📂Abstract Algebra

Vector Spaces in Abstract Algebra

Definition ¹

A field $F$ and an abelian group $V$ are said to form a vector space over $F$ if all elements $\alpha , \beta \in F$ and $x, y \in V$ satisfy the following conditions. Elements of $F$ are called scalars, and elements of $V$ are called vectors.

• (i): $\alpha x \in V$
• (ii): $\alpha ( \beta x) = ( \alpha \beta ) x$
• (iii): $\alpha (x + y) = \alpha x + \alpha y$
• (iv): $1 x = x$

Let’s denote the index set as $I$ and define $\left\{ x_{i} \right\}_{i \in I} \subset V$.

1. For some $\left\{ \alpha_{i} \right\}_{ i \in I} \subset F$, $\displaystyle \sum_{i \in I} \alpha_{i} x_{i}$ is said to be a linear combination of $\left\{ x_{i} \right\}_{i \in I}$.
2. If all elements of $V$ can be represented as linear combinations of $M$, $ \left\{ x_{i} \right\}_{i \in I}$ is said to generate $V$, denoted as $\text{span} \left\{ x_{i} \right\}_{i \in I} = V$.
3. If there exists a $\left\{ x_{i} \right\}_{i \in I}$ satisfying $\text{span} \left\{ x_{i} \right\}_{i \in I} = V$ for a finite set $I$, then $V$ is said to be finite-dimensional.
4. If the only case satisfying $\displaystyle \sum_{i \in I} \alpha_{i} x_{i} = 0$ for all $\left\{ x_{i} \right\}_{i \in I}$ is $\alpha_{i} = 0$, then $\left\{ x_{i} \right\}_{i \in I}$ is linearly independent over $F$; otherwise, it is linearly dependent.
5. When $\text{span} \left\{ x_{i} \right\}_{i \in I} = V$, if $\left\{ x_{i} \right\}_{i \in I}$ is linearly independent, then $\left\{ x_{i} \right\}_{i \in I}$ is the basis of $V$.
6. When the basis of a finite-dimensional vector space $V$ is denoted by $M$, the cardinality of $M$ is called the dimension of $V$, represented as $\dim V$.

Description

This concept might already be familiar from linear algebra and can be described as an abstract algebra without the ‘algebra’ name attached. For a simple example, the ring of polynomial functions $\mathbb{R} [ x ]$ easily verifies to be a vector space with $1 , x , \cdots , x^{n}$ as its basis.

See Also

• Vector space in linear algebra
• Vector space in abstract algebra

The $F$-vector spaces discussed in the documents below are essentially no different from the vector spaces mentioned above. However, the perspective can vary: in linear algebra, a vector space is an abstraction of the intuitive Euclidean space, whereas in abstract algebra, it brings that abstraction into the true meaning of ‘algebra’.

On the contrary, the $R$-module generalizes the scalar field $F$ of the $F$-vector space to a scalar ring $R$, and thus, its significance lies in generalization. From the perspective of group $G$, since a new operation $\mu$ is added to ring $R$, it also falls under the category of modules.

• R-module in abstract algebra
• $F$-vector space in abstract algebra