1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.

    Hmmmm....

    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).

    Help?
     
  2. jcsd
  3. Jul 28, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: If W is a subset of V, then dim(W) ≤ dim(V)
Loading...