Can a Set of 5 Vectors Span All of R6?

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SUMMARY

A set of 5 vectors cannot span R6 because the dimension of R6 is 6, which requires at least 6 linearly independent vectors to span the space. The discussion highlights the importance of understanding the relationship between the number of vectors and the dimension of the space they inhabit. Specifically, for any vector space of dimension n, a spanning set must contain at least n vectors. Therefore, to span R6, one needs a minimum of 6 vectors.

PREREQUISITES
  • Understanding of vector spaces and their dimensions
  • Knowledge of linear independence and spanning sets
  • Familiarity with Rn notation
  • Basic concepts of linear algebra
NEXT STEPS
  • Study the concept of linear independence in vector spaces
  • Learn about the dimension of vector spaces and its implications
  • Explore examples of spanning sets in Rn
  • Review the definitions and properties of vector spaces in linear algebra
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Students of linear algebra, educators teaching vector space theory, and anyone seeking to deepen their understanding of dimensions and spanning sets in mathematics.

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1. Can a set of 5 vectors in R6 span all of R6?I want to say that it does span, because i remember my teacher saying "don't think of the physical world," but I'm not entirely sure if it does.
 
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Neither of those are very good reasons for answering one way or another. Why do you think 5 vectors can span R^6?
 
Dick said:
Neither of those are very good reasons for answering one way or another. Why do you think 5 vectors can span R^6?


I'm not sure, to be honest.
 
How many vectors span R2? R3?
 
venom192 said:
How many vectors span R2? R3?
Um, I'm guessing that vectors span R2 and 3 Vectors span R3.
so... that must mean that 6 vectors span R6.
 
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Why are you doing all of this guessing? It looks to me like you need to review the basic definitions. You should know immediately that R6 has dimension 6. What does that mean. What can you say about any spanning set in a space of dimension n? What can you say about any linearly independent set in a space of dimension n?
 

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