Results 1 to 2 of 2

Math Help - Prove 2 bases have the same cardinality

  1. #1
    Member
    Joined
    May 2011
    Posts
    169

    Prove 2 bases have the same cardinality

    My attempt

    Let V be a finite- dimensional vector space. Let B = {{v_1},...{v_n}} and B'= {{u_1,...,u_m}} be 2 bases for B. By the (one half of) exchange lemma, a spanning set has at least as many elements as a lin. independent set. Hence m is greater than or equal to n and n is greater than or equal to m. So m=n

    When I set out to do this proof I thought I would have to use all the exchange lemma so I'm a bit unsure of my 'proof'. Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Prove 2 bases have the same cardinality

    Quote Originally Posted by Duke View Post
    My attempt

    Let V be a finite- dimensional vector space. Let B = {{v_1},...{v_n}} and B'= {{u_1,...,u_m}} be 2 bases for B. By the (one half of) exchange lemma, a spanning set has at least as many elements as a lin. independent set. Hence m is greater than or equal to n and n is greater than or equal to m. So m=n

    When I set out to do this proof I thought I would have to use all the exchange lemma so I'm a bit unsure of my 'proof'. Thanks
    Yes, that's a good proof.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove that two sets have the same cardinality
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: March 17th 2010, 04:04 AM
  2. Prove Cardinality of the sets
    Posted in the Discrete Math Forum
    Replies: 12
    Last Post: February 25th 2010, 10:46 PM
  3. Replies: 1
    Last Post: November 5th 2009, 07:14 PM
  4. Prove the 2^N has the same cardinality as the reals
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: January 3rd 2009, 01:54 PM
  5. Replies: 1
    Last Post: May 27th 2008, 08:56 AM

/mathhelpforum @mathhelpforum