Product matrix as a Linear Combination

[Saladsamurai]

Problem Statement

Let

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

And

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

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 + ... + y_ma_m1) (y₁a₁₂ + y₂a₂₂ + ... + y_ma_m2) (y₁a_1n + y₂a_2n + ... + y_ma_mn)]

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?

 

[aPhilosopher]

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.

 

[Saladsamurai]

I think that I got it!

I just wrote the summation of the rows of A :

[a₁₁ a₁₂ ... a_1n] + [a₂₁ a₂₂ ... a_2n] + ... + [a_m1 a_m2 ... a_mn]

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

y₁[a₁₁ a₁₂ ... a_1n] + y₂[a₂₁ a₂₂ ... a_2n] + ... + y_m[a_m1 a_m2 ... a_mn]

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

thanks!