Slides_2010_01_25

# Slides_2010_01_25 - Applied linear algebra and numerical...

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Applied linear algebra and numerical analysis Session 9 Prof. Ulrich Hetmaniuk Department of Applied Mathematics January 25, 2010

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Range of a Matrix Consider a matrix A R m × n Deﬁnition The range is the subspace of all vectors “produced” by the matrix A R ( A ) = { Ax | ∀ x R n } = span ( a ( 1 ) , ··· , a ( n ) ) R m (1) The dimension of the range of A is called the rank of A : rank ( A ) = dim ( R ( A )) . (2) rank ( A ) min ( m , n ) ; Among the n columns of A , there exists at least one group of rank ( A ) columns that are linearly independent.
Examples Compute the rank of the matrices A = 2 6 - 5 3 1 2 A = 3 0 - 1 2 - 8 2 - 1 - 2 1 A = 1 1 1 1 1 1 1 1 1

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Question 1 1 1 2 - 1 5 - 1 0 - 2 = 1 0 2 0 1 - 1 0 0 0 ?
A = 1 1 1 2 - 1 5 - 1 0 - 2 = 1 0 2 0 1 - 1 0 0 0 ? NO rank 1 0 2 0 1 - 1 0 0 0 = 2 ... then rank ( A ) = 2 ... Why? Explain. Give details.

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Slides_2010_01_25 - Applied linear algebra and numerical...

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