Difference between vector basis and bases?

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SUMMARY

The discussion clarifies the distinction between "basis" and "bases" in vector spaces, emphasizing that "bases" is simply the plural form of "basis." It illustrates that a vector space, such as R², can have multiple bases, including the standard basis {(1, 0), (0, 1)} and the alternative basis {(1, 1), (1, -1)}. Furthermore, any two non-scalar multiple vectors in R² can form a valid basis, demonstrating the flexibility in choosing bases for vector spaces.

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  • Understanding of vector spaces
  • Familiarity with linear independence
  • Knowledge of standard basis vectors
  • Basic concepts of linear algebra
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Homework Statement



Could someone please explain the difference? Maybe show some examples?
Than you!

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The Attempt at a Solution

 
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I am not sure I understand you correctly, "bases" is the plural of "basis".
 
A given vector space can have many bases. For example, R2 has the standard basis {(1, 0), (0, 1)}, but the following set is also a basis for R2--{(1, 1), (1, -1)}. In fact, any two vectors in R2 that aren't scalar multiples of each other would serve as a basis for R2.
 

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