Math Problem Statement
prove the farkas lemma
Solution
Farkas' Lemma is a fundamental result in linear algebra and optimization. It provides a condition for the solvability of a system of linear inequalities. The lemma states:
Farkas' Lemma: Given a matrix and a vector , exactly one of the following two statements is true:
- There exists a vector such that and (component-wise).
- There exists a vector such that and .
Proof of Farkas' Lemma:
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Suppose (1) is true:
- If there exists such that , we need to show that (2) cannot be true.
- Assume is such that and .
- Then, , since .
- However, since and , , contradicting the assumption that .
- Therefore, if (1) holds, (2) must be false.
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Suppose (1) is false:
- This means there is no such that .
- We aim to prove that (2) must then be true, which would mean that there exists such that and .
- Consider the set . This is a convex cone in .
- Since , by the separating hyperplane theorem, there exists a hyperplane defined by a non-zero vector that strictly separates from .
- Specifically, satisfies for all , meaning , and .
- This establishes (2).
Thus, either exists satisfying the first condition, or exists satisfying the second condition, but not both.
Would you like to explore this proof further or have any questions?
Related Questions
- How does Farkas' Lemma relate to the theory of linear programming?
- Can you provide an example where Farkas' Lemma is applied?
- What is the geometric interpretation of Farkas' Lemma?
- How does Farkas' Lemma connect with the Duality Theorem in linear programming?
- What are some generalizations of Farkas' Lemma?
Tip:
Understanding the geometric interpretation of Farkas' Lemma can provide deeper insights into linear programming and optimization problems.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Optimization
Formulas
-
Theorems
Farkas' Lemma
Suitable Grade Level
Advanced Undergraduate
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