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 maximal 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, 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. 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.

$\Box$