Basis is finite set of vectors that are linearly independant

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SUMMARY

The basis of a vector space is defined as a finite set of vectors that are linearly independent and span the vector space. In the context of row space, the basis is represented as the set {u1, u2, ..., um}, not as span{u1, u2, ..., um}. This distinction is crucial because while the span of a set of vectors forms a vector space, a basis itself is merely the set of vectors, which is essential for understanding linear algebra concepts accurately.

PREREQUISITES
  • Understanding of linear independence
  • Familiarity with vector spaces
  • Knowledge of the concept of spanning sets
  • Basic principles of linear algebra
NEXT STEPS
  • Study the properties of linear independence in vector spaces
  • Explore the relationship between bases and dimension in linear algebra
  • Learn about the row space and column space in matrix theory
  • Investigate the implications of spanning sets in vector space theory
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Students of linear algebra, educators teaching vector space concepts, and anyone seeking to clarify the distinction between a basis and the span of a set of vectors.

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I know that the basis is finite set of vectors that are linearly independent and SPANS that set. But why is when you find the basis for the row space for example the answer is {[u1,u2,...,um}} and not span{[u1,u2,...,um}]. I got this wrong in a test. I don't see why eventhough the definition states that it spans that set.
 
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If {u1, u2, ..., um} spans the vector space V, then span{u1, u2, ..., um} = V, right? And V is obviously not a basis for V.
 


A basis is a set of vectors. The span of a set of vectors is a vector space, not a basis.
 

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