Activity 2.3.2
Let's look at two examples to develop some intuition for the concept of span.

First, we will consider the set of vectors
\begin{equation*} \vvec = \twovec{1}{2}, \wvec = \twovec{2}{4}\text{.} \end{equation*}SPAN1

What vector is the linear combination of \(\vvec\) and \(\wvec\) with weights:
\(a = 2\) and \(b=0\text{?}\)
\(a = 1\) and \(b=1\text{?}\)
\(a = 0\) and \(b=1\text{?}\)
Can the vector \(\twovec{2}{4}\) be expressed as a linear combination of \(\vvec\) and \(\wvec\text{?}\) Is the vector \(\twovec{2}{4}\) in the span of \(\vvec\) and \(\wvec\text{?}\)
Can the vector \(\twovec{3}{0}\) be expressed as a linear combination of \(\vvec\) and \(\wvec\text{?}\) Is the vector \(\twovec{3}{0}\) in the span of \(\vvec\) and \(\wvec\text{?}\)
Describe the set of vectors in the span of \(\vvec\) and \(\wvec\text{.}\)

For what vectors \(\bvec\) does the equation
\begin{equation*} \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 4 \end{array}\right] \xvec = \bvec \end{equation*}have a solution?


We will now look at an example where
\begin{equation*} \vvec = \twovec{2}{1}, \wvec = \twovec{1}{2}\text{.} \end{equation*}SPAN2

What vector is the linear combination of \(\vvec\) and \(\wvec\) with weights:
\(a = 2\) and \(b=0\text{?}\)
\(a = 1\) and \(b=1\text{?}\)
\(a = 0\) and \(b=1\text{?}\)
Can the vector \(\twovec{2}{2}\) be expressed as a linear combination of \(\vvec\) and \(\wvec\text{?}\) Is the vector \(\twovec{2}{2}\) in the span of \(\vvec\) and \(\wvec\text{?}\)
Can the vector \(\twovec{3}{0}\) be expressed as a linear combination of \(\vvec\) and \(\wvec\text{?}\) Is the vector \(\twovec{3}{0}\) in the span of \(\vvec\) and \(\wvec\text{?}\)
Describe the set of vectors in the span of \(\vvec\) and \(\wvec\text{.}\)

For what vectors \(\bvec\) does the equation
\begin{equation*} \left[\begin{array}{rr} 2 \amp 1 \\ 1 \amp 2 \end{array}\right] \xvec = \bvec \end{equation*}have a solution?
