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Linear Algebra Proofs.

  1. Feb 21, 2009 #1
    Hey people. I find myself getting through my course but currently with not as much understanding as I would like. We've got to some proofs and i either vaguely understand them or do not know how to prove them.


    1. The problem statement, all variables and given/known data
    The first would be to prove the Dimension theorem that.

    dimU + dimV = dim(U + V) + dim( U intersection V )

    The second is if U,V are finite dimensional vector spaces over a field, and T: V -> W is a bijection

    Show dimU = dimW

    The last is to do with the Rank-Nullity Theorem and a basic linear transformation T: Rn -> Rm

    That if m < n prove that T is not injective
    And if m = n prove that T is injective iff T is surjective.

    Some general insight into these proofs would be very very helpful. All feedback would be very much appreciated.

    Thanks.
     
  2. jcsd
  3. Feb 22, 2009 #2
    Let {u1, ... , uk} be a basis for U and {v1, ... , vp} a basis for V. Now consider the intersection U[tex]\cap[/tex]V. If U[tex]\cap[/tex]V = {0}, what can you say about the linear depence of the set {u1, ... , uk, v1, ... , vp}? How about the situation U[tex]\cap[/tex]V [tex]\neq[/tex] {0}?

    If T: V -> W is a bijective map, then what can you say about its range and kernel? How is this related to the first problem?

    Again, consider range and kernel.
     
  4. Feb 22, 2009 #3

    HallsofIvy

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    I would do this: subtract dim(U intersect V) from both sides to get the equivalent
    dim(U+ V)= dimU + dimV- dim( U intersection V )
    Choose a basis for U intersect V, then extend it to a basis of U. Extend that same basis for U intersect V to a basis for V. Show that the union of those two bases is a basis for U+ V.


    Choose a basis for U, {u1, u2, ..., un} and show that {Tu1, Tu2, ... Tun} are independent. Thus, [itex]dimU\le dimW[/itex]. Choose a basis for W, {w1, w2, ..., wm} and show that {T-1w1, T-1w2, ..., T-1wn} are independent.

    Again choose a basis for Rn, {v1, v2, ..., vn} and look at {Tv1, Tv2, ..., Tvn}.

     
    Last edited by a moderator: Feb 22, 2009
  5. Feb 22, 2009 #4
    Thanks for the feedback it all helps. But still doesn't clear a lot up for me.

    For this questions i've proved U+V is a subspace. What your saying is the next step but it doesn't make much sense to me. Can you explain further?
     
  6. Feb 22, 2009 #5
    Also can anyone suggest a decent book or website where i can read up on these?
     
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