An excellent reference for this type of problem is Anton and Rorres' "Elementary Linear Algebra". You will probably be able to find a copy in your university's library or even in your local public library. You are much better off looking in a good textbook than on the net or on wikipedia as it is often hard to find good stuff on the net and wikipedia is not designed as a teaching tool.

seems to be the eigenspace of for the eigenvalue 2

QUESTION 1

What does it mean by : V_1(2), V_2(2), V_3(2) etc.

The kernel of a matrix A is the set of all x such that Ax = 0QUESTION 2

What exactly is the kernal in this example and how is the span calculated from the kernal matrices....??

In this example, we are using the fact that the kernel of is the eigenspace of A for the eigenvalue .

This follows simply from the definition of eigenvalues and eigenvectors.

The span of a set of vectors is the set of all linear combinations of the vectors.

QUESTION 3

which part of this whole question/example is the eigenspace?

I really do recommend that you get hold of that textbook, or failing that, another elementary linear algebra textbook, because you sound like you are more in need of a thorough introduction to linear algebra than the answers to specific questions that you will get here.