Linear Algebra: Proving or Disproving Span Equivalence

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SUMMARY

The discussion centers on the linear algebra concept of span equivalence, specifically addressing the assertion that if span(S) = span(P), then S must equal P. The user provided a counterexample with sets S={v} where v=(1,2) and P={u} where u=(2,4), demonstrating that both spans are equivalent (a line through the origin parallel to vector (1,2)), yet S does not equal P. This confirms that the original statement is false, illustrating a fundamental property of vector spans.

PREREQUISITES
  • Understanding of vector spaces and spans in linear algebra
  • Familiarity with the concept of linear independence
  • Knowledge of counterexamples in mathematical proofs
  • Basic proficiency in manipulating vectors and their representations
NEXT STEPS
  • Study the properties of vector spans in linear algebra
  • Learn about linear independence and dependence of vectors
  • Explore more counterexamples in linear algebra to strengthen proof skills
  • Investigate the implications of span equivalence in higher-dimensional spaces
USEFUL FOR

Students studying linear algebra, educators teaching vector space concepts, and anyone interested in mathematical proofs and counterexamples.

*best&sweetest*
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I have just had linear algebra exam, and one of the questions was to prove or disprove (give a counterexample) that if span(S) = span(P) then S=P.

I gave this example:

S={v} where vector v=(1,2), and P={u} where u= (2,4).
Then, span(S) = span (P) (line through origin parallel to vector (1,2)),
but S does not equal P.

Am I right?
Thank you!
 
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Yes, that's perfectly correct. (And it was kind of a silly question!)
 
Thank you! I spent 15 minutes proving that, and then decided to find a counterexample. I was confused because it seemed so easy once I got the idea. Thanks again!
 

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