True/False Question on Linear Combos

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Homework Help Overview

The discussion revolves around a linear algebra problem concerning the existence of solutions to the equation Ax=b, given two known solutions, x1 and x2. Participants are exploring the implications of having multiple solutions and the conditions under which additional solutions may exist.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the possibility of finding another solution based on the linear combination of known solutions. There is uncertainty about whether this applies to nonhomogeneous equations and whether specific conditions, such as the nature of vector b, affect the uniqueness of solutions.

Discussion Status

The discussion is active, with participants questioning the assumptions related to the problem, such as the nature of the solutions and the implications of linear combinations. Some guidance is offered regarding the context of homogeneous versus nonhomogeneous equations, but no consensus has been reached.

Contextual Notes

There is a mention of the condition that vector b may not be the zero vector, which raises questions about the uniqueness of solutions and the rank of the matrix involved.

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Homework Statement


If the system Ax=b has two known solutions, x1 and x2, true or false that another solution exists. If it does, find another solution.

Homework Equations


The Attempt at a Solution


I think it's true. But after that, I don't know.

I'm really lost, but I think I remember learning that given two solutions, any linear combination of the two is also a solution. Does that mean that x1+x2 is a solution?

I'm worried that that is actually a solution to Ax=2b, though.
 
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Perhaps that is only true for homogeneous equations, now that I think about it.

Would nonhomogeneous equations require some sort of weighted average linear combination?
 
NullSpace0 said:

Homework Statement


If the system Ax=b has two known solutions, x1 and x2, true or false that another solution exists. If it does, find another solution.

Homework Equations



The Attempt at a Solution


I think it's true. But after that, I don't know.

I'm really lost, but I think I remember learning that given two solutions, any linear combination of the two is also a solution. Does that mean that x1+x2 is a solution?

I'm worried that that is actually a solution to Ax=2b, though.
Is it true that (1/2)Ax = A((1/2)x) ?
 
NullSpace0 said:

Homework Statement


If the system Ax=b has two known solutions, x1 and x2, true or false that another solution exists. If it does, find another solution.

.

Is it given b is not the zero vector? Think about uniqueness, rank, and free variables.
 

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