Proving the Existence of a Vector for a Matrix with Linearly Independent Rows

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Homework Help Overview

The problem involves a matrix A with linearly independent rows and asks to prove the existence of a vector p such that Ap equals a specific vector e_1. The context is within an Optimization class.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of the problem statement, with one questioning the completeness of the problem and another suggesting a connection to the surjectivity of linear maps.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and clarifying the nature of the matrix A. Some guidance regarding the properties of linear maps has been suggested, though no consensus has been reached.

Contextual Notes

There is a mention of the zero matrix as a potential counterexample, which raises questions about the assumptions underlying the problem. The original poster notes that the problem is from an Optimization class, which may influence the interpretation of the matrix properties.

retspool
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So i have a problem in front of me


Let A be a m x n matrix whose rows are linearly independent. Prove that there exists a vector p such taht Ap = e_1 where e_1 =( 1, 0 , 0, 0, 0,0 ,0 ... 0)T



i don't even know where to begin
 
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Is that the complete problem statement? As it is, it's easy to find a counterexample, e.g., the zero matrix.
 


Yeah that is the complete problem statement.

but it is a question from my Optimization class and not Linear Algebra

I don't understand how the zero matrix will fit the bill
 


Sorry, I misread "independent" as "dependent." Never mind.
 


Maybe you know something about surjectivity of linear maps and the properties of the matrices that represent them?
Since you don't have detailed information about A, you can't explicitly find p, but also it doesn't really matter whether it is e_1 or any other vector of the right dimension.
 

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