1 / 18

Example: Given a matrix defining a linear mapping Find a basis for the null space and a basis for the range

Example: Given a matrix defining a linear mapping Find a basis for the null space and a basis for the range. Pamela Leutwyler. Let M be the matrix for the linear mapping T ( ie: ). Let M be the matrix for the linear mapping T

riordan
Download Presentation

Example: Given a matrix defining a linear mapping Find a basis for the null space and a basis for the range

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Example: Given a matrix defining a linear mapping Find a basis for the null space and a basis for the range Pamela Leutwyler

  2. Let M be the matrix for the linear mapping T ( ie: )

  3. Let M be the matrix for the linear mapping T ( ie: ) Note: This vector is in the null space of T The vectors in the null space are the solutions to

  4. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the null space of T you must solve:

  5. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the null space of T you must solve:

  6. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the null space of T you must solve:

  7. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the null space of T you must solve: Every vector in the null space looks like:

  8. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the null space of T you must solve: A basis for the null space = Every vector in the null space looks like:

  9. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the range of T, remember that the columns of M span its range . Every vector in the range looks like:

  10. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the range of T, remember that the columns of M span its range . Every vector in the range looks like: a linear combination of the columns of M

  11. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the range of T, remember that the columns of M span its range . This is not a basis because the vectors are not independent

  12. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the range of T, remember that the columns of M span its range . This is not a basis because the vectors are not independent + =

  13. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the range of T, remember that the columns of M span its range . This is not a basis because the vectors are not independent =

  14. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the range of T, remember that the columns of M span its range . Find the largest INDEPENDENT subset of the set of the columns of M . These 2 vectors still span the range and they are independent.

  15. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the range of T, remember that the columns of M span its range . Find the largest INDEPENDENT subset of the set of the columns of M . { , } A basis for the range of T

  16. Let M be the matrix for the linear mapping T ( ie: ) To find a basis for the range of T, remember that the columns of M span its range . Find the largest INDEPENDENT subset of the set of the columns of M . Hint: wherever you see a FNZE in the reduced echelon form of the matrix, choose the original column of the matrix to include in your basis for the range. 1 { , } 1 the reduced echelon form of M (see slide 7 ) A basis for the range of T

  17. Let M be the matrix for the linear mapping T ( ie: ) A basis for the null space = A basis for the range = the dimension of the null space = 2 the dimension of the range = 2

  18. Let M be the matrix for the linear mapping T ( ie: ) A basis for the null space = A basis for the range = The domain = R4 The domain = R4 the dimension of the null space = 2 the dimension of the null range = 2 the dimension of the domain = 4 The dimension of the null space + the dimension of the range =the dimension of the domain

More Related