Linearly Independent Vectors: Same Plane?

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SUMMARY

Three vectors in V^3(O) are linearly independent if and only if they are not on the same plane. Conversely, three vectors are linearly dependent if and only if they are on the same plane. This relationship is definitive and is expressed through the term "if and only if" (iff), indicating a biconditional relationship between the independence and dependence of vectors based on their spatial arrangement.

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  • Understanding of linear algebra concepts
  • Familiarity with vector spaces, specifically V^3(O)
  • Knowledge of linear independence and dependence
  • Basic grasp of geometric interpretations of vectors
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Three vectors are linearly independent iff The vectors are not on the same plane. Three vectors are linearly dependent ⇔ The vectors are on the same plane. Is this true.
 
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kingyof2thejring said:
Three vectors are linearly independent iff The vectors are not on the same plane. Three vectors are linearly dependent ⇔ The vectors are on the same plane. Is this true.

Yes, it is true, if you are talking about vectors in V^3(O), i.e. radius vectors.
 
kingyof2thejring said:
Three vectors are linearly independent iff The vectors are not on the same plane. Three vectors are linearly dependent ⇔ The vectors are on the same plane. Is this true.
That's what "if and only if" (iff) means! "Three vectors are linearly independent iff The vectors are not on the same plane" means two things:
If the vectors are not on the same plane then the vectors are linearly independent" and "Three vectors are linearly independent only if the vectors are not on the same" which is the same as "if three vectors are linearly independent, then they are not on the same plane", exactly the statement you are asking about.
 

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