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Homework Help: Linear algebra, subspace

  1. Feb 9, 2010 #1
    Let V be a 9 dimensional vector space and let U and W
    be five dimensional subspaces of V with the bases Bu
    and Bw respectively,
    (a) show that if Bu intersect Bw is empty then
    Bu union Bw is linearly dependen
    (b)use part (a) to prove U intersect W is not
    equal to the 0 vector
    now i have already done part (a), now i have already
    done part (a). can you please help me..

    for part a.. this is what i have briefly....
    we know Bu intersect Bw has nothing in common.
    Since Bu and Bw is a basis we know that it is linearly independent.
    Therefore Let bu be a linearly independent
    subset of a vector space V and let Bw be a
    vector in V that is not in Bu, then Bu union Bw
    is linearly dependent iff Bw is in the span of Bu... (by a theorem)
    by definition of a basis we know, span ( Bu) = v
    therefore, the question now is if Bw is in V
    which is true ( definition of a basis)... therefore
    Bu union Bw is linearly dependent if Bu intersect Bw is nothing...
     
  2. jcsd
  3. Feb 9, 2010 #2
    my homework is due tomorrow... if some one can help me before then or even give me a clue.. it will be appreciated.... :)
     
  4. Feb 9, 2010 #3
    Your argument is severely confused.
    Here you have just defined [tex]B_U[/tex] for the second time, which won't fly. Either [tex]B_U[/tex] is a basis for [tex]U[/tex], or [tex]B_U[/tex] is an arbitrarily chosen linearly independent subset of [tex]V[/tex]. The first is what the problem says.
    No. For a single vector [tex]w[/tex], [tex]B_U \cup \{w\}[/tex] is linearly dependent if and only if [tex]w \in \mathop{\mathrm{span}} B_U[/tex]. It is false that [tex]B_U \cup B_W[/tex] is linearly dependent iff [tex]B_W \subset \mathop{\mathrm{span}} B_U[/tex], because [tex]B_W[/tex] has more than one vector in it (try to construct some simple examples in [tex]\mathbb{R}^3[/tex] with [tex]B_U[/tex] and [tex]B_W[/tex] having two elements each).
    No. [tex]B_U[/tex] is a basis for [tex]U[/tex], not [tex]V[/tex]; [tex]\mathop{\mathrm{span}} B_U = U[/tex].
    It seems like you are thinking much too hard. The dimension of [tex]V[/tex] is 9. If [tex]B_U[/tex] and [tex]B_W[/tex] are disjoint, what is the size of the set [tex]B_U \cup B_W[/tex]? Can this set be linearly independent?
     
  5. Feb 18, 2010 #4
    sorry about the late reply... I am kinda new to physics forum.....
    I am actually completely lost with this problem...... well i would guess the Bu U Bw is linearly independent.. right? because any basis by definition is linearly independent..... so wouldnt their union be independent as well? i dont know.. um just confused abou this problem..
    if you could help me out that would be really great :)
     
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