Let V be a vector space over a field k. (i) Prove that V is...

Let $V$ be a vector space over a field $k$.
(i) Prove that $V$ is a free $k$-module.
(ii) Prove that a subset $B$ of $V$ is a basis of $V$ considered as a vector space if and only if $B$ is a basis of $V$ considered as a free $k$-module.

Key Concepts

Linear Independence and Spanning

These two properties are fundamental for a set to be considered a basis. Linear independence ensures that no vector in the set can be written as a linear combination of the others, while the spanning property ensures that every vector in the space can be constructed from the set. Together, they guarantee a unique representation of every vector in the space, which is central to the structure of vector spaces and free modules.

Basis

A basis of a vector space or free module is a set of vectors that is both linearly independent and spans the entire space. This concept is critical because it allows every element of the space to be uniquely represented as a linear combination of the basis elements, which is essential for both theoretical considerations and practical computations.

Free Module

A free module is a module that possesses a basis, meaning it is isomorphic to a direct sum of copies of the ring. In the case where the ring is a field, every vector space (or module) automatically is free because it always has a basis, and thus every vector space can be thought of as a free module.

Module

A module generalizes the notion of a vector space by allowing scalars to come from a ring instead of a field. In modules over rings that are not fields, many properties familiar from vector spaces may fail. However, when the ring is a field, the structure of a module coincides with that of a vector space.

Field

A field is an algebraic structure with two operations, commonly addition and multiplication, where every nonzero element has a multiplicative inverse. This property plays a crucial role in the behavior of vector spaces since the invertibility of nonzero scalars guarantees that every vector space over a field has well-behaved linear combinations.

Vector Space

A vector space is a collection of objects called vectors, equipped with operations of addition and scalar multiplication that satisfy certain axioms such as associativity, commutativity, and distributivity. It is defined over a field, which ensures that every nonzero scalar has an inverse, thereby enabling many convenient properties and results including the existence of bases and dimension theory.

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