# One Linear Algebra Problem (Dual Spaces)

• June 4th 2010, 09:07 AM
eyke
One Linear Algebra Problem (Dual Spaces)
• June 4th 2010, 05:09 PM
tonio
Quote:

Originally Posted by eyke

What's the problem? You need three polynomials $p_i(x)\,,\,i=1,2,3$ s.t. $f_i(p_j(x))=\delta_{ij}$.
For example, you need that $\int_0^1p_1(x)\,dx=1\,,\,\,\int^2_0p_1(x)\,dx=0\,, \,\int^{-1}_0p_1(x)\,dx=0$ . Putting $p_1(x)=ax^2+bx+c$ , this means :

$1=\int^1_0(ax^2+bx+c)dx=\frac{a}{3}+\frac{b}{2}+c$

$0=\int^2_0(ax^2+bx+c)dx=\frac{8a}{3}+2b+2c$

$0=\int^{-1}_0(ax^2+bx+c)dx=-\frac{a}{3}+\frac{b}{2}+c$

Solving the resulting linear system we get $p_1(x)=\frac{3}{2}x^2-5x+3$ . Do the same with the other two polynomials.

Tonio