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Homework Help: If W is a subset of V, then dim(W) ≤ dim(V)

  1. Jul 28, 2011 #1
    1. The problem statement, all variables and given/known data

    I need to prove this:

    W, V are linear subspaces
    W is a subset of V

    -----> dimension(W) ≤ dimension(V)

    2. Relevant equations

    dimension(X): # of linearly independent vectors in any basis of X

    3. The attempt at a solution

    I'm trying to think this through, but getting stalled.


    Suppose dim(W) > dim(V). Given any basis of W and any basis of V, there will be some vector w* such that w* is contained in the basis of W but not in the basis of V.

    ..... somehow I'm supposed to deduce a contradiction (if this is even the most efficient way to the conclusion).

  2. jcsd
  3. Jul 28, 2011 #2
    Hi Jamin2112! :smile:

    Take a basis of W, can you extend this basis to form a basis of V??
  4. Jul 28, 2011 #3
    Are you talking about my supposition where dim(W)>dim(V)?
  5. Jul 29, 2011 #4
    No, I'm not. I doubt that a proof by contradiction will be the most efficient route here :frown:
  6. Jul 29, 2011 #5
    If dim(W) = dim(V), yes;
    if dim(W) < dim(V), no.
  7. Jul 30, 2011 #6
    Can't we just use the fact that every element in W is in V??
  8. Jul 30, 2011 #7


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    Why not? The x-axis is a subset of [itex]R^2[/itex] and a vector space of dimension 1. It has {<1, 0>} as basis. Adding <0, 1> to that set gives {<1, 0>, <0, 1>}, extending the first basis to a basis of [itex]R^2[/itex].
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