1. Not finding help here? Sign up for a free 30min 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!

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

    User Avatar
    Staff Emeritus
    Science Advisor

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




Similar Discussions: Linear Algebra Proofs.
  1. Linear Algebra Proof (Replies: 8)

  2. Linear Algebra proof (Replies: 21)

Loading...