linear combination of rows

• Jan 19th 2011, 08:28 PM
Jskid
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.
• Jan 19th 2011, 09:02 PM
snowtea
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.
• Jan 19th 2011, 10:36 PM
Jskid
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?
• Jan 19th 2011, 11:19 PM
FernandoRevilla
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