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Prove that for any m x n matrix A and vector b = Rm, the vector AT6 belongs to the image of matrix AT A. (This would prove that the normal equation AT Ax = ATb always has a solution, and, therefore, a least squares solution of Ax = b always exists.) Hint: use the result ker(ATA) = ker(A), and the Fundamental Theorem of Linear Algebra.

Prove that for any m x n matrix A and vector b = Rm, the vector AT6 belongs to the image of matrix AT A. (This would prove that the normal equation AT Ax = ATb always has a solution, and, therefore, a least squares solution of Ax = b always exists.) Hint: use the result ker(ATA) = ker(A), and the Fundamental Theorem of Linear Algebra.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove that for any m × n matrix A and vector b = Rm, the vector AT6 belongs to the image of matrix AT A. (This would prove that the normal equation AT Ax ATb always has a solution, and, therefore, a least squares solution of Ax = b always exists.) Hint: use the result ker(ATA) = ker(A), and the Fundamental Theorem of Linear Algebra. =
Transcribed Image Text:Prove that for any m × n matrix A and vector b = Rm, the vector AT6 belongs to the image of matrix AT A. (This would prove that the normal equation AT Ax ATb always has a solution, and, therefore, a least squares solution of Ax = b always exists.) Hint: use the result ker(ATA) = ker(A), and the Fundamental Theorem of Linear Algebra. =
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