TITLE: Linear Algebra Proof theorem
QUESTION [0 upvotes]: I understand the first part of this theorem, but I don't understand how in the second part you subtract the two variables
Theorem: Assume $AX=B$ is a linear system and let $X_1$ be one solution to the system. Then $X$ is a solution to $AX=B$ if and only if $X = X_1 - X_0$ where $X_1$ is the above solution, and $X_0$ is a solution to the associated homogeneous system $AX=0$.
Part 1.) Assume  $X = X_1 + X_0, X_1$ being a solution to $AX=B$ and $X_0$ being a solution to $AX=0$.
$AX=A(X_1 + X_0) = AX_1 + AX_0 = B + 0 = B$.
$X_1 + X_0$ is a solution to $AX=B$.
Part 2.) Assume $X$ is a solution to $AX=B$.
$A(X-X_1) = AX - AX_1 = B - B = 0$.
$X - X_1$ is a solution to $AX=0$.

REPLY [1 votes]: It's not that you substract the second variable. But you want to verify that if you have a solution $X$ to $AX=B$ then it is of the form $X_1+X_0$. A simple way to do this is to verify that $X-X_1=X_0$ : is a solution to $AX=0$ which is what was done in the proof.
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