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Math Help - linear combination of rows

  1. #1
    Member Jskid's Avatar
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    linear combination of rows

    Let a be a 1 x n matrix and B an n x p matrix. Show that the matrix product aB can be written as a linear combination of rows of B where the coefficients are the entries of A.

    I don't get how you can have a combination of rows.
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  2. #2
    Senior Member
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    Linear combination of rows just means treating each row as a vector, and the final resulting product aB is a vector which is some linear combination of all the row vectors.

    Hint: Notice that during multiplication, each element on the same row is multiplied by the same factor before being added.
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  3. #3
    Member Jskid's Avatar
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    aB = [1(2)+-2(-2)+3(4) 1(1)+-2(-2)+3(5) 1(-4)+-2(3)+3(-2)]
    I see how the rows are inside the elements but how do I write this as an linear combination?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Write:

    a=  (a_1,\ldots,a_n),\quad B=\begin{pmatrix}R_1\\ \vdots\\{R_n}\end{pmatrix}

    Then,

    aB=(a_1,\ldots,a_n)\begin{pmatrix}R_1\\ \vdots\\{R_n}\end{pmatrix}=a_1R_1+\ldots+a_nR_n


    Fernando Revilla
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