# Question on proof of Solutions of Linear Equations

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?

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The first part proves that all vectors of the form $x_0 + u$ with $u \in \ker T$ are solutions. This provides a sufficient condition for a vector to be a solution. But note that this does not mean that all solutions are of this form. This is what is proved in the second part.