Linear Dependence/Independence

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Discussion Overview

The discussion revolves around the concept of linear dependence and independence in n-dimensional Euclidean space, exploring the conditions under which vectors can be considered linearly independent or dependent. It includes theoretical considerations and clarifications regarding the definitions and implications of vector independence in various dimensions.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants assert that for n-dimensional Euclidean space, at least n vectors are required for linear independence.
  • Others propose that an n-dimensional space can contain (n-1)-dimensional subspaces, which may lead to confusion about the independence of fewer vectors.
  • A participant provides examples from three-dimensional space, illustrating that one vector is independent, two vectors can be independent, and three vectors can be independent, but four vectors must be dependent.
  • Another participant acknowledges a misunderstanding regarding the phrase "at least" in the context of vector independence.
  • It is stated that having more than n vectors in Rn cannot result in independence, while having fewer than n can still allow for independence.

Areas of Agreement / Disagreement

Participants generally agree on the principle that more than n vectors cannot be independent in n-dimensional space, but there is some debate regarding the implications of subspaces and the interpretation of linear independence in relation to fewer vectors.

Contextual Notes

The discussion reflects varying interpretations of the definitions of linear independence and dependence, particularly concerning the relationship between n-dimensional spaces and their subspaces. There are unresolved nuances regarding the implications of the term "at least" in this context.

Gear300
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It is stated that for n-dimensional Euclidean space, n vectors are needed at least for linear independence. But if an n-dimensional Euclidean space also includes (n-1)-dimensional Euclidean space, then why can't it also include a family of n-1 linearly independent vectors?
 
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Gear300 said:
It is stated that for n-dimensional Euclidean space, n vectors are needed at least for linear independence. But if an n-dimensional Euclidean space also includes (n-1)-dimensional Euclidean space, then why can't it also include a family of n-1 linearly independent vectors?

It can. For example, in three dimensional Euclidean space the single vector (1,0,0) is linearly independent, as are the two vectors {(1,0,0), (0,1,0)} and the three vectors {(1,0,0), (0,1,0), (0,0,1)}. However, any four vectors in this space will necessarily be linearly dependent. Does that help?
 
I get what you're saying. I mixed stuff up (thought too hard about the words "at least"). Thanks.
 
More than n vectors cannot be independent in Rn. Less can be.
 

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