Difference between space, subspace, subset, span, and ambient space
1. Overview
2. Description
2.1 Space
2.2 Subspace
- Be closed under addition and scalar multiplication
- Contain the zero vector
$\forall \boldsymbol{v},\boldsymbol{w}\in V$; $\forall \lambda ,\alpha \in \mathbb{R}$; $ \lambda\boldsymbol{v} +\alpha\boldsymbol{w} \in V$
2.3 Subset
2.3.1 Definition
2.3.2 How to distinguish subset and subspace
Step 1: Determine whether the origin is in the set
Step 2: Try to write down the criteria in terms of scalars and vectors of the form $\alpha v+\beta w$
2.4 span
2.4.1 Definition
Given a vector space V over a field K, the span of a set S of vectors (not necessarily infinite) is defined to be the intersection W of all subspaces of V that contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W, and we say that S spans W.
Alternatively, the span of S may be defined as the set of all finite linear combinations of elements (vectors) of S, which follows from the above definition.
$$span({\boldsymbol{v}_{1},\cdots ,\boldsymbol{v}_{n}})=\alpha _{1}\boldsymbol{v}_{1}+\cdots +\alpha _{n}\boldsymbol{v}_{n},\: \alpha \in \mathbb{R}$$
2.4.2 theorems
Theorem 1: The subspace spanned by a non-empty subset S of a vector space V is the set of all linear combinations of vectors in S.
This theorem is so well known that at times it is referred to as the definition of span of a set.
Theorem 2: Every spanning set S of a vector space V must contain at least as many elements as any linearly indenpendent set of vectors from V.
Theorem 3: Let V be a finite-dimensional vector space. Any set of vectors that spans V can be reduced to a basis for V by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the axiom of choice holds, this is true without the assumption that V has finite dimension.
This also indicates that a basis is a minimal spanning set when V is finite-dimensional.
3. Example
3.1 Subset
3.2 Subset 2
3.3 Subset and subspace
3.4 Subset
3.5 More vectors $\neq $ more dimensions
3.6 Span
3.6.1 Span $\mathbb{R}^{2}$
Let $S_{1}=\begin{Bmatrix}
\begin{pmatrix}
1\\
1\\
0
\end{pmatrix},\begin{pmatrix}
1\\
7\\
0
\end{pmatrix} &
\end{Bmatrix}$, $S_{2}=\begin{Bmatrix}
\begin{pmatrix}
1\\
1\\
0
\end{pmatrix}, & \begin{pmatrix}
1\\
7\\
0
\end{pmatrix} & ,\begin{pmatrix}
-1\\
2\\
1
\end{pmatrix}
\end{Bmatrix}$
$$span(S_{1})=span(S_{2})=\mathbb{R}^{2}$$
4. References
https://en.wikipedia.org/wiki/Subset
https://en.wikipedia.org/wiki/Space_(mathematics)
https://en.wikipedia.org/wiki/Linear_subspace