Question

Fix a matrix A and a vector b. Suppose that y is any solution of the homogeneous system Ax=0 and that z is any solution of the system Ax=b. Show that y+z is also a solution of the system Ax=b.

Answer 1

Answer:

Step-by-step explanation:

Let $y$ be a solution vector of the homogeneous linear system $Ax = 0$, then, $Ay = 0$.

On the other hand, suppose that $z$ is a solution vector of the non-homogeneous linear system $AX = b$, that is, $Az = b$.

Now, considering the previous assumptions, you have:

$A (y + z) = Ay + Az = 0 + b = b$

The above demonstrates that the vector $(y + z)$ satisfies the system $Ax = b$. Then $(y + z)$ is a solution of the non-homogeneous linear system $Ax = b$