Question

how do you find the basis for an eigenspace

Answer 1

this is one of the best explanations with examples:
http://www.math.hmc.edu/calculus/tutorials/eigenstuff/eigenstuff.pdf

Answer 2

Thanks so much that was a big help. I am taking an io online course so all the help is needed.

Answer 3

How do you find the algebratic and geometric multiplicity of an eigenvalue.

Answer 4

Given an n×n matrix A and an eigenvalue λi of this matrix, there are two numbers measuring, roughly speaking, the number of eigenvectors belonging to λi. They are called multiplicities: the algebraic multiplicity of an eigenvalue is defined as the multiplicity of the corresponding root of the characteristic polynomial. The geometric multiplicity of an eigenvalue is defined as the dimension of the associated eigenspace, i.e. number of linearly independent eigenvectors with that eigenvalue. Both algebraic and geometric multiplicity are integers between (including) 1 and n. The algebraic multiplicity ni and geometric multiplicity mi may or may not be equal, but we always have mi ≤ ni.

I would recommend to check (good as a first reference):
http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors