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

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`