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### 2.7. Vector Spaces and Modules

Vector spaces and linear transformations between them are the central objects of study in linear algebra. In this section, we investigate the basic properties of vector spaces. We also generalize the concept of vector spaces to get another useful class of objects called modules. A module which also carries a (compatible) ring structure is referred to as an algebra. Study of algebras over fields (or more generally over rings) is of importance in commutative algebra, algebraic geometry and algebraic number theory.

#### 2.7.1. Vector Spaces

Unless otherwise specified, K denotes a field in this section.

##### Definition 2.42.
 A vector space V over a field K (or a K-vector space, in short) is an (additively written) Abelian ...

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