Example Given a matrix defining a linear mapping

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 ( ie: ) Note: This vector is in the null space of T The vectors in the null space are the solutions to

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

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

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

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:

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:

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:

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

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

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 + =

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 =

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 columns of M. These 2 vectors still span the range and they are independent.

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 columns of M. { , } A basis for the range of T

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 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

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

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