# Thread: solutions

1. ## solutions

i dont really understand this statement. can someone explain to me?

what does it mean:

Ax=b has solutions iff APy=b has solutions.

in my notes, it states that that means that column operations dont change the span of the columns...but i still dont get what that means

2. The statement is true in general for $P$ invertible.

In fact, if $x_0$ is a solution of $Ax=b$ then

$AP(P^{-1}x_0)=Ax_0=b$

This implies that $y_0=P^{-1}x_0$ is a solution for $APy=b$

It's easy to prove the reciproque.

Regards

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Fernando Revilla