Actions of Field on Vector Space

In summary, depending on the field F, there may be multiple ways to define the action of F on V, but for prime fields like Q and Z_p, there is only one possible action.
  • #1
WWGD
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Hi, Algebraists:

Say V is finite-dimensional over F . Is there more than one way of defining the
action of F on V (of course, satisfying the vector space axioms.) By different
ways, I mean that the two actions are not equivariant.

Thanks.
 
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  • #2
WWGD said:
Hi, Algebraists:

Say V is finite-dimensional over F . Is there more than one way of defining the
action of F on V (of course, satisfying the vector space axioms.) By different
ways, I mean that the two actions are not equivariant.

Thanks.
It depends upon the field F. If F is Q (the field of rationals) or more generally a prime field (i.e. either Q or Z_p for some prome p), then there is only one possible field action of F on V: The vector space axioms determine the value of nv for each natural number n and each v in V, and this can only be extended in one way to Z and then to Q if the axioms shall hold.

But in general, there can be more than one possibility. For example, If V is a vector space over C (the complex numbers), and if a' denotes the conjugate of a complex number a, then we define a new action x of C upon V by a x v = a'v (where a'v is computed by the original action). It can be shown that V with this new action and the same addition as before satisfies the vector space axioms.
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of elements (vectors) and operations (such as addition and scalar multiplication) that satisfy certain properties. These properties include closure, associativity, commutativity, and distributivity.

2. What is the significance of vector spaces in science?

Vector spaces are essential in many scientific fields, including physics, engineering, and computer science. They provide a powerful tool for representing and manipulating quantities that have both magnitude and direction, such as forces, velocities, and electric fields.

3. How does a field act on a vector space?

A field acts on a vector space by defining a set of operations (such as addition and scalar multiplication) that can be performed on the vectors in the space. These operations must satisfy the properties of a vector space to ensure that the space remains closed under the field's actions.

4. What are some examples of fields that act on vector spaces?

Examples of fields that act on vector spaces include the real and complex numbers, as well as other mathematical structures such as matrices and functions. These fields provide the necessary operations for performing calculations and transformations on vectors in a vector space.

5. How do actions of field on vector space relate to linear transformations?

The actions of a field on a vector space are closely related to linear transformations. In fact, every action of a field on a vector space corresponds to a unique linear transformation, and vice versa. This connection allows us to apply the powerful tools of linear algebra to understand and analyze the actions of fields on vector spaces.

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