Results 1 to 4 of 4

Math Help - Direct sums, Polynomials

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    21

    Direct sums, Polynomials

    Hi,
    Thank you so much for the help before. I appreciate it.
    I have 2 new problems to solve. Please, help me!

    1. Prove or disprove if U, V, W are subspaces of V for which
    U (dir sum) W = V (dir sum) W then U=V

    2. Suppose that p_0,...p_3 are polynomials in the space P_m(F) (of polynomials over F with degree at most m) such that p_j(2)=0 for eachj. Prove that the set {p_0,...,p_m} is not lin. independant in P_m(F).

    Thank you in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by mivanova View Post
    1. Prove or disprove if U, V, W are subspaces of V for which
    U (dir sum) W = V (dir sum) W then U=V[/tex]
    ok, there's a problem with the names you gave your subspaces: you already assumed V is the vector sapce. so the subspaces should be named, say, U, W, and Z.

    as a vector space over \mathbb{R} let V=\mathbb{R}^2 and e_1=(1,0), \ e_2=(0,1), \ e_3=e_1+e_2=(1,1). let U=\mathbb{R}e_1, \ W=\mathbb{R}e_2, \ Z=\mathbb{R}e_3. then V=U \oplus W = Z \oplus W. but U \neq Z.


    2. Suppose that p_0,...p_m are polynomials in the space P_m(F) (of polynomials over F with degree at most m) such that p_j(2)=0 for eachj. Prove that the set {p_0,...,p_m} is not lin. independant in P_m(F).
    let p_j(x) \in P_m(\mathbb{F}), \ j=0, \cdots, m, with a common root \alpha. (in your problem \alpha=2.) then q_j(x)=\frac{p_j(x)}{x-\alpha} \in P_{m-1}(\mathbb{F}), \ j=0, \cdots , m. but \dim_{\mathbb{F}} P_{m-1}(\mathbb{F})=m, and

    the set \{q_j(x): \ j=0, \cdots, m \} has m + 1 elements. so it's linearly dependent, i.e. there exist c_j \in \mathbb{F}, \ j=0, \cdots, m, not all zero such that \sum_{j=0}^m c_jq_j(x)=0. \ \ \ \ (1)

    now multiply both sides of (1) by x - \alpha to get \sum_{j=0}^m c_jp_j(x)=0. thus p_j(x), \ j=0, \cdots , m, are linearly dependent. Q.E.D.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2008
    Posts
    21

    Direct Sums 2

    Thank you so much for the help.
    How would you prove the 1st problem
    (1. Prove or disprove if U, V, W are subspaces of V for which
    U (dir sum) W = V (dir sum) W then U=V[/tex])
    in the general case (not only in R^2) but for any subspace of a vector sace?
    Thanks!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by mivanova View Post
    Thank you so much for the help.
    How would you prove the 1st problem
    (1. Prove or disprove if U, V, W are subspaces of V for which
    U (dir sum) W = V (dir sum) W then U=V[/tex])
    in the general case (not only in R^2) but for any subspace of a vector sace?
    Thanks!
    i did not prove it! i disproved it by giving a counter-example.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Direct Sums of Modules
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: August 30th 2011, 03:09 AM
  2. Direct sums and bases
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: August 26th 2009, 02:05 PM
  3. Eigenvalues and direct sums
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 13th 2009, 09:15 PM
  4. need help...finite direct sums
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 1st 2008, 05:32 PM
  5. Direct Sums
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 11th 2006, 07:46 PM

Search Tags


/mathhelpforum @mathhelpforum