Question

The book defines the completes solution of Ax=b as X(particular) + X(nullspace). Can anyone elaborate further?

Answer 1

x(particular) solves the given equation:
Ax(particular) = b.
And x(nullspace) satisfies:
Ax(nullspace) = 0.
So
Ax(particular) + Ax(nullspace)
= Ax(particular) + 0
= b.
Hows that?

Answer 2

By definition, any vectors in the nullspace N(A), satisfy Ax = 0. For example, say n was in the nullspace then An =0. Now, if you go back to the problem where you are looking for a solution x to the problem Ax = b and lets say you found one, call it p. In other words Ap = b.

What the statement says it that using that solution we can find another one by using the vector in the nullspace. For example what about the vector p + n?
A(p + n) = Ap + An = b + 0 = 0
We could also have used 10n or -3n or any other scalar value.

When it talks about the "complete" solution, it means what are all of the possible solutions to Ax = b. This turns out to be any vector that is the sum of a particular solution and any combination of vectors in the nullspace N(A) since they all go factor to zero. In other words, if you find any particular solution, and you find a basis for the nullspace, then you've found all possible solutions.

Dunno if that helps.