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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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
What have you been able to do on this problem so far?Quote:
Originally Posted by bernoli
-Dan
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.
(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
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