a) T((P(x))=P(x−1)
T(1)=0,T(x)=x−1,
T(x) 2 =x 2−1
With respect to U
0=a(1)+b(x)+c(x 2)
a=0,b=0,c=0,
[0]u=[0,0,0]T
x−1=a(1)+b(x)+c(x2)
a=−1,b=1,c=0
[x−1]u=[−1,1,0]T
x2−1=a(1)+b(x)+c(x2)
a=−1,b=0,c=1
[x2−1]u=[−1,0,1]T
Matrix U → u(T)=⎝⎛000−110−101⎠⎞
Part B
Coordu(P(x))=⎣⎡010⎦⎤
Coord u(T(P(x)))=
Coordu (P(x−1))=⎣⎡−110⎦⎤
⎝⎛000−110−101⎠⎞ ⎝⎛010⎠⎞ =⎝⎛−110⎠⎞
Verified
Part C
Expressing elements of B in terms of the elements of U
1+x+x2=a(1)+b(x)+c(x2)
a=1,b=1,c=1
[1+x+x2]u=[1,1,1]T
2x+x2=a(1)+b(x)+c(x2)
a=0,b=2,c=1
[2x+x2]u=[0,2,1]T
x+x2=a(1)+b(x)+c(x2)
a=0,b=1,c=1
[x+x2]u=[0,1,1]T
Change of base matrix PB→U=
⎣⎡111021011⎦⎤
Part D
Expressing elements of Coordu (P(x)) in terms of basis B
x=a(1+x+x2)+b(2x+x2)
+c(x+x2)
x=a+(a+2b+c)x+(a+b+c)x2
⟹a=0,a+2b+c=1
⟹2b+c=1.......(i)
a+b+c=0
⟹b+c=0......(ii)
Solving (i) and(ii)
b=1,c=−1
CoordB(P(x))=⎝⎛01−1⎠⎞
Alternatively (method 2)
Coordu (P(x))=⎝⎛010⎠⎞
Change of basis matrix Pu→B=
(MatrixPB→u)−1
∴⎝⎛111021011⎠⎞ −1 ( coordB(P(x)))
⎝⎛10−101−10−12⎠⎞ ⎝⎛010⎠⎞ =⎝⎛01−1⎠⎞
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