2. Find EIGENVALUES and EIGENVECTORS for the matrix:

3. The EIGENVALUES are 2 and

5. The NULL SPACE of contains solutions to: The EIENVECTORS belonging to 2 are nonzero multiples of

7. The NULL SPACE of contains solutions to: The EIENVECTORS belonging to -1 are nonzero multiples of

8. has EIGENVALUES : 2 and -1 and EIGENVECTORS: Cosets of eigenspace??

9. has EIGENVALUES : 2 and -1 and EIGENVECTORS: Consider the EIGENSPACE W =

10. W Add the same vector to every point on W to get a COSET OF W

11. Consider the EIGENSPACE W = W Add the same vector to every point on W to get a COSET OF W - a line parallel to W

12. Consider the EIGENSPACE W = W The COSETS of the EIGENSPACE W are lines parallel to W.

13. Consider the EIGENSPACE W = W The blue coset can be obtained by adding the vector to each vector in W

14. Consider the EIGENSPACE W = W The blue coset can be obtained by adding the vector to each vector in W

15. Consider the EIGENSPACE W = W The blue coset can be obtained by adding the vector to each vector in W If v is a vector in the blue coset then v = k +

16. Consider the EIGENSPACE W = W If v is a vector in the blue coset then v = k + v

17. Consider the EIGENSPACE W = W If v is a vector in the blue coset then v = k + Av = kA + A A vector on W

18. Consider the EIGENSPACE W = W If v is a vector in the blue coset then v = k + Av = kA + A A vector on W Every point on the blue coset has an image on the green coset.

19. W If v is a vector in the blue coset then v = k + Av = kA + A A vector on W Consider the EIGENSPACE W = If B is a coset of an eigenspace W then A maps every point on B onto G, another coset of W.