Let \(S\) be a set of vectors, \(S=\{\vec{v}_1,\vec{v}_2,\ldots\}\text{.}\) We define the span of \(S\text{,}\) denoted \(Span(S)\text{,}\) as the set of all linear combinations of vectors from \(S\text{.}\) That is,
Look back at Activity 1.6.2 and Activity 1.6.3 and restate each of the questions in terms of span. For instance, part 1 of Activity 1.6.2 could be stated as "Show that \(\colvec{3\\-4}\) is in the span of \(\left\{\colvec{1\\1}, \colvec{-1\\1}\right\}\text{.}\)"
Note that the set \(S\) might not be finite but the number of vectors involved in the summation for a linear combination is finite. Also, remember to treat \(Span(S)\) as a set and not a vector. Remember that the use of span in \(Span(S)\) is a noun.
Activity1.7.2.
How many vectors are in \(Span(\{ \langle 1,1 \rangle ,\langle 1,-1 \rangle \})\text{?}\)
Is there any vector in \(\mathbb{R}^2\) that is not in \(Span(\{ \langle 1,1 \rangle ,\langle 1,-1 \rangle \})\text{?}\)
How many vectors are in \(Span(\{ \langle 1,1,1 \rangle ,\langle 1,-1,1 \rangle \})\text{?}\)
Is there any vector in \(\mathbb{R}^3\) that is not in \(Span(\{ \langle 1,1,1 \rangle ,\langle 1,-1,1 \rangle \})\text{?}\)
Try to write out the set of vectors in \(Span(\{ \langle 1,1,1 \rangle ,\langle 1,-1,1 \rangle \})\text{?}\) Hint: write the corresponding system of equations, then use the solution set of this system to write out the exact vector form of \(Span(\{ \langle 1,1,1 \rangle ,\langle 1,-1,1 \rangle \})\text{.}\)
Is there any vector in \(\mathbb{R}^2\) that is not in \(Span(\{ \langle 1, 0\rangle ,\langle 0,1 \rangle, \langle 1,1 \rangle \})\text{?}\)
The following are equivalent questions:
Is a vector \(\vec{b}\) in \(Span(S)\text{?}\)
Does \(\vec{b} = \sum_i (c_i \vec{v}_i)\) have a solution?
A few other related questions are:
When will there be a solution to \(\vec{b} = \sum_i (c_i \vec{v}_i)\text{?}\)
When will there be a UNIQUE solution to \(\vec{b} = \sum_i (c_i \vec{v}_i)\text{?}\)
How can we describe \(Span(S)\) as a collection of vectors?
Subsection1.7.2Span as a Verb
Definition1.7.2.
A set of vectors \(\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_m\}\) spans a vector space \(V\) if \(Span(\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_m\})=V\text{.}\) In other words, \(\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_m\}\) spans a vector space \(V\) if every vector in \(V\) can be written as a linear combination from the set \(\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_m\} \text{.}\)
Activity1.7.3.
(a)
Does \(\{ \langle 1,2\rangle,\langle 3,6\rangle\}\) span \(\mathbb{R}^2\text{?}\)
(b)
Does \(\{ \langle 1,2\rangle,\langle 3,4\rangle\}\) span \(\mathbb{R}^2\text{?}\)
(c)
Does \(\{ \langle 1,2,3\rangle,\langle 4,5,6\rangle\}\) span \(\mathbb{R}^3\text{?}\)
(d)
Does \(\{ \langle 1,2,3\rangle,\langle 4,5,6\rangle , \langle 0,1,0\rangle\}\) span \(\mathbb{R}^3\text{?}\)
