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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 Pamela Leutwyler

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

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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: )

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

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

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

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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:

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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: A basis for the null space =

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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:

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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

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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

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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 +=

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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 =

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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. These 2 vectors still span the range and they are independent. Find the largest INDEPENDENT subset of the set of the columns of M.

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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. A basis for the range of T Find the largest INDEPENDENT subset of the set of the columns of M. {, }

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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. A basis for the range of T 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. the reduced echelon form of M (see slide 7 ) 1 1

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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

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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 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

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4 © 2012 Pearson Education, Inc. Vector Spaces 4.4 COORDINATE SYSTEMS.

4 © 2012 Pearson Education, Inc. Vector Spaces 4.4 COORDINATE SYSTEMS.

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