Solution of a Ax=b exists iff b is in CS (A) ?

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Homework Help Overview

The discussion revolves around the conditions under which the equation Ax = b has a solution, specifically focusing on the relationship between b and the column space of matrix A.

Discussion Character

  • Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants explore the definition of the column space and its implications for the existence of solutions to the equation Ax = b. There is an attempt to break down the problem into two parts: one assuming a solution exists and the other assuming b is in the column space.

Discussion Status

The discussion is actively exploring foundational concepts and definitions related to the column space and its role in determining the solvability of the equation. Some guidance has been provided on how to approach the problem from both directions, but no consensus or resolution has been reached.

Contextual Notes

Participants are working with the assumption that a clear understanding of the column space is necessary to address the problem, and there may be varying interpretations of how to apply this concept to the equation Ax = b.

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Homework Statement



Show that Ax = b has a solution if and only if b is in CS(A).

Homework Equations





The Attempt at a Solution



Ax = b
b \in CS(A) means
d1A1 + d2A2+ ...+ dnAn = b

and I am lost
 
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Let's go back to basics. What is the definition of Column space?
 
the subspace of Rn spanned by the column vectors of A
 
Yep, let's break this down into parts:

1)Assume Ax = b has a solution, then b can be written as a linear combination with the vectors from the columns of A. Since the span is the linear combination of the vectors a1 a2 a3 ... an then b is in the span. So then use your definition.

2)Now work the other way. Assume b is in Col A, and work towards showing that Ax = b has a solution because of that.
 

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