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F. Let a = (a1, a2, .., an) E R" where ||a|| +0 and let b e R. Let SC R" where S = {(x1, x2, .., xn) E R" : a1x1+a2X2+... + aŋXn = b} Show that the shortest distance from 0 = (0, 0, 0, .., 0) e R" to the set S is %3D Hint: Let x = (x1, x2, ..., xn) E S. We wan to minimize ||x – ở|| subject to A1X1+a2X2+... + aŋXn = b Let f(x1, X2, .., "n) = }|(x1, x2, .., rn)|l² and g(x1, x2, .., m) = a1x1+a2X2+...+a„Xn-b ..., In and apply Lagrange Multiplier.

F. Let a = (a1, a2, .., an) E R" where ||a|| +0 and let b e R. Let SC R" where S = {(x1, x2, .., xn) E R" : a1x1+a2X2+... + aŋXn = b} Show that the shortest distance from 0 = (0, 0, 0, .., 0) e R" to the set S is %3D Hint: Let x = (x1, x2, ..., xn) E S. We wan to minimize ||x – ở|| subject to A1X1+a2X2+... + aŋXn = b Let f(x1, X2, .., "n) = }|(x1, x2, .., rn)|l² and g(x1, x2, .., m) = a1x1+a2X2+...+a„Xn-b ..., In and apply Lagrange Multiplier.

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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F. Let a = (a1, a2, ..., an) e R" where ||a|| 7 0 and let b e R. Let S C R" where S = : {(x1, x2, ..., xn) E R" : a1x1 + a2x2 + ... + anXn = b} Show that the shortest distance from 0 = (0, 0, 0, .., 0) E R" to the set S is %3D ||| Hint: Let x = : (x1, x2, ..., xn) E S. We wan to minimize ||x – ở|| subject to ɑ1X1+ a2x2+ ... + aŋx, = b Let f(x1, x2, ..., xn) = }||(x1, x2, .. xn)||² and g(x1, x2, ..., "n) = a1x1+a2x2+...+a„Xn=b %3D and apply Lagrange Multiplier.
Transcribed Image Text:F. Let a = (a1, a2, ..., an) e R" where ||a|| 7 0 and let b e R. Let S C R" where S = : {(x1, x2, ..., xn) E R" : a1x1 + a2x2 + ... + anXn = b} Show that the shortest distance from 0 = (0, 0, 0, .., 0) E R" to the set S is %3D ||| Hint: Let x = : (x1, x2, ..., xn) E S. We wan to minimize ||x – ở|| subject to ɑ1X1+ a2x2+ ... + aŋx, = b Let f(x1, x2, ..., xn) = }||(x1, x2, .. xn)||² and g(x1, x2, ..., "n) = a1x1+a2x2+...+a„Xn=b %3D and apply Lagrange Multiplier.
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