prove given an m times n matrix a and a vector b in mathbbrm either i there exists a vector x in

Name: prove given an m times n matrix a and a vector b in mathbbrm either i there exists a vector x in mathbbrn with x geq 0 and ax b or ii there exists a vector y in mathbbrm with ytop a geq math 75564
Uploaded: 2024-12-23T23:36:24-08:00
Duration: 4 min 10 s
Channel: Khushbu Rani
Description: prove given an m times n matrix a and a vector b in mathbbrm either i there exists a vector x in mathbbrn with x geq 0 and ax b or ii there exists a vector y in mathbbrm with ytop a geq math 75564

Question

Khushbu Rani · Accepted Answer

Step 1: This is a statement of Farkas&#x27; Lemma, specifically the version for non-negative solution

Question

Prove: Given an $m \times n$ matrix $A$ and a vector $b \in \mathbb{R}^m$, EITHER (i) there exists a vector $x \in \mathbb{R}^n$ with $x \geq 0$ and $Ax = b$, OR (ii) there exists a vector $y \in \mathbb{R}^m$ with $y^\top A \geq \mathbf{0}$ and $y^\top b < 0$, NOT BOTH.

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