Why a Field V Can be Considered a Vector Space Over Itself in Linear Algebra

Introduction to Linear Algebra and Vector Spaces

Linear algebra is a fundamental branch of mathematics that deals with vector spaces, linear equations, and related structures. At its core, a vector space is defined as a set of elements, known as vectors, which can be added together and multiplied by scalars (numbers) while satisfying specific axioms. A key concept in this context is understanding why a field V can be considered a vector space over itself.

The Basics: Fields and Vector Spaces

In linear algebra, a field V is a set of elements where two operations, addition and multiplication, are defined, and these operations adhere to a set of axioms that ensure a well-structured system. When we say that a field V is a vector space over itself, it means that all the elements of the field V can also serve as vectors and scalars in a vector space. This article delves into why this is true.

Vector Spaces and Necessary Conditions

To qualify as a vector space, a set must satisfy a multitude of axioms with respect to two operations: vector addition and scalar multiplication. For a field V to be a vector space over itself, it must fulfill these criteria:

Set and Scalars

The vectors in the vector space are the elements of the field V. The scalars for scalar multiplication are also elements from the field V.

Vector Addition

The addition operation on the vector space is defined as the addition operation in the field. This means that for any two elements u, v in V, their sum u v is also an element of V.

Scalar Multiplication

The scalar multiplication operation is defined as the multiplication operation in the field. For any scalar c in V and vector u in V, the product c · u is also an element of V.

Vector Space Axioms

The vector space must satisfy the following axioms:

Closure

Closure under addition and scalar multiplication: The sum and product of any two elements in V must also be in V. This holds because V is a field, meaning every operation on its elements results in an element within the set.

Associativity of Addition

For all u, v, w in V, the associativity axiom ensures that (u v) w u (v w).

Commutativity of Addition

For all u, v in V, the commutativity axiom ensures that u v v u.

Existence of Additive Identity

There exists an element 0 in V such that for every u in V, u 0 u.

Existence of Additive Inverses

For every u in V, there exists an element -u in V such that u (-u) 0.

Distributive Properties

The distributive properties hold for all a, b in V and u in V. They ensure that a · (u v) a · u a · v and (a b) · u a · u b · u.

Associativity of Scalar Multiplication

For all a, b in V and u in V, scalar multiplication is associative, meaning (a · b) · u a · (b · u).

Identity Element of Scalar Multiplication

There exists an element 1 in V such that for every u in V, 1 · u u. This ensures that the identity element scales any vector back to itself.

Dimensionality of V as a Vector Space Over Itself

The dimension of V as a vector space over itself is 1. This is because any element of V can be expressed as a scalar multiple of the multiplicative identity 1. For example, if c is any element in V, then u c · 1.

Conclusion

From these properties, it is clear that the field V meets the criteria for being a vector space over itself. This validates that a field can indeed serve as a vector space, showcasing the rich interplay between these fundamental mathematical structures.