# matrix and basis

• Jun 15th 2011, 08:35 AM
bernoli
matrix 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 AM
topsquark
Re: matrix and basis
Quote:

Originally Posted by bernoli
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

What have you been able to do on this problem so far?

-Dan
• Jun 15th 2011, 09:08 AM
HallsofIvy
Re: 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 AM
bernoli
Re: 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 AM
HallsofIvy
Re: 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
$\begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0\end{bmatrix}$