Consider a) f1=1, f2=sinx , f3=cosxb) f1=1, f2=ex , f3=e2xc)f1=e2x , f2=xe2x f3=x2e2xin each part B={f1,f2,f3} is a basis for a subspace V of the vector space.Find the matrix with respect to B of the differentiation operator D:V→V
No, the matrix consists of the coeficients, not the functions.
f1(x)= 1 so f1'(x)= 0(1)+ 0(sin(x))+ 0(cos(x)): <0, 0, 0>
f2(x)= sin(x) so f2'(x)= 0(1)+ 0(sin(x))+ 1(cos(x)): <0, 0, 1>
f3(x)= cos(x) so f2'(x)= 0(1)+ (-1)(sin(x))+ 0(cos(x)): <0, -1, 0>
The matrix is
$\displaystyle \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{bmatrix}$