Now consider this basis 1 ex eminusx e2x eminus2x Write down

Now consider this basis: {1, e^x, e^minusx, e^2x, e^minus2x} Write down a matrix that represents transformation F under this basis. Try your new PrimeintegratorPrime matrix on the function 1 + 2e^x + 3e^minusx + 4e^2x + 5e^minus2x to verify it works (do the integral manually, and sec if the matrix gives the same result). Is it possible to come up with a basis for the whole space C, so that ANY integral can be written as a matrix multiplication? Why or why not? Is differentiation also a linear transformation (which would mean that it could be written as a matrix multiplication, once a basis is chosen).

Solution

Solution : 8)

Differentiation is a linear transformation because it satisfies the definition of a linear operator. Namely, the derivative of the sum of two (differentiable) functions is the sum of their derivatives. Also, the derivative of a constant multiplying a function is equal to the constant multiplied by the derivative of the function.