Answer to Question #103515 in Linear Algebra for Arun Kumar

Question

For the vector space P2,find the dual basis of {1+x,1+2x,1+x+x2}?

Answer 1

Dual basis:

f1(x)=a1+a2*x+a3*x^2

f2(x)=b1+b2*x+b3*x^2

f3(x)=c1+c2*x+c3*x^2

y1=1+x

y2=1+2x

y3=1+x+x^2

f_i(y_i)=1; f_i(y_j)=0 (i $\ne$ j)

1 = a1+a2

0 = b1+b2

0 = c1+c2

1 = b1+2b2

0 = a1+2a2

0 = c1+2c2

1 = c1+c2+c3

0 = a1+a2+a3

0 = b1+b2+b3

Solving this system we obtain

a1=2, a2=-1, a3=-1, b1=-1, b2=1, b3=0, c1=0, c2=0, c3=1

Answer:

f1=2-x-x^2

f2=-1+x

f3=x^2