Vector space/Basis/Introduction/Section

Definition

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Then a linearly independent generating system ${}v_{i}\in V$, ${}i\in I$,

of ${}V$ is called a basis of ${}V$.

Example

The standard vectors in ${}K^{n}$ form a basis. The linear independence was shown in example. To show that they also form a generating system, let

${}v={\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}\in K^{n}\,$

be an arbitrary vector. Then we have immediately

${}v=\sum _{i=1}^{n}b_{i}e_{i}\,.$

Hence, we have a basis, which is called the standard basis of ${}K^{n}$.

Theorem

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}v_{1},\ldots ,v_{n}\in V$

be a family of vectors. Then the following statements are equivalent.

1. The family is a basis of ${}V$.
2. The family is a minimal generating system; that is, as soon as we remove one vector ${}v_{i}$, the remaining family is not a generating system any more.
3. For every vector ${}u\in V$, there is exactly one representation

   ${}u=s_{1}v_{1}+\cdots +s_{n}v_{n}\,.$

4. The family is maximally linearly independent; that is, as soon as some vector is added, the family is not linearly independent any more.

Proof

This proof was not presented in the lecture.

$\Box$

Remark

Let a basis ${}v_{1},\ldots ,v_{n}$ of a ${}K$-vector space ${}V$ be given. Due to fact  (3), this means that for every vector ${}u\in V$, there exists a uniquely determined representation

${}u=s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n}\,.$

The elements ${}s_{i}\in K$ (scalars) are called the coordinates of ${}u$ with respect to the given basis. Thus, for a fixed basis, we have a (bijective) correspondence between the vectors from ${}V$, and the coordinate tuples ${}(s_{1},s_{2},\ldots ,s_{n})\in K^{n}$. We express this by saying that a basis determines a linear coordinate system.

Theorem

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space with a finite generating system. Then ${}V$ has a finite

basis.

Proof

Let ${}v_{i}$, ${}i\in I$, be a finite generating system of ${}V$ with a finite index set ${}I$. We argue with the characterization from fact  (2). If the family is minimal, then we have a basis. If not, then there exists some ${}k\in I$ such that the remaining family, where ${}v_{k}$ is removed, that is, ${}v_{i}$, ${}i\in I\setminus \{k\}$, is also a generating system. In this case, we can go on with this smaller index set. With this method, we arrive at a subset ${}J\subseteq I$ such that ${}v_{i}$, ${}i\in J$, is a minimal generating set, hence a basis.

$\Box$