Showing that the matrix transformation T ( f ) = x ∗ f ′ ( x ) + f ″ ( x ) is linear

Name: Showing that the matrix transformation T ( f ) = x ∗ f ′ ( x ) + f ″ ( x ) is linear
Uploaded: 2022-02-23T17:22:57+00:00
Description: Showing that the matrix transformation T ( f ) = x ∗ f ′ ( x ) + f ″ ( x ) is linear

Question

Showing that the matrix transformation   T  (  f  )  =  x  ∗      f    ′    (  x  )  +      f    ″    (  x  ) is linear

Layla Love · Accepted Answer

By the definition of T, you have for every function h in your domain of T, that

T (h) = x \cdot h^{'} (x) + h^{″} (x)

If now

h = f + g

, we get

T (f + g) = x \cdot (f + g)^{'} (x) + (f + g)^{″} (x),

which equals your calculated result, which moreover used the linearity of differentiation.

Showing that the matrix transformation T ( f ) = x ∗ f ′ (...

Answered question

Answer & Explanation