1. Not finding help here? Sign up for a free 30min 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!

Isomorph homework question

  1. Mar 8, 2008 #1
    1. The problem statement, all variables and given/known data

    http://i25.tinypic.com/j8i4278.jpg


    2. Relevant equations&3. The attempt at a solution

    isomoprism is bijective

    i have no clue whatsoever..im gonna research a little bit now
    ill appreciate any help
     
    Last edited: Mar 8, 2008
  2. jcsd
  3. Mar 8, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You are given that F is an isomorphism from X to Y and G is an isomorphism from Y->Z. You are asked to show that G[itex]\circ[/itex]F is an isomorphism from X to Z. I can guess that X, Y, and Z are vector spaces since another part of the problem talks about a "spanning set" but you should have said that! Proving an isomorphism of groups, rings, fields, etc. is quite different. An isomorphis is, as you say, bijective: you must prove this is surjective: that if z is any member of Z, then there exist x in X such that G[itex]\circ[/itex]F(x)= z. Since G is an isomorphism from Y to Z it is surjective: what does that tell you? Once you have that, you know that F is surjective from X to Y. Use that.

    An isomorphism must be injective also. If x1 and x2 are such that G[itex]\circ[/itex]F(x1)= G[itex]\circ[/itex](x2) then you must prove that x1= x2. Use can use the fact that F and G are each injective to prove that. Of course, a isomorphism must also "preserve" the operations. You need to show that G[itex]\circ[/itex]F(au+ bv)= aG[itex]\circ[/itex]F(u)+ bG[itex]\circ[/itex]F(v).

    As for b) using the definition of "span" together with that last statement: G[itex]\circ[/itex]F(au+ bv)= aG[itex]\circ[/itex]F(u)+ bG[itex]\circ[/itex]F(v) should be enough.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Isomorph homework question
  1. Isomorphism homework (Replies: 5)

  2. Isomorphism question (Replies: 2)

  3. Isomorphism question (Replies: 0)

Loading...