1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 1-1 and onto linear transformation question

  1. Mar 23, 2012 #1
    1. The problem statement, all variables and given/known data
    Assume that T:C^0[-1,1] ---> ℝ. assume that T is a linear transformation that maps from the set of all continuous functions to the set of real numbers.
    T(f(x)) = ∫f(x)dx from -1 to 1. is T one to one, is it onto, is it both or is it neither.

    2. Relevant equations
    one to one: if v(1) != v(2) then T(v(1)) != T(v(2)). the numbers in parenthesis are subscripts.
    onto: if the range (image) of T is the whole of W; if every w ε W is the image under T of at least one vector v ε V

    3. The attempt at a solution
    i was able to show that it was not one to one because there are multiple values of f(x) that map to the same real number. however i am not sure if my justification for it being onto is correct. i said that because the dimension of C^0[-1,1] is infinite and the dimension of ℝ is one, we can say that dim[V] > dim[W] which implies that T is onto.

    there is a little corollary to a theorem in my book that says that if T is onto, dim[V] >= dim[W]. because i showed that dim[V] > dim[W], can i say that T is onto?
  2. jcsd
  3. Mar 23, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    This argument is incomplete at best. Consider e.g. the map P:ℝ3→ℝ2 defined by P(x,y,z)=(0,z). Clearly the dimensions of the domain and codomain are 3 and 2 respectively, but P is not surjective (onto). This P is also not injective. Your argument might work for injective functions, but I'm too tired to think about that right now.

    It's very easy to prove that your T is surjective directly from the definition. Just let r be an arbitrary real number, and find a continuous function f such that Tf=r.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook