Understanding Linear Independence: Proving Non-Zero Status of One-Element Sets

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Discussion Overview

The discussion centers on the concept of linear independence, specifically regarding one-element sets in vector spaces. Participants explore the conditions under which a one-element set is considered linearly independent and the proof required to establish this status.

Discussion Character

  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant questions whether a one-element set is linearly independent only when it is non-zero.
  • Another participant suggests that to prove linear independence, one must show that if a linear combination of the one element equals zero, then all coefficients must be zero.
  • A participant expresses confusion about creating a linear combination with a one-element set and seeks clarification on the concept.
  • Further clarification is provided that a linear combination of one element is simply a scalar multiple of that element.
  • One participant acknowledges confusion between linear combinations and linear dependency, indicating a shift in understanding after receiving explanations.

Areas of Agreement / Disagreement

Participants generally agree on the condition for linear independence of one-element sets, but there is some confusion regarding the definitions and applications of linear combinations and linear dependency.

Contextual Notes

There are limitations in the discussion regarding the definitions of linear combinations and linear dependency, as well as the assumptions made about the participants' prior knowledge of linear algebra concepts.

Who May Find This Useful

This discussion may be useful for students beginning their studies in linear algebra, particularly those grappling with the concepts of linear independence and linear combinations.

EvLer
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Hi all,
when is a one-element set is linearly independent? Just when it's non-zero?
I am not sure how to prove this on one element set.

Thanks in advance.
 
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(Assuming you speak of a vector space)

You're right on the condition. To prove it, you need to show that if any linear combination of your one element is zero, then all the coefficients are zero.
 
Hurkyl said:
To prove it, you need to show that if any linear combination of your one element is zero, then all the coefficients are zero.
Ok, is that what you mean?
S = {A},
Code: 
A = 3 3 3
    3 3 3
A = cA?
I do not know how to create a linear combination on one element, it doesn't make sense to me. Maybe because I started Linear Algebra course two days ago.

Thanks.
 
Last edited:
A linear combination of one element is just some scalar multiple of it. What is your definition of linear combination? Can't you apply it to a one element set? And what is A, it looks like an array? It is a simple exercise that if v is a non-zero vector, and tv=0 for some t in the underlying field then t=0. Which is what they're asking you to prove.
 
I think I got confused linear combination and linear dependency. Now, I see the difference. Thank you for the explanations.
 

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