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Linear transformation proof

  1. Dec 5, 2004 #1
    Let f: R --> R and let T: P2 --> F, and T(p) = p(f). Prove that T is a linear transformation.

    P2 is the set of polynomials of degree 2 or less, and F is the set of all functions.

    It seems to me that I can treat f as really just a real number, in which case it's no different from proving T(p) = p(x) for all x in R. Is it this simple?

    Last edited: Dec 5, 2004
  2. jcsd
  3. Dec 5, 2004 #2


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    You can treat f similarly to a real number, because you can add and multiply real-valued functions, but you can't treat it as a real number.

    Secondly, you're not trying to prove T(p) = p(f): that's the definition of T. You're trying to prove that T is a linear transformation from P2 to F.

    Finally, F is is not just a set, it's a vector space.
  4. Dec 5, 2004 #3
    To prove that T(p) is a linear transformation, you must show that it has the properties of a linear transformation, that is:
    (a) T(a+b)=T(a)+T(b) for a,b in P2
    (b) T(ca)=cT(a) for a in P2 and c in R
  5. Dec 5, 2004 #4
    So let p1 and p2 be elements of P2
    Then T(p1 + p2) = (p1 + p2)(f) = p1(f) + p2(f) = T(p1) + T(p2)
    and T(cp1) = cp1(f) = cT(p1)

    Is this sufficient to show that T is a linear transformation or am I leaving something out?
  6. Dec 5, 2004 #5
    That looks right. Good job!
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