Can a Matrix be Written as a Linear Combination of Another Matrix's Columns?

[brunette15]

So i have the following:
[a b; c d] = [e f ; g h] * [p q ; r s]

I have to show that the if the original matrices are written as A = EP then the columns of A are linear combinations of E.

I was able to prove [a;c] = p[e;g] + r[f;h] and the same for [b;d] but i don't know where to go from here :/

Any help would really be appreciated!
Thanks in advance!

 

[I like Serena]

So i have the following:
[a b; c d] = [e f ; g h] * [p q ; r s]

I have to show that the if the original matrices are written as A = EP then the columns of A are linear combinations of E.

I was able to prove [a;c] = p[e;g] + r[f;h] and the same for [b;d] but i don't know where to go from here :/

Any help would really be appreciated!
Thanks in advance!

Hi again! (Wave)

Let's write out the right hand side:
$$\begin{pmatrix}e&f \\ g&h\end{pmatrix}
\begin{pmatrix}p&q \\ r&s\end{pmatrix}
=\begin{pmatrix}ep+fr&eq+fs \\ gp+hr&gq+hs\end{pmatrix}
$$
Taking a look at the first column, we can write it as:
$$p\begin{pmatrix}e \\ g\end{pmatrix} + r\begin{pmatrix}f \\ h\end{pmatrix}
$$
This is a linear combination of the columns of E! (Happy)

 

[brunette15]

Hi again! (Wave)

Let's write out the right hand side:
$$\begin{pmatrix}e&f \\ g&h\end{pmatrix}
\begin{pmatrix}p&q \\ r&s\end{pmatrix}
=\begin{pmatrix}ep+fr&eq+fs \\ gp+hr&gq+hs\end{pmatrix}
$$
Taking a look at the first column, we can write it as:
$$p\begin{pmatrix}e \\ g\end{pmatrix} + r\begin{pmatrix}f \\ h\end{pmatrix}
$$
This is a linear combination of the columns of E! (Happy)

Thankyou so much!