Results 1 to 5 of 5

Math Help - Isomorphism...

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    28

    Isomorphism...

    Let B be an n x n invertible matrix. Define Ф: Mnxn (F) -> Mnxn (F) by Ф (A) = B^-1 AB. Prove that Ф is an isomorphism
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by leungsta View Post
    Let B be an n x n invertible matrix. Define Ф: Mnxn (F) -> Mnxn (F) by Ф (A) = B^-1 AB. Prove that Ф is an isomorphism
    Just check three conditions: (i)it is one-to-one, (ii)it is onto, (iii) \phi (AB) = \phi (A) \phi (B).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2008
    Posts
    120
    Quote Originally Posted by ThePerfectHacker View Post
    Just check three conditions: (i)it is one-to-one, (ii)it is onto, (iii) \phi (A<b>B</b>) = \phi (A) \phi (<b>B</b>).
    Quick note: Make sure it is a different B from the one defined in the question. It is also a ring isomorphism.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by whipflip15 View Post
    Quick note: Make sure it is a different B from the one defined in the question. It is also a ring isomorphism.
    Thank you for correcting the notational mistake.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Sep 2008
    From
    Freiburg i. Brgs. Germany
    Posts
    45
    Quote Originally Posted by ThePerfectHacker View Post
    Just check three conditions: (i)it is one-to-one, (ii)it is onto, (iii) \phi (AB) = \phi (A) \phi (B).
    This is correct. Another way for showing is to proof as the matrix is invertible. That is a constraint for isomorphism.

    If you have a linear map which is bijective then this linear map always has a inverse map.

    greetings
    Herbststurm
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: October 27th 2010, 01:08 AM
  2. isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 30th 2010, 10:52 AM
  3. isomorphism
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: March 10th 2010, 09:50 AM
  4. Replies: 4
    Last Post: February 14th 2010, 04:05 AM
  5. Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 29th 2009, 12:13 AM

Search Tags


/mathhelpforum @mathhelpforum