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Same dimensions, same space

  1. Dec 14, 2013 #1
    Is it correct to state that if a space [itex]E[/itex] has dimension 3 then:
    [itex]E = ℝ^{3}[/itex] and that the two spaces are isomorph?
     
  2. jcsd
  3. Dec 14, 2013 #2

    jgens

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    If E is a 3-dimensional real vector space, then yes E is isomorphic as a vector space to R3. The statement that E = R3 is false however.
     
  4. Dec 14, 2013 #3
    Then I wonder why in my textbook, every time there is an omomorphism [itex]f[/itex] whose image [itex]Im(f)[/itex] (row space) has dimension 3 it writes [itex]Im(f) = ℝ^3[/itex].

    Am I missing something?
     
  5. Dec 14, 2013 #4

    jgens

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    Two possibilities for this: First if your linear transformation has codomain R3 and its image has dimension 3, then its image literally is R3. And two sometimes the equality sign is used to mean "isomorphic to" and in that case im f is definitely isomorphic to R3.
     
  6. Dec 14, 2013 #5
    I think I should have add that all the homomorphisms in the textbook are always [itex]f : ℝ^m \rightarrow ℝ^n[/itex]
     
  7. Dec 14, 2013 #6

    jgens

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    In that case it is a slight abuse of notation to write im f = R3, but also a very common one.
     
  8. Dec 14, 2013 #7
    I see, thank you for your answer.
     
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