Proving Nonzero Vector Intersection in 3D Subspaces of R5

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SUMMARY

In the discussion, participants address the problem of proving that two three-dimensional subspaces, V and W, of R5 must share a nonzero vector. The subspaces are represented with specific vectors, V = { (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0) } and W = { (0, 0, 0, 1, 0), (0, 0, 0, 0, 1), (1, 1, 0, 0, 0) }. The discussion emphasizes the importance of understanding linear independence and the dimensionality of vector spaces, leading to the conclusion that the intersection of V and W must contain at least one nonzero vector due to the properties of linear combinations in R5.

PREREQUISITES
  • Understanding of vector spaces in linear algebra
  • Knowledge of linear independence and basis vectors
  • Familiarity with the concepts of dimensionality in R5
  • Ability to perform vector operations and linear combinations
NEXT STEPS
  • Study the properties of linear independence in vector spaces
  • Learn about the intersection of subspaces in linear algebra
  • Explore the concept of basis vectors in R5
  • Investigate the implications of the dimension theorem for vector spaces
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Students and educators in linear algebra, mathematicians interested in vector space theory, and anyone seeking to understand the intersection properties of subspaces in higher-dimensional spaces.

RyanV
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Homework Statement


Prove that if V and W are three dimensional subspaces of R5, then V and W must have a nonzero vector in common.


Homework Equations


NA


The Attempt at a Solution


I've attempted to set up the problem by writing out,

V = { (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0) }
W = { (0, 0, 0, 1, 0), (0, 0, 0, 0, 1), (1, 1, 0, 0, 0) }

After that, I'm lost.

I don't really like vector spaces because I don't understand it very well. So could whoever explain please explain thoroughly? =P I would help a lot because I want to know what's going on! hehe, thanks in advance =)
 
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rather than narrowing in on a single case, think aobut linear independence... what is the maximal number of linearly independent vectors, in any subspace of R^3?

similarly, how many vectors are in a the basis for R^5?
 
Last edited:

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