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Dimension of a vector space

  1. Jan 28, 2010 #1
    1. The problem statement, all variables and given/known data
    [tex]V=R^{4}\ and\ a^{\rightarrow}, b^{\rightarrow}, c^{\rightarrow}, d^{\rightarrow}, e^{\rightarrow} \in V. [/tex]

    (I'll drop the vector signs for easier typing...)

    [tex]a = (2,0,3,0), b = (2,1,0,0), c = (-2,0,3,0), d = (1,1,-2,-2), e = (3,1,-5,-2) [/tex]

    [tex]Let\ U \subseteq V be\ spanned\ by\ a\ and\ b.\ Let\ W \subseteq V\ be\ spanned\ by\ c,d,e[/tex]

    [tex]Compute\ dim_{F}U, dim_{F}W, dim_{F}(U \cap W)[/tex]

    2. The attempt at a solution

    I guess start with the dimension. We know the vectors a and b span U and by inspection they are linearly independent. Now I'm confused, is the dimension 3 or 4? I think 4 because the vectors have four 'slots' but I also think 3 since the last 'slot' is zero for both.

    Also, for [tex]U \cap W[/tex] I would have to prove that a,b,c,d,e are linearly independent before I can find the dimension, no?
  2. jcsd
  3. Jan 28, 2010 #2
    What is the definition of basis?
  4. Jan 28, 2010 #3


    Staff: Mentor

    For dim U, the answer is neither 3 nor 4. You have two vectors that span U and are linearly independent. As rochfor1 asked, what is the definition of a basis?
  5. Jan 28, 2010 #4
    Yeah, I just realized my error, so for V and W it would be 2 (since c=d+e)
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