1. ## Eigenspaces

I needed to find the eigenvalues for the matrix
5 0 1
1 1 0
-7 1 0

Which I found to be 2 with multiplicity 3.

Now I am required to find the eigenspace, and therein describe it geometrically.

I believe that I am supposed to use this:

det(2I-A)=0

and then subsequently row reduce the augmented matrix to find the solution.
I end up with an identity matrix with a column of zeroes, which would then mean that the matrix has only the trivial solution. Have I done this incorrectly? And if not, how do I describe this geometrically?

2. You would like to find all $\vec v$ such that

$A \vec v=2\vec v$

or equivalently

$(2I-A)\vec v=0$.

Now you should see that you need to find

$null(2I-A)$

and this is your eigenspace

3. So should it be written as 0(2I-A) or as null(2I-A)?

Because null equals zero?

4. Nope, that is the null space. Have you dealt with null spaces before?

5. No, I can't say I have. As a matter of fact, I haven't dealt with much of this linear algebra stuff at all. I am generally confused beyond... well... I'm so confused I don't even have an apt metaphor to describe it...

6. Thanks. I believe that we are going to be doing null spaces very shortly, as we are just starting to do linear transformations...