Linear Subspace and Span
- Linear Algebra 1.3
from 「Linear Algebra: A Modern Introduction」: David Poole
Linear subspace
If $V$ is a $K$-vector space $W$ is a subset of $V$, then $W$ is a linear subspace of $V$ if $W$ is a vector space over $K$ under the operations of $V$. Equivalently, a nonempty subset $W$ is a subspace of $V$ if arbitary linear combinations of some vectors of $W$ are also an element of $W$.
- $ W \subset V $
- $ \b{w}_1,\b{w}_2\in W,\; \alpha,\beta\in F \Rightarrow \alpha\b{w}_1+\beta\b{w}_2\in W$
It’s sometimes denoted as $W \le V$, but but these aren’t common enough to be used without explicitly specifying their meaning first.
As a corollary, all vector spaces are equipped with at least two linear subspaces:
- The zero vector space
- The vector space itself
These are called the trivial subspaces of the vector space.
Linear span
The linear span(i.e. span / linear hull) of a set $S$, which is a subset of a vector space over a field $F$, is defined as the set of all possible linear combinations of the vectors in $S$.
- $ \span(S) \coloneqq \Set{ \dsum_{i=1}^k \lambda_i\b{v}_i | k\in\N,\; \b{v}_i\in S,\; \lambda_i\in F} $
Two vectors(black arrows) spanning the blue plane
To express that a vector space $V$ is a linear span of a subset $S$, one commonly uses the following phrases:
- $S$ spans $V$
- $S$ is a spanning set of $V$
- $V$ is spanned/generated by $S$
- $S$ is a generator or generator set of $V$
If $S$ is a subset of a vector space $V$, then the span of $S$ is a subspace of $V$.
- $ S \subset V \Rightarrow \span(S) \le V $
