1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: If m<n prove that y_1, ,y_m are linear functionals

  1. Mar 4, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that if m<n, and if [tex]y_1,\cdots,y_m[/tex] are linear functionals on an n-dimensional vector space V, then there exists a non-zero vector x in V such that [tex][x,y_j]=0[/tex] for [tex]j=1,\cdots,m[/tex]. What does this result say about the solutions of linear equations?

    2. Relevant equations


    3. The attempt at a solution

    Let [tex]X=\{x_1,\cdots,x_n\}[/tex] be a basis for V.
    By a theorem in the book I know that there exists a uniquely determinet basis X' in V',
    [tex] X'=\{y_1,\cdots,y_n\} [/tex] with the property that [tex][x_i,y_j]=\delta_{ij}[/tex].

    Every [tex]x\in V[/tex] can be written as [tex] x=\xi_1 x_1+\cdots+\xi_n x_n[/tex].
    If we let [tex]j=1,\cdots,m[/tex] we get,
    [tex][x,y_j]=\xi_1[x_1,y_j]+\cdots+\xi_m[x_m,y_j][/tex] and since [tex][x_i,y_j]=\delta_{ij} [/tex], there exists some x such that [tex][x,y_j]=0[/tex].

    The result says that the solution of linear equations consists of a homogeneous solution and a particular solution.

    Any input is very welcome, Thanks!
  2. jcsd
  3. Mar 6, 2010 #2
    Gentle bump.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook