3. Domain Range

7. definition: T is a linear transformation, EIGENVECTOR EIGENVALUE

8. A is the matrix for a linear transformation T relative to the STANDARD BASIS

10. T T

11. T T The matrix for T relative to the  basis

12. T T Eigenvectors for T Diagonal matrix

13. The matrix for a linear transformation T relative to a basis of eigenvectors will be diagonal

14. To find eigenvalues and eigenvectors for a given matrix A: Solve for and A = A = I A = I - A ) = I - (

15. To find eigenvalues and eigenvectors for a given matrix A: Solve for and A ) = I - ( Remember: is a NONZERO vector in the null space of the matrix: A )I - (

16. is a NONZERO vector in the null space of the matrix: A )I - ( The matrix has a nonzero vector in its null space iff: A )I - ( I - ( det = 0

17. A )I - ( det = 0 A =I - -

18. A )I - ( det =

19. A )I - ( det =

20. A )I - ( det = This is called the characteristic polynomial

21. A )I - ( det = = 0 the eigenvalues are 2 and -1

22. A )I - ( = the null space of 2I - A = 2 the eigenvectors belonging to 2 are nonzero vectors in the null space of 2I - A

23. A )I - ( = the null space of -1I - A = the eigenvectors belonging to -1 are nonzero vectors in the null space of -1I - A

25. Matrix for T relative to standard basis

26. Matrix for T relative to columns of P

27. Basis of eigenvectors

28. eigenvalues