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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

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

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  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

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