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Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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4. Let A be a 7 × 5 matrix with rank equal to 4 and let b be a vector in R. The four fundamental sub- spaces associated with A are R(A), N(A"), R(A"), and N(A). (a) What is the dimension of N(A'), and which of the other fundamental subspaces is the ortho- gonal complement of N(A")? (b) If x is a vector in R(A) and A'x = 0, then what can you conclude about the value of ||x||? Explain. (c) What is the dimension of N(A'A)? How many solutions will the least squares system Ax = b have? Explain. S
Transcribed Image Text:4. Let A be a 7 × 5 matrix with rank equal to 4 and let b be a vector in R. The four fundamental sub- spaces associated with A are R(A), N(A"), R(A"), and N(A). (a) What is the dimension of N(A'), and which of the other fundamental subspaces is the ortho- gonal complement of N(A")? (b) If x is a vector in R(A) and A'x = 0, then what can you conclude about the value of ||x||? Explain. (c) What is the dimension of N(A'A)? How many solutions will the least squares system Ax = b have? Explain. S
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