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.
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.
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.
Write:
$\displaystyle a= (a_1,\ldots,a_n),\quad B=\begin{pmatrix}R_1\\ \vdots\\{R_n}\end{pmatrix}$
Then,
$\displaystyle aB=(a_1,\ldots,a_n)\begin{pmatrix}R_1\\ \vdots\\{R_n}\end{pmatrix}=a_1R_1+\ldots+a_nR_n$
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