A question about linear algebra...

The E-spaces are just vector spaces. It is the dimension of those vector spaces.

They are defined as the null spaces of certain linear transformations i.e.

$E_\lambda^k$ is defined (on the same page) to be the null space of the linear transformation $(A-\lambda I)^k$.

So for example, $E_\lambda^k$ is the null space of $(A-\lambda I)$ i.e. the eigenspace for $\lambda$.

So that's just the number of free varaibles right?

It depends what you mean by "the number of free variables".

If you mean the number of zero-rows in the reduced echelon form of the matrix $(A-\lambda I)^k$, then yes - that is one way to calculate the nullity i.e. the dimension of the nullspace of $(A-\lambda I)^k$.

By "the number of free variables" I mean the number of columns that do not have a pivot...so I'm gussing that that's equal to the number of zero rows...