Vector Space: Scalar Restriction in ℤ2

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SUMMARY

The discussion centers on the concept of scalar restriction in vector spaces, specifically within ℤ2. It establishes that while vector spaces require a field as their scalar domain, one can limit scalars to a ring, such as ℤ, resulting in a structure known as a ℤ-module. This limitation alters the properties of the vector space; for example, restricting scalars to rational numbers transforms the real line into an infinite-dimensional vector space over that field. Thus, scalar restrictions fundamentally change the nature of the vector space.

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  • Familiarity with ℤ-modules and their characteristics
  • Concept of dimensionality in vector spaces
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In a vector space can you limit the scalar. For example, if I have Vector space in ℤ2 can i only multiple integer scalars?
 
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Austin Chang said:
In a vector space can you limit the scalar. For example, if I have Vector space in ℤ2 can i only multiple integer scalars?
In this case it is called a ##\mathbb{Z}-##module, i.e. the scalars are from a ring, in your example ##\mathbb{Z}##. Vector spaces are required to have a field as scalar domain, that is we have invertible elements as scalars (and of course ##0##). However, the field doesn't have to be "unlimited". E.g. ##\{0,1\}## is also a field.
 
Austin Chang said:
In a vector space can you limit the scalar. For example, if I have Vector space in ℤ2 can i only multiple integer scalars?
You can limit the scalars to a subfield of the field of scalars but then you have a different vector space. For instance the real line is a one dimensional vector space over the field of real numbers. If you limit the scalars to the field of rational numbers, then the real line is an infinite dimensional vector space over the field of rational numbers.
 

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