Section 2.6 Span
Definition 2.6.1.
A point \(\mathbf{P}\) is said to be a linear combination of the points: \(\mathbf{P}_1 , \mathbf{P}_2,\ldots,\mathbf{P}_n\) if \(\mathbf{P} = t_1\mathbf{P}_1+ t_2\mathbf{P}_2+ \cdots+ t_n\mathbf{P}_n\) for some list of real numbers \(t_i\text{.}\)
Definition 2.6.2.
The set of all linear combinations using points from a non-empty set
\(\mathbb{S}=\{ \mathbf{P}_1, \mathbf{P}_2, \ldots, \mathbf{P}_n \}\) is called the span of \(\mathbb{S}\text{,}\) and written:
Definition 2.6.3.
A set \(\mathbb{S}\) of points in \(\mathbb{L}\) is said to span \(\mathbb{L}\) if \(span\mathbb{S}=\mathbb{L}\text{.}\) This means that every point in \(\mathbb{L}\) is a linear combination of the points in \(\mathbb{S}\text{.}\) (Note: Span has again been defined as a noun and as a verb.)
Checkpoint 2.6.4.
Describe the span of \(\mathbb{S} = \{ \ot{1}{1}{0},\ot{0}{0}{1}\}\text{.}\) Does \(\mathbb{S}\) span \(\mathbb{R}^3\text{?}\)
Checkpoint 2.6.5.
Does \(\mathbb{S} = \{ x,x^2\}\) span \(\mathbb{P}_3\text{?}\) If not, find a set that does.
Checkpoint 2.6.6.
Find two different sets that span \(\mathbb{R}^3\text{.}\)
Theorem 2.6.7.
If \(\mathbb{S}\) is a nonempty subset of a linear space \(\mathbb{L}\text{,}\) then \(span\mathbb{S}\) is a subspace of \(\mathbb{L}\text{.}\) Moreover, \(span\mathbb{S}\) is the smallest subspace of \(\mathbb{L}\) containing \(\mathbb{S}\text{,}\) in other words, any subspace of \(\mathbb{L}\) containing \(\mathbb{S}\) must also contain \(span\mathbb{S}\text{.}\)
Discussion Question 2.6.8.
What definition for \(span\{\}\) would make Theorem 2.6.7 true without the condition that \(\mathbb{S}\) be non-empty?