Undergrad Vector Space: Scalar Restriction in ℤ2

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In a vector space, scalars are typically drawn from a field, which includes invertible elements. When considering a vector space in ℤ2, limiting scalars to integers creates a structure known as a ℤ-module. While it is possible to restrict scalars to a subfield, this results in a different vector space. For example, the real line is one-dimensional over the reals but becomes infinite-dimensional over the rationals when scalars are limited. Thus, scalar restrictions fundamentally alter the nature of the vector space.
Austin Chang
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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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