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Math Help - Linear mappings if polynomials

  1. #1
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    Linear mappings if polynomials

    Hi everyone,
    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.

    Also,
    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!
    Thanks
    Last edited by s0791264; February 23rd 2010 at 09:12 AM. Reason: extra notes
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