Vector space/Generating system/Introduction/Section

The plane generated by two vectors ${}v_{1}$ and ${}v_{2}$ consists of all linear combinations ${}u=sv_{1}+tv_{2}$.

Definition

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}v_{1},\ldots ,v_{n}$ denote a family of vectors in ${}V$. Then the vector

$s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n}{\text{ with }}s_{i}\in K$

is called a linear combination of this vectors

(for the coefficient tuple ${}(s_{1},\ldots ,s_{n})$).

Two different coefficient tuples can define the same vector.

Definition

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. A family ${}v_{i}\in V$, ${}i\in I$, is called a generating system (or spanning system) of ${}V$, if every vector ${}v\in V$ can be written as

${}v=\sum _{j\in J}s_{j}v_{j}\,,$

with a finite subfamily ${}J\subseteq I$, and with

${}s_{j}\in K$.

In ${}K^{n}$, the standard vectors ${}e_{i}$, ${}1\leq i\leq n$, form a generating system. In the polynomial ring ${}K[X]$, the powers ${}X^{n}$, ${}n\in \mathbb {N}$, form an (infinite) generating system.

Definition

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. For a family ${}v_{i}$, ${}i\in I$, we set

${}\langle v_{i},\,i\in I\rangle ={\left\{\sum _{i\in J}s_{i}v_{i}\mid s_{i}\in K,\,J\subseteq I{\text{ finite subset}}\right\}}\,,$

and call this the linear span of the family, or the generated linear subspace.

The empty set generates the null space. The null space is also generated by the element ${}0$. A single vector ${}v$ spans the space ${}Kv={\left\{sv\mid s\in K\right\}}$. For ${}v\neq 0$, this is a line, a term we will make more precise in the framework of dimension theory. For two vectors ${}v$ and ${}w$, the "form“ of the spanned space depends on how the two vectors are related to each other. If they both lie on a line, say ${}w=sv$, then ${}w$ is superfluous, and the linear subspace generated by the two vectors equals the linear subspace generated by ${}v$. If this is not the case (and ${}v$ and ${}w$ are not ${}0$), then the two vectors span a "plane“.

We list some simple properties for generating systems and linear subspaces.

Lemma

Let ${}K$ be a field, and let ${}V$ be a

${}K$-vector space. Then the following hold.

1. Let ${}U_{j}$, ${}j\in J$, be a family of linear subspaces of ${}V$. Then the intersection

   ${}U=\bigcap _{j\in J}U_{j}\,$

   is a linear subspace.

2. Let ${}v_{i}$, ${}i\in I$, be a family of elements of ${}V$, and consider the subset ${}W$ of ${}V$ which is given by all linear combinations of these elements. Then ${}W$ is a linear subspace of ${}V$.
3. The family ${}v_{i}$, ${}i\in I$, is a system of generators of ${}V$ if and only if

   ${}\langle v_{i},\,i\in I\rangle =V\,.$

Proof

See exercise.

$\Box$