Our lecturer gave us the following proof for the Solutions of Linear Equations but I don't understand the point of what he does in the second part. What is the point of the second part where he has $T(z - x_0) = 0$, so $z - x_0$ is an element of Ker(T). I don't see what this is proving.
It seems to me that the statement is proved in the first part that ends with 'so x is a solution'. As he has shown T(x) = y is satisfied by the particular solution $x_0$ along with the general solution of the homogeneous equation u. As in T(u) = 0.
Anyone able to explain this solution to me?