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

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- Jun 15th 2011, 08:35 AMbernolimatrix and basis
**Consider a) f1=1, f2=sinx , f3=cosx****b) f1=1, f2=ex , f3=e2x****c)f1=e2x , f2=xe2x f3=x2e2x****in 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** - Jun 15th 2011, 08:54 AMtopsquarkRe: matrix and basis
- Jun 15th 2011, 09:08 AMHallsofIvyRe: matrix and basis
Apply the operator to each basis function in turn and write the result as a linear combination of the basis functions. The coefficients in that linear combination form one column in the matrix representation.

- Jun 16th 2011, 01:39 AMbernoliRe: matrix and basis
(Thinking)f '=0 f'2=cos x f'3=-sinx

0 0 0

A= 0 cosx 0

0 0 -sinx A is matrix

is this okay - Jun 18th 2011, 03:34 AMHallsofIvyRe: matrix and basis
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

$\displaystyle \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{bmatrix}$