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Homework Help: Help with Linear Algebra exercise!

  1. Jul 7, 2010 #1
    1. The problem statement, all variables and given/known data
    Hi guys,
    I am new to this forum. I got a final exam tomorrow and the professor told us to solve some exercise before it. I came up with one exercise that I dont know how to do, at all.
    Hope you guys can give me some light. Here it goes.

    Know that the multiplication of a matrix by a vector can be write as :

    [tex]\left[ \begin{array}{c} b1 \\ . \\ bn \end{array} \right]\; =\; \left[ \begin{array}{ccc} A11 & . & A1n \\ . & . & . \\ Am1 & . & Amn \end{array} \right]\; .\; \left[ \begin{array}{c} x1 \\ . \\ xn \end{array} \right][/tex]

    or like [tex]bi=\sum_{j=1}^{n}{\; } \mbox{Ai}jxj\; ,\; i\; =1,2,3,...,m\; ,\; j=\; 1,2,3,\; ...,\; n[/tex]


    a) prove that :
    ||[tex]\vec{d}[/tex] - [tex]\vec{w}[/tex] [tex]\underline{x}[/tex]||2 = [tex]\sum_{i=1}^{m}{\; }\left( d_{i}\; -x_{i}^{T}w\; \; \right)^{2}[/tex]

    where [tex]\vec{d}[/tex]= [d1 d2 d3...dm]T

    W[tex]\in[/tex]Rn

    [tex]\vec{x}[/tex]i=[xi1 xi2 ... xin ]T

    [tex]\underline{x}[/tex]= [[tex]\vec{x}[/tex]1 [tex]\vec{x}[/tex]2 ... [tex]\vec{x}[/tex]m ]

    b) Prove that [tex]\underline{x}[/tex]T[tex]\underline{x}[/tex] is real and simetric.

    Obs: [tex]\underline{x}[/tex] means the matrix x

    2. Relevant equations



    3. The attempt at a solution

    None of my attempts were close to something

    Hope you guys understand the question and give me a hand !
     
    Last edited: Jul 7, 2010
  2. jcsd
  3. Jul 7, 2010 #2
    anyone, Please?
     
  4. Jul 8, 2010 #3

    hunt_mat

    User Avatar
    Homework Helper

    If x is a (1,n) row matrix, then x^T*x will be a (n,n) matrix, I presume that it is this that your lecturer wanted to prove that this matrix is symmetric?
    The way I would go about it is this:
    1) Write down the definition of the product of two matrices
    2) Specialise to the case which you're interested (i.e. one row vector and column vector)
    3) Compute the transpose of this matrix.
    4) Compare with the original matrix to see if the elements are the same.

    I can't help you with the first part as you have either misquoted it and not told me the norm you're using.

    Mat
     
  5. Jul 8, 2010 #4
    vslo are you sure you copied question one correctly?
     
  6. Jul 8, 2010 #5

    Mark44

    Staff: Mentor

    The b vector in this equation -
    [tex]\left[ \begin{array}{c} b1 \\ . \\ bn \end{array} \right]\; =\; \left[ \begin{array}{ccc} A11 & . & A1n \\ . & . & . \\ Am1 & . & Amn \end{array} \right]\; .\; \left[ \begin{array}{c} x1 \\ . \\ xn \end{array} \right][/tex]

    should have m entries, not n.
     
  7. Jul 9, 2010 #6

    Yes, he is right. the b vector has M entries, not N... Does it helps ?
    Thank you for your observation Mark !
     
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