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Product matrix as a Linear Combination

  1. Oct 4, 2009 #1
    Problem Statement

    Let

    [tex]\mathbf{y} = [y_1\, y_2\, ...\, y_m][/tex]

    And

    [tex]A =
    \left[\begin{array} {cccc}
    a_{11}&a_{12}&...&a_{1n}\\
    a_{21}&a_{22}&...&a_{2n}\\
    a_{m1}&a_{m2}&...&a_{mn}
    \end{array}\right]
    [/tex]

    Show that the product yA can be expressed as a linear combination of the row matrices of A
    with the scalar coefficients coming from y



    Attempt at Solution

    I thought that I would write out the actual product, which is a row vector. I thought that
    something might jump out at me from here:

    yA = [(y1a11 + y2a21 + ... + ymam1) (y1a12 + y2a22 + ... + ymam2) (y1a1n + y2a2n + ... + ymamn)]

    I am not sure where to go from here. I know that it is going to be a summation of the rows of A .... but what I have now is just written column-wise... and it is not a summation.

    A hint maybe?
     
  2. jcsd
  3. Oct 4, 2009 #2
    I think that you have a row vector. You can treat it just like a column vector and break it down into a sum. For example (a + b + c, d + b, b + c) = (a, d, b) + (b, b, c) + (c, 0, 0).

    Pull out terms that have the same factor from y.
     
  4. Oct 4, 2009 #3
    I think that I got it!

    I just wrote the summation of the rows of A :

    [a11 a12 ... a1n] + [a21 a22 ... a2n] + ... + [am1 am2 ... amn]

    and then noted that each term needs to multiplied by each of the elements of y :

    y1[a11 a12 ... a1n] + y2[a21 a22 ... a2n] + ... + ym[am1 am2 ... amn]

    I guess I just thought the solution would have been a little more 'graceful' as opposed to 'guess and check.' :yuck:

    thanks!
     
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