(adsbygoogle = window.adsbygoogle || []).push({}); 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

definition:

one to one: if v(1) != v(2) then T(v(1)) != T(v(2)). the numbers in parenthesis are subscripts.

W

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?

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# Homework Help: 1-1 and onto linear transformation question

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