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vector space q

  • 15-10-2012 03:40PM
    #1
    eoins23456
    Registered Users, Registered Users 2 Posts: 603 ✭✭✭


    Assume the following lemma , Let V be a finite dimensional vector space. Let L ={l1,....ln} be a linear independent set in v let S= {s1,....sm} be a second subset of V which spans v , then n<=m .

    Use this lemma to conclude that any two bases for a finite dimensional vector space must have the same number of elements.

    Could anybody give me a good starting point?


Comments

  • #2
    Snotzenfartz
    Closed Accounts Posts: 49 ✭Snotzenfartz


    Use the Pythagoras theorem.


  • #3
    Michael Collins
    Moderators, Science, Health & Environment Moderators Posts: 1,855 Mod ✭✭✭✭Michael Collins


    Snotzenfartz wrote: »
    Use the Pythagoras theorem.

    Not sure how you would use Pythagoras' theorem here, there's no mention of an inner product space being defined.
    eoins23456 wrote: »
    Assume the following lemma , Let V be a finite dimensional vector space. Let L ={l1,....ln} be a linear independent set in v let S= {s1,....sm} be a second subset of V which spans v , then n<=m .

    Use this lemma to conclude that any two bases for a finite dimensional vector space must have the same number of elements.

    Could anybody give me a good starting point?

    Sure, can you list the two main properties that a basis of a vector space has?


  • #4
    eoins23456
    Registered Users, Registered Users 2 Posts: 603 ✭✭✭eoins23456


    its vectors are linear independent and span the vector space ?


  • #5
    Michael Collins
    Moderators, Science, Health & Environment Moderators Posts: 1,855 Mod ✭✭✭✭Michael Collins


    eoins23456 wrote: »
    its vectors are linear independent and span the vector space ?

    Exactly. Now let L and S be two different bases for the same vector space V. Since L is a linear independent set it satisfies the same L from the lemma, and since S is a linear independent spanning set, it satisifies S from the lemma. Now, what does the the lemma say about the number of elements in each?


  • #6
    eoins23456
    Registered Users, Registered Users 2 Posts: 603 ✭✭✭eoins23456


    Michael Collins wrote: »
    Exactly. Now let L and S be two different bases for the same vector space V. Since L is a linear independent set it satisfies the same L from the lemma, and since S is a linear independent spanning set, it satisifies S from the lemma. Now, what does the the lemma say about the number of elements in each?
    that they have to be less then or equal to eachother


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  • #7
    Michael Collins
    Moderators, Science, Health & Environment Moderators Posts: 1,855 Mod ✭✭✭✭Michael Collins


    Yeh, well specifically the number of elements in S must be greater than or equal to the number of elements in L i.e. #S >= #L.

    Now swap the two vector spaces, i.e. put L in place of S, and S in place of L, into the lemma. Do they satisfy the conditions of the lemma first? If so, what is the out come this time?


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