Question on Remark in Axler's Linear Algebra

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SUMMARY

The discussion centers on the criteria for a vector subspace as introduced by Axler in his linear algebra text. Specifically, it addresses the example of the vector space defined by (x1, x2, x3, x4) in F^4, where x3 = x4 + b. The consensus is that this example does not constitute a subspace unless b = 0, as the zero vector must be included in the subspace. The realization that the zero vector's inclusion necessitates b = 0 is confirmed as a correct understanding of the subspace criteria.

PREREQUISITES
  • Understanding of vector spaces and subspaces
  • Familiarity with the concepts of closure under addition and scalar multiplication
  • Basic knowledge of linear algebra terminology
  • Experience with examples in linear algebra texts, particularly Axler's work
NEXT STEPS
  • Study the properties of vector subspaces in detail
  • Explore examples of vector spaces in Axler's "Linear Algebra Done Right"
  • Learn about the implications of the zero vector in vector space theory
  • Investigate the concept of linear combinations and their role in defining subspaces
USEFUL FOR

Students of linear algebra, educators teaching vector space concepts, and anyone seeking to deepen their understanding of subspace criteria as outlined in Axler's "Linear Algebra Done Right".

Group_Complex
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Hello, i am studying vector subspacess and Axler introduces the two criteria for a vector subspace (closure under addition and scalar multiplication).
He then proceeds to give an example; (x1,x2,x3,x4) belonging to F^4 : x3=x4+b, where b is an element of F. Axler states that this example is not a subspace unless b=0, yet this is the same space as V and i was under the impression (Axler states it himself) that V is a subspace of itself? Should not any value of b in F be possible?
 
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I think i have realized where i went wrong. The zero vector must be contained within the subspace, thus b=0 is the only solution which allows this. Is that a suitable method to complete the example or am i missing something else?
 
Group_Complex said:
I think i have realized where i went wrong. The zero vector must be contained within the subspace, thus b=0 is the only solution which allows this. Is that a suitable method to complete the example or am i missing something else?

that is correct.
 

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