Activity2.3.2

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

  1. First, we will consider the set of vectors

    \begin{equation*} \vvec = \twovec{1}{2}, \wvec = \twovec{-2}{-4} \text{.} \end{equation*}

    SPAN1

    1. 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 \(b=-1\text{?}\)

    2. 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 \(\wvec\text{?}\)

    3. 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 \(\wvec\text{?}\)

    4. Describe the set of vectors in the span of \(\vvec\) and \(\wvec\text{.}\)

    5. 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?

  2. We will now look at an example where

    \begin{equation*} \vvec = \twovec{2}{1}, \wvec = \twovec{1}{2} \text{.} \end{equation*}

    SPAN2

    1. 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 \(b=-1\text{?}\)

    2. 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 \(\wvec\text{?}\)

    3. 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 \(\wvec\text{?}\)

    4. Describe the set of vectors in the span of \(\vvec\) and \(\wvec\text{.}\)

    5. 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?

in-context