Span#

Definition: Span

Let \((V, F, +, \cdot)\) be a vector space and let \(S \subseteq V\).

The span of \(S\) is the set of all linear combinations which can be constructed from vectors in \(S\):

\[ \{c_1 \mathbf{v}_1 + \cdots + c_n \mathbf{v}_n \mid n \ge 1, c_k \in F, \mathbf{v}_k \in S\} \]

Let \(\mathbf{v}_1, \mathbf{v}_2, \dotsc\) be vectors in a vector space \((V, F, +, \cdot)\).

Notation

\[ \langle S \rangle \qquad \mathop{\operatorname{span}} (S) \]

If \(S\) contains only countably many elements \(\mathbf{s}_1, \mathbf{s}_2, \dotsc\), we can also write

\[ \langle \{ \mathbf{s}_1, \mathbf{s}_1, \cdots \} \rangle \qquad \langle \mathbf{s}_1, \mathbf{s}_1, \cdots \rangle \qquad \operatorname{span}(\{ \mathbf{s}_1, \mathbf{s}_1, \cdots \}) \qquad \operatorname{span}( \mathbf{s}_1, \mathbf{s}_1, \cdots) \]

If \(S\) contains only finitely many elements \(\mathbf{s}_1, \dotsc, \mathbf{s}_n\), we can also write

\[ \langle \{ \mathbf{s}_1, \dotsc, \mathbf{s}_n \} \rangle \qquad \langle \mathbf{s}_1, \dotsc, \mathbf{s}_n \rangle \qquad \operatorname{span}(\{ \mathbf{s}_1, \dotsc, \mathbf{s}_n \}) \qquad \operatorname{span}( \mathbf{s}_1, \dotsc, \mathbf{s}_n) \]

Theorem: Equivalent Definition

The span of \(S\) is the intersection of all subspaces containing \(S\):

\[ \mathop{\operatorname{span}}(S) = \bigcap \{U \mid U \text{ is a subspace of } V \text{ and } S \subseteq U \} \]

Proof

Let \(L\) be the set of all linear combinations which can be constructed from vectors in \(S\):

\[ L = \{c_1 \mathbf{v}_1 + \cdots + c_n \mathbf{v}_n \mid n \ge 1, c_k \in F, \mathbf{v}_k \in S\} \]

Let \(I\) be the intersection of all subspaces containing \(S\):

\[ I = \bigcap \{U \mid U \text{ is a subspace of } V \text{ and } S \subseteq U \} \]

TODO

Theorem: Span is a Subspace

If \(S\) is a subset of a vector space \(V\), then the span of \(S\) is a subspace of \((V, F, +, \cdot)\).

Proof

Non-emptiness:

Obviously, \(\mathbf{0} \in \operatorname{\mathop{span}}(S)\).

Closure under vector addition:

Let \(\mathbf{u}, \mathbf{v} \in \operatorname{\mathop{span}}(S)\). By definition, we have \(\mathbf{u}, \mathbf{v} \in U\) for all subspaces \(U\) of \(V\) with \(S \subseteq U\). Since these are subspaces we know that \(\mathbf{u} + \mathbf{v} \in U\) for all subspaces \(U\) of \(V\) with \(S \subseteq U\). Since \(\mathbf{u} + \mathbf{v}\) belongs to all such subspaces, it must also belong to their intersection:

\[ \mathbf{u}, \mathbf{v} \in \bigcap_{S \subseteq U} U = \operatorname{\mathop{span}}(S) \]

Closure under scalar multiplication:

Let \(\mathbf{u} \in \operatorname{\mathop{span}}(S)\). By definition, we have \(\mathbf{u} \in U\) for all subspaces \(U\) of \(V\) with \(S \subseteq U\). Since these are subspaces we know that \(\lambda \mathbf{u} \in U\) for all subspaces \(U\) of \(V\) with \(S \subseteq U\). Since \(\lambda \mathbf{u}\) belongs to all such subspaces, it must also belong to their intersection:

\[ \lambda \mathbf{u} \in \bigcap_{S \subseteq U} U = \operatorname{\mathop{span}}(S) \]

We have thus shown that \(\operatorname{\mathop{span}}(S)\) satisfies the definition of a subspace.

Theorem: Span of Subset is Subset of Span

Let \(V\) be a vector space and let \(S, T \subseteq V\).

If \(S\) is a subset of \(T\), then the span of \(S\) is a subset of the span of \(T\):

\[ S \subseteq T \implies \mathop{\operatorname{span}} S \subseteq \mathop{\operatorname{span}} T \]

Proof

TODO

Theorem: Span of Subspace Union

If \(U_1, \dotsc, U_n\) are subspaces of a vector space \(V\), then the span of their union is equal to their sum:

\[ \mathop{\operatorname{span}}(U_1 \cup \cdots \cup U_n) = U_1 + \cdots U_n \]

Proof

TODO

Spanning Sets#

Definition: Spanning Set (Generator)

Let \((V,F,+,\cdot)\) be a vector space.

A subset \(S \subseteq V\) is a spanning set (or a generator) of \((V,F,+,\cdot)\) if the span of \(S\) is \(V\):

\[ \operatorname{span}(S) = V \]

Tip

This essentially means that each vector in \(V\) can be expressed as a linear combination of some vectors in \(S\).

Definition: Minimality

A spanning set \(S\) is minimal if there is no vector \(\mathbf{v} \in S\) such that \(S \setminus \{\mathbf{v}\}\) is still a spanning set.

Tip

This means that there is no way to remove a vector from a minimal spanning set and still obtain a spanning set of the vector space.

Definition: Finitely Generated Vector Space

A vector space is finitely generated if there exists a finite spanning set for it.