Question

Let V be the subspace of C[infinity](R) spanned by the functions f(x)= sin(x) and g(x)=cos(x),

V:=spanR​{f,g}⊆C[infinity](R) Consider the linear transformation
T:V→V given by the rule
T(h(x))=h(x)+3h′(x) (for all h∈V ).
Write down the matrix representation [T]B​ with respect to the ordered basis B={f,g}.

Answer 1

The matrix representation [T]B with respect to the ordered basis B={sin(x),cos(x)} of the linear transformation T(h(x))=h(x)+3h'(x) is a 2x2 matrix with entries [[1, 3], [-3, 1]].

Step-by-step explanation:

We need to determine the matrix representation [T]B of the linear transformation T(h(x))=h(x)+3h'(x) for all h in the subspace V. The basis B given is {f,g} where f(x)=sin(x) and g(x)=cos(x). First, we apply T to each basis element:

• T(f(x)) = f(x) + 3f'(x) = sin(x) + 3cos(x)
• T(g(x)) = g(x) + 3g'(x) = cos(x) - 3sin(x)

These expressions can be written as:

• T(f(x)) = 1 · f(x) + 3 · g(x)
• T(g(x)) = -3 · f(x) + 1 · g(x)

The coefficients in front of f(x) and g(x) are then the entries of the matrix [T] in the order of the basis B:

[T]B =

[[ 1 3] \\ [ -3 1 ]]