# Eigenspace and basis of eigenvectors

1. Oct 17, 2011

### Locoism

1. The problem statement, all variables and given/known data
Given the matrix
0 1 0
0 0 1
-3 -7 -5

Find the eigenspaces for the various eigenvalues
Prove that there cannot be a basis of R3 consisting entirely of eigenvectors of A

2. Relevant equations

3. The attempt at a solution
The eigenvectors are not a problem, I end up with (λ+3)(λ+1)2 so my eigenvalues are -3 and -1. Substituting in I get [1, -3, 9] and [1, -1, 1]. Now would the eigenspace simply be {[1, -3, 9], [1, -1, 1], [0, 0, 0]} or am I missing some other step?

Also, how can I prove there cannot be a basis of R3 consisting of eigenvectors of A? Could it just be because there must be n vectors in a basis of Rn?

Last edited: Oct 17, 2011
2. Oct 17, 2011

### Staff: Mentor

Don't include the zero vector. It can't be a vector in a basis.
For R3, any basis must contain 3 linearly independent vectors. Since your basis contains only two vectors, these two vectors could not possibly span R3, and so aren't a basis for R3.

3. Oct 17, 2011

### Dick

Your characteristic polynomial is wrong. So the eigenvalues and eigenvectors are too. Did you put the correct matrix in the problem statement?

4. Oct 17, 2011

### Staff: Mentor

Dick makes a good point. After putting in all that effort to find eigenvalues and eigenvectors, you should at least check your work. If x is an eigenvector with eigenvalue $\lambda$, then it should be true that Ax = $\lambda$x.

5. Oct 17, 2011

### Locoism

Aaahhh sorry, I missed a 1, it's corrected now

6. Oct 17, 2011

### Dick

That's better. Now listen to Mark44 on the basis part of the question.

7. Oct 17, 2011

### Locoism

Thank you. So I know I can't have the zero vector as part of a basis, but should it be included in the eigenspace? or is the eigenspace juste a basis of the eigenvectors? The terminology confuses me a little.

8. Oct 17, 2011

### Staff: Mentor

The eigenspace is the set of all linear combinations of the basis vectors. The eigenspace is a vector space, which like all vector spaces, includes a zero vector.

No one is asking you to list the eigenspace (an impossible task) - just a basis for it.

9. Oct 17, 2011

### Dick

Be careful. You have two different eigenspaces here. One corresponding to the eigenvalue -3 and another to the eigenvalue -1. What's a basis for each?

10. Oct 17, 2011

### Locoism

Oh ok so basis for lambda=-3 is span(1, -3, 9) and for -1 it is span(1, -1, 1).
Why is it that there can't be a basis for R3 of only eigenvectors?

11. Oct 17, 2011

### Dick

It's basically what you said. To span R3 you need three linearly independent eigenvectors. The eigenspaces only give you two.

12. Oct 17, 2011

### Locoism

Alright thank you so much guys, this was really helpful. Is there some way to +rep or something?

13. Oct 17, 2011

### Dick

If you mean a ratings boost, no, we don't have that. Thanks is enough. Very welcome.

14. Oct 17, 2011

### Staff: Mentor

Same here. Posters don't always say "thank you," but it's appreciated when they do.