Linear Combinations: Will Two Always Produce b=(0,1)?

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Homework Help Overview

The discussion revolves around the concept of linear combinations of vectors in the context of producing a specific vector, b=(0,1), using three vectors: u, v, and w. The participants explore the conditions under which such combinations exist and the implications of vector relationships in R^2.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster questions whether two different combinations of three vectors can produce the vector (0,1) and considers the implications of the vectors being non-parallel. Other participants inquire about the nature of the solutions, particularly the existence of infinite solutions when at least one combination is found.

Discussion Status

The discussion is active, with participants exploring the conditions under which combinations of vectors can yield the target vector. Some guidance has been offered regarding the relationship between the vectors and the potential for infinite solutions, though no consensus has been reached on the specifics.

Contextual Notes

Participants are working under the assumption that the vectors are not all parallel and are considering the implications of this assumption on the existence of solutions. The nature of the vector space R^2 is also a relevant factor in the discussion.

porschedude
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Will there always be two different combinations that produce b=(0,1) of three vectors: u, v, and w?

I'm pretty certain that the answer is no, but am I right in saying that with three vectors, assuming they are not all parallel, will always have at least one combination that produces (0,1)
 
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You've got that right. If you have three vectors in R^2 then either there are no combinations that produce (0,1) (if they are all parallel and not parallel to (0,1)) or there are an infinite number.
 
How can there be an infinite number?
 
porschedude said:
How can there be an infinite number?

Say u=(0,1), v=(1,0) and w=(1,1). There are an infinite number of solutions to the equation a*u+b*v+c*w=(0,1). You should be able to show that. It's maybe a little harder to show that if there is one solution, then there are an infinite number.
 

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