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2. (Fredholm Alternative) Let A € Rmxn and b = Rm. Show that (I) either the system Ax = b has a solution or (exclusive) (II) there exists y such that y¹A = 0, y¹b ‡ 0. Note: "exclusive or” means either I holds or II holds, but not both. So, you need to show that both I and II cannot hold simultaneously.

2. (Fredholm Alternative) Let A € Rmxn and b = Rm. Show that (I) either the system Ax = b has a solution or (exclusive) (II) there exists y such that y¹A = 0, y¹b ‡ 0. Note: "exclusive or” means either I holds or II holds, but not both. So, you need to show that both I and II cannot hold simultaneously.

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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Matrix Analysis practice question, please show clear thanks.
2. (Fredholm Alternative) Let A € Rmxn and b € Rm. Show that (I) either the system Ax = b has a solution or (exclusive) (II) there exists y such that y¹A = 0, y¹b ‡ 0. Note: "exclusive or" means either I holds or II holds, but not both. So, you need to show that both I and II cannot hold simultaneously. Hint 1. Assume that I does not hold and show that II holds. Hint 2: Make use of the identity: Rm = R(A) + N(AT).
Transcribed Image Text:2. (Fredholm Alternative) Let A € Rmxn and b € Rm. Show that (I) either the system Ax = b has a solution or (exclusive) (II) there exists y such that y¹A = 0, y¹b ‡ 0. Note: "exclusive or" means either I holds or II holds, but not both. So, you need to show that both I and II cannot hold simultaneously. Hint 1. Assume that I does not hold and show that II holds. Hint 2: Make use of the identity: Rm = R(A) + N(AT).
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