Proving Equality of Spans of Vectors in Rn

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Homework Help Overview

The discussion revolves around proving the equality of spans of vectors in Rn, specifically comparing the spans of the sets {u, v} and {u+v, u-v}. Additionally, there is a second problem concerning the relationship between two finite subsets S1 and S2 in Rn, focusing on the definition of span.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore how to demonstrate that a vector in the span of {u+v, u-v} can be expressed as a linear combination of {u, v}. There is also discussion about the reverse direction of this proof. For the second problem, participants question the clarity of the problem statement and seek confirmation on its interpretation.

Discussion Status

Some participants have provided initial thoughts on how to approach the problems, particularly focusing on rewriting linear combinations. There is acknowledgment of uncertainty regarding the next steps in the reasoning process, especially for the first problem. The second problem's statement has been confirmed, but further exploration is needed.

Contextual Notes

Participants note a potential lack of clarity in the problem statement for the second question, which may affect their approach. There is also an emphasis on using only the definition of span in the context of the problems presented.

mat175
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Homework Statement



1. let u and v be any vectors in Rn. Prove that the spans of {u,v,} and {u+v, u-v} are equal.

2. Let S1 and S2 be finite subsets of Rn such that S1 is contained in S2. Use only the definition of span s1 is contained in span s2.

Homework Equations


The Attempt at a Solution



1. w in the span (u+v, u-v) show that w is in the span (u,v)
w is a linear combination of (u+v, u-v)
w= c1 (u+v) + c2(u-v)------
w=(c1+c2)u+(c1-c2)v

How do you prove it going the other way though?


2. I am not sure how to start this one, other than having the definition of span there.
 
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For 1, start with a vector w that is in the span of {u, v}, which means that w = c1u + c2v. See if you can get creative on the two constants to rewrite them as the sum of two numbers and the difference of two numbers, respectively. If haven't done this, but that's the direction I would take.

For 2, you are missing some words in the problem statement. I'm guessing that this is the actual problem statement: "Use only the definition of span to show that span s1 is contained in span s2." Please confirm that this is the correct interpretation.
 
Yes that is correct.

For 1, I did have what you have then I don't know for sure where to go from there.

Thanks.
 
mat175 said:
Yes that is correct.

For 1, I did have what you have then I don't know for sure where to go from there.

Thanks.
Show me what you've done for the 2nd half of this problem.
 
All I have is let y be a linear combination of (u,v)
which is written y=c1u1+c2u2
 
mat175 said:
All I have is let y be a linear combination of (u,v)
which is written y=c1u1+c2u2
Well, no. It would be y = c1u + c2v, right? You want y to be a linear combination of u and v, not u1 and u2.

Starting with y as I have written it, you want to end up with y (the same y) as a linear combination of u + v and u - v, right? What does that look like?

Now you know what you're starting with, and you sort of know what you want to end up with. Can you work backward from the end to the beginning and fill in the middle?
 

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