- #1
member 428835
Homework Statement
Let ##C## be the space of continuous real functions on ##[0,\pi]##. With any function ##f\in C##, associate another function ##g=T(f)## defined by $$g=T(f)\equiv \int_0^\pi \cos(t-\tau) f(\tau) \, d \tau$$
a) Show ##T## is a linear transformation from ##C## to ##C##.
b)What is the image of ##T##? Find a basis for it.
c) List a set of linearly independent vectors that are in the null space of ##T##. What is ##dimN(T)##?
Homework Equations
Linear algebra stuff. Too much to write I think.
The Attempt at a Solution
a)##T(af + bg) = \int\cos(t-\tau)(af + bg) = \int\cos(t-\tau)af + \int\cos(t-\tau)bg = a\int\cos(t-\tau)f + b\int\cos(t-\tau)g = aT(f)+bT(g)##.
b) I want to say the space of continuous real functions on ##[0,\pi]##. Does this even make sense though?
c)By inspection, ##a\sin(t-\tau)## but this is all I can see. Are there any others? In this case, I would say ##dimN(T) = 1## so far since I can only think of the one example. Obviously ##0## is as well but this is not linearly independent of what I have listed.