2+t3) C=(1,−t,t2−t3) Find tha...

Solution: ${\displaystyle A=(1,1+t,1+t+t^2,1+t+t^2+t^3)\text{and}C=(1,-t,t^2,-t^3)}$ To find the chance of coordinate matrix ${\displaystyle I_{CA}}$:
${\displaystyle 1=a_1\left(1\right)+a_2(1+t)+a_3(1+t+t^2)+a_4(1+t+T^2+t^3)}$
${\displaystyle \Rightarrow a_1+a_2+a_3+a_4=1,a_2+a_3+a_4=0,a_3+a_4=0\text{and}{a}_4=0}$
${\displaystyle \Rightarrow a_1=1}$
${\displaystyle -t=b_1\left(1\right)+b_2(1+t)+b_3(1+t+t_2)+b_4(1+t+t^2+t^3)}$
${\displaystyle Ri>\leftrightarrow ow{b}_1+b_2+b_3+b_4=0,b_2+b_3+b_4=-1,b_3+b_4=0\text{and}{b}_4=0}$
${\displaystyle \Rightarrow b_1=1\text{and}{b}_2=-1}$
${\displaystyle t^2=c_1\left(1\right)+c_2(1+t)+c_3(1+t+t^2)+c_4(1+t+t^2+t^3)}$
${\displaystyle \Rightarrow c_1+c_2+c_3+c_4=0,c_2+c_3+c_4=0,c_3+c_4=1\text{and}{c}_4=0}$
${\displaystyle \Rightarrow c_2=-1,c_3=1}$