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Algebra span proof question

  1. Jun 21, 2011 #1
    there is an orthonormal group {u1,..,uk} in R^n

    there is vector v which belongs to R^n

    prove that if

    ||v||^2=(v*u1)^2 +..+(v*u_k)^2

    then v belongs to the sp{u1..uk}

    *-is dot product



    how i tried to solve it:

    i expanded the orthonormal group {u1,..,uk} to

    the orthonormal group {u1,..,uk,..un}

    then v is its combination

    v=a1u1+a2u2+..anun

    i put v in the given formula

    ||v||^2=((a1u1+a2u2+..anun)*u1)^2 +..+((a1u1+a2u2+..anun)*u_k)^2

    =(a1u1^2)^2 +..(akuk^2)^2=a1^2+..a^k^2

    u1..uk are orthonormal

    so u1^2=1.. uk^2=1

    what now?
     
  2. jcsd
  3. Jun 21, 2011 #2

    micromass

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    Hi nhrock3! :smile:

    Can you write out [itex]\|v\|^2[/itex]? You can use

    [tex]\|v\|^2=<v,v>=<a_1u_1+...+a_nu_n,a_1u_1+...+a_nu_n>[/tex]
     
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