Problem:
In each of Exercises 40 through 46 following, a linear space V is given and a mapping T : V→V is defined as indicated. In each case determine whether T is a linear mapping. If T is linear, determine the kernel (or null space) and range, and compute the dimension of each of these subspaces wherever they are finite-dimensional.
1. V is the (real) linear space of all real polynomials p on R. If p E V, then T(p) is defined by setting
T(p)(x) = p(x+1), x E R
2. V is the linear space of all real functions f defined and differentiable on the open interval (0,1). If f E V, then T(f) is defined by setting
T(f)(x) = xf'(x), x E (0,1)
3. V is the linear space of all real functions f defined and continuous on the closed interval [0, 2π]. If f E V, then T(f) is defined by setting
T(f)(x) = 0∫2πf(t)sin(x-t)dt, x E [0, 2π]
4. V is the linear space of all real functions f defined and continuous on the closed interval [0, 2π]. If f E V, then T(f) is defined by setting
T(f)(x) = 0∫2πf(t)cos(x-t)dt, x E [0, 2π]