Linear Independence. express each vector as a lin. combo

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Homework Help Overview

The discussion revolves around the concept of linear independence and linear combinations in the context of vectors in R4. Participants are tasked with expressing each vector as a linear combination of the others after establishing that the set is linearly dependent.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss how to express each vector as a linear combination of the others, questioning the starting point for this task. There is an exploration of the definition of linear combinations and how to apply it to the given vectors.

Discussion Status

Some participants have provided guidance on how to approach the problem, suggesting that one can express each vector in terms of the others using constants. Multiple interpretations of the task are being explored, with a focus on finding the appropriate constants for each vector's representation.

Contextual Notes

Participants are working under the assumption that the vectors are linearly dependent, which influences their approach to expressing them as combinations of one another. There is also mention of adhering to a template for posting questions, indicating a structured approach to the discussion.

burton95
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V1 = (1,2,3,4) V2 = (0,1,0,-1) V3 = (1,3,3,3)

a) I already expressed them a linearly dependent set in R4

b) Express each vector in part (a) as a linear combination of the other two

linear combo is just {c1v1 + c2v2...cnvn} right? But I don't get where to start to prove this
 
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burton95 said:
V1 = (1,2,3,4) V2 = (0,1,0,-1) V3 = (1,3,3,3)

a) I already expressed them a linearly dependent set in R4

b) Express each vector in part (a) as a linear combination of the other two

linear combo is just {c1v1 + c2v2...cnvn} right? But I don't get where to start to prove this

There is nothing to prove; that is just a *definition* of "linear combination". You are being asked to represent v1 as a linear combination of v2, v3, and so forth. To start: just write things down in detail: figure out what are the components of c2v2 + c3v3 for constants c2 and c3. How can that combination be equal to v1?
 
so are you saying take v1 = c2 (v2) + c3 (v3) using the constants that I found through proving that they are linearly dependent?
 
What Ray is saying has three parts.
1) Find constants c2 and c3 for which v1 = c2v2 + c3v3
2) Find constants c1 and c3 for which v2 = c1v1 + c3v3
3) Find constants c1 and c2 for which v3 = c1v1 + c2v2
 
BTW, when you post a question, do not throw away the three parts of the template. They are there for a reason.
 
Thx. I will leave the template. So I am left with 3 different equations in the form of vn = cx(vx) + cy(vy) as my final answer?
 
burton95 said:
so are you saying take v1 = c2 (v2) + c3 (v3) using the constants that I found through proving that they are linearly dependent?

That would be one way of doing it.
 

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