Domain Range Domain Range Domain Range Domain Range
![](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-1.jpg)
![Domain Range Domain Range](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-2.jpg)
![Domain Range Domain Range](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-3.jpg)
![Domain Range Domain Range](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-4.jpg)
![Domain Range Domain Range](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-5.jpg)
![definition: T is a linear transformation , EIGENVECTOR EIGENVALUE definition: T is a linear transformation , EIGENVECTOR EIGENVALUE](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-6.jpg)
![A is the matrix for a linear transformation T relative to the STANDARD BASIS A is the matrix for a linear transformation T relative to the STANDARD BASIS](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-7.jpg)
![](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-8.jpg)
![T T T T](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-9.jpg)
![The matrix for T relative to the basis T T The matrix for T relative to the basis T T](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-10.jpg)
![The matrix for T relative to the basis Diagonal matrix Eigenvectors for T T The matrix for T relative to the basis Diagonal matrix Eigenvectors for T T](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-11.jpg)
![The matrix for a linear transformation T relative to a basis of eigenvectors will The matrix for a linear transformation T relative to a basis of eigenvectors will](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-12.jpg)
![To find eigenvalues and eigenvectors for a given matrix A: Solve for and A To find eigenvalues and eigenvectors for a given matrix A: Solve for and A](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-13.jpg)
![To find eigenvalues and eigenvectors for a given matrix A: Solve for and Remember: To find eigenvalues and eigenvectors for a given matrix A: Solve for and Remember:](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-14.jpg)
![is a NONZERO vector in the null space of the matrix: ( The matrix is a NONZERO vector in the null space of the matrix: ( The matrix](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-15.jpg)
![](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-16.jpg)
![det ( I - A) = det ( I - A) =](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-17.jpg)
![det ( I - A) = det ( I - A) =](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-18.jpg)
![det ( I - A) = det This is called the characteristic polynomial det ( I - A) = det This is called the characteristic polynomial](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-19.jpg)
![det ( I - A) = det =0 the eigenvalues are 2 and -1 det ( I - A) = det =0 the eigenvalues are 2 and -1](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-20.jpg)
![( 2 I - A) = the eigenvectors belonging to 2 are nonzero vectors ( 2 I - A) = the eigenvectors belonging to 2 are nonzero vectors](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-21.jpg)
![( -1 I - A ) = the eigenvectors belonging to -1 are nonzero ( -1 I - A ) = the eigenvectors belonging to -1 are nonzero](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-22.jpg)
![](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-23.jpg)
![Matrix for T relative to standard basis Matrix for T relative to standard basis](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-24.jpg)
![Matrix for T relative to columns of P Matrix for T relative to columns of P](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-25.jpg)
![Basis of eigenvectors Basis of eigenvectors](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-26.jpg)
![eigenvalues eigenvalues](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-27.jpg)
- Slides: 27
![](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-1.jpg)
![Domain Range Domain Range](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-2.jpg)
Domain Range
![Domain Range Domain Range](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-3.jpg)
Domain Range
![Domain Range Domain Range](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-4.jpg)
Domain Range
![Domain Range Domain Range](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-5.jpg)
Domain Range
![definition T is a linear transformation EIGENVECTOR EIGENVALUE definition: T is a linear transformation , EIGENVECTOR EIGENVALUE](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-6.jpg)
definition: T is a linear transformation , EIGENVECTOR EIGENVALUE
![A is the matrix for a linear transformation T relative to the STANDARD BASIS A is the matrix for a linear transformation T relative to the STANDARD BASIS](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-7.jpg)
A is the matrix for a linear transformation T relative to the STANDARD BASIS
![](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-8.jpg)
![T T T T](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-9.jpg)
T T
![The matrix for T relative to the basis T T The matrix for T relative to the basis T T](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-10.jpg)
The matrix for T relative to the basis T T
![The matrix for T relative to the basis Diagonal matrix Eigenvectors for T T The matrix for T relative to the basis Diagonal matrix Eigenvectors for T T](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-11.jpg)
The matrix for T relative to the basis Diagonal matrix Eigenvectors for T T T
![The matrix for a linear transformation T relative to a basis of eigenvectors will The matrix for a linear transformation T relative to a basis of eigenvectors will](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-12.jpg)
The matrix for a linear transformation T relative to a basis of eigenvectors will be diagonal
![To find eigenvalues and eigenvectors for a given matrix A Solve for and A To find eigenvalues and eigenvectors for a given matrix A: Solve for and A](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-13.jpg)
To find eigenvalues and eigenvectors for a given matrix A: Solve for and A = I =( I - A)
![To find eigenvalues and eigenvectors for a given matrix A Solve for and Remember To find eigenvalues and eigenvectors for a given matrix A: Solve for and Remember:](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-14.jpg)
To find eigenvalues and eigenvectors for a given matrix A: Solve for and Remember: is a NONZERO vector in the null space of the matrix: ( =( I I - - A) A)
![is a NONZERO vector in the null space of the matrix The matrix is a NONZERO vector in the null space of the matrix: ( The matrix](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-15.jpg)
is a NONZERO vector in the null space of the matrix: ( The matrix ( I det I - A) A ) has a nonzero vector in its null space iff: ( I - A) = 0
![](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-16.jpg)
![det I A det ( I - A) =](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-17.jpg)
det ( I - A) =
![det I A det ( I - A) =](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-18.jpg)
det ( I - A) =
![det I A det This is called the characteristic polynomial det ( I - A) = det This is called the characteristic polynomial](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-19.jpg)
det ( I - A) = det This is called the characteristic polynomial
![det I A det 0 the eigenvalues are 2 and 1 det ( I - A) = det =0 the eigenvalues are 2 and -1](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-20.jpg)
det ( I - A) = det =0 the eigenvalues are 2 and -1
![2 I A the eigenvectors belonging to 2 are nonzero vectors ( 2 I - A) = the eigenvectors belonging to 2 are nonzero vectors](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-21.jpg)
( 2 I - A) = the eigenvectors belonging to 2 are nonzero vectors in the null space of 2 I - A =
![1 I A the eigenvectors belonging to 1 are nonzero ( -1 I - A ) = the eigenvectors belonging to -1 are nonzero](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-22.jpg)
( -1 I - A ) = the eigenvectors belonging to -1 are nonzero vectors in the null space of -1 I - A =
![](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-23.jpg)
![Matrix for T relative to standard basis Matrix for T relative to standard basis](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-24.jpg)
Matrix for T relative to standard basis
![Matrix for T relative to columns of P Matrix for T relative to columns of P](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-25.jpg)
Matrix for T relative to columns of P
![Basis of eigenvectors Basis of eigenvectors](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-26.jpg)
Basis of eigenvectors
![eigenvalues eigenvalues](https://slidetodoc.com/presentation_image_h2/d9da43a0de6ccaad3a670160a3f9a8ad/image-27.jpg)
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