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Proving that something is a subspace of all the infinite sequences

  1. Oct 14, 2009 #1
    1. The problem statement, all variables and given/known data

    let V be the vector space consisting of all infinite real sequences. Show that the subset W consisting of all such sequences with only finitely many non-0 entities is a subspace of V


    3. The attempt at a solution
    Ok so i have to show 1.Closure under Addition,2. Closure under Scalar Multiplcation

    let x=(x[tex]_{n}[/tex]),y=(y[tex]_{n}[/tex]) [tex]\in[/tex] W
    x + y [tex]\in[/tex] W closed under addition and since W is a set of finite sequences [tex]\in[/tex] V as it consists of infinite sequences

    [tex]\lambda[/tex] is a scalar and [tex]\lambda[/tex]x [tex]\in[/tex] W closed under scalar multiplication and since W is a set of finite sequences [tex]\in[/tex] V as it consists of infinite sequences

    I think this proves it

    ...But if [tex]\lambda[/tex] is negative this proof doesnt work,is there something i am missing or have i provided the required proof?
    Thanks
     
    Last edited: Oct 14, 2009
  2. jcsd
  3. Oct 14, 2009 #2

    CompuChip

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    Re: subspace

    You want to check your definitions for V and W.
    W is not closed under addition because the sequences in W are finite and those in V are infinite... you actually told us yourself that V consists of finite sequences.
    What you need to check is that when you add two sequences with finitely many non-zero entries, you get something with finitely many non-zero entries.

    So either look carefully at your proof, or at your definitions, because it looks like the proof doesn't belong to the exercise.
     
  4. Oct 14, 2009 #3
    Re: subspace

    whoops really sorry V consists of all infinite real sequences.

    if W only consists of finitely many entries and V consists of infinite does it not follow that W has to be a sub space?
     
  5. Oct 16, 2009 #4

    CompuChip

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    Re: subspace

    Yes, it follows, but you still have to show it by checking the properties of a subspace.

    For example: is W closed? I.e., if you take two infinite sequences with only finitely many non-zero entries, and you add them, do you get again a sequence with only finitely many non-zero entries?

    What else do you have to show?
     
  6. Oct 16, 2009 #5

    HallsofIvy

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    Re: subspace

    Suppose u has n non-zero entries and v has m non zero entries. What is the maximum possible number of non-zero entries of u+ v?
     
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