Evaluating Vector Spaces: V = {(0,1), (1,0)}

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SUMMARY

The discussion evaluates whether the set V = {(0, 1), (1, 0)} constitutes a vector space over the real numbers, ℝ. Participants clarify that V is a finite set containing only two vectors, which does not include the zero vector (0, 0). Consequently, V fails to meet the criteria for a vector space, as it does not satisfy the requirements for closure under addition and scalar multiplication. The consensus is that V is not a vector subspace of ℝ².

PREREQUISITES
  • Understanding of vector space definitions
  • Familiarity with scalar multiplication in vector spaces
  • Knowledge of vector addition properties
  • Basic concepts of linear algebra, particularly in ℝ²
NEXT STEPS
  • Study the properties of vector spaces in linear algebra
  • Learn about the concept of span and its implications for vector sets
  • Investigate the role of the zero vector in vector spaces
  • Explore examples of valid and invalid vector spaces
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Students of linear algebra, educators teaching vector space concepts, and anyone seeking to understand the foundational principles of vector spaces in mathematics.

Kosh11
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Homework Statement



Are the following vector spaces over \Re, with the usual notion of
addition and scalar multiplication: V = {(0, 1), (1, 0)}

Homework Equations



definition of vector space

The Attempt at a Solution



I'm a little confused by what this means. Am I correcting in thinking in drawing a line from the origin to (0,1) and another from the origin to (1,0) and all the vectors contained in those two lines belong to V? Then I'm unsure as to whether or nor this a vector space. Addition and scalar multiplication seem to hold, but I'm not sure and I'm lost on how to prove it either true or false.
 
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as its written, i would read that as the subset of R^2, with two elements, the vectors ( 0,1) and (1,0), which is clearly not a vector subspace

maybe you should write the question just as it is aksed for clarity
 
I would interpret that exactly as it is given- as set notation- that (0, 1) and (1, 0) are the only objects in the set. Nothing is said about a "span" or combinations of those vectors. In particular, the 0 vector, (0, 0), is not in the set.
 

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