Linear mappings if polynomials
I need to show whether the following mapping/transformations are linear:
a. U = P2, V = P5; (Tp)(t) = tp(t^2) + p(1)
b. U = P2, V = P5; (Tp)(t) = tp(t^2) + 1
where Pn = set of polynomials with degree <=n
My instinct tells me that the first is linear but the second isn't, if only because of the "+1" in the 2nd. I think my main issue is actually understanding what the tp(t^2) etc notation really means.
for p in P3 let Tp be the polynomial defined by
(Tp)(t) = p(t + 1) - 2'p(t)
obtain the matrix of T w.r.t. the standard basis of P3.
Here, I know that I need to express T(1), T(t), T(t^2) and T(t^3) as linear combinations of degree 3 t, t^2, t^3 i.e. the standard basis and go from there, but again I'm stuck on the p(t + 1) etc notation.
I'd really appreciate any help!