Is a subspace still valid without the zero vector?

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SUMMARY

A subspace in vector space theory must include the zero vector to satisfy the closure properties required for addition and scalar multiplication. Specifically, if a vector v is part of a subspace, then its negative -v must also be included, leading to the necessity of the zero vector being present. This is corroborated by the definition of a subspace, which states that it must be closed under addition and scalar multiplication. Some textbooks explicitly state that a subspace must contain the zero vector, while others may define it as non-empty.

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  • Understanding of vector spaces and their properties
  • Familiarity with the concepts of closure under addition and scalar multiplication
  • Knowledge of the definitions of subspaces in linear algebra
  • Basic comprehension of vector operations and their implications
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Students and educators in mathematics, particularly those focusing on linear algebra, as well as anyone seeking to deepen their understanding of vector spaces and subspace properties.

jeffreylze
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If a set of vectors does not contain the zero vector is it still a subspace?
 
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No, because the subspace will have negatives of elements,
i.e., for all v an element of V, (-1)v or -v will be an element.
For the subspace to be closed under addition (a necessary requirement)
v + (-v) = 0 must be an element which implies the zero vector must be in a subspace of vectors.
 
Some textbooks include "contains the 0 vector" as part of the definition of "subspace".
Others just say "is non-empty". As DorianG pointed out, if some vector, v, is in the subspace, then so is -v (a subspace is "closed under scalar multiplication") and so is v+ (-v)= 0 (a subspace is "close under addition").
 

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