Preview Activity2.4.1

Let's start this activity by studying the span of two different sets of vectors.

  1. Consider the following vectors in \(\real^3\text{:}\)

    \begin{equation*} \vvec_1=\threevec{0}{-1}{2}, \vvec_2=\threevec{3}{1}{-1}, \vvec_3=\threevec{2}{0}{1} \text{.} \end{equation*}

    Describe the span of these vectors, \(\span{\vvec_1,\vvec_2,\vvec_3}\text{.}\)

  2. We will now consider a set of vectors that looks very much like the first set:

    \begin{equation*} \wvec_1=\threevec{0}{-1}{2}, \wvec_2=\threevec{3}{1}{-1}, \wvec_3=\threevec{3}{0}{1} \text{.} \end{equation*}

    Describe the span of these vectors, \(\span{\wvec_1,\wvec_2,\wvec_3}\text{.}\)

  3. Show that the vector \(\wvec_3\) is a linear combination of \(\wvec_1\) and \(\wvec_2\) by finding weights such that

    \begin{equation*} \wvec_3 = a\wvec_1 + b\wvec_2 \text{.} \end{equation*}
  4. Explain why any linear combination of \(\wvec_1\text{,}\) \(\wvec_2\text{,}\) and \(\wvec_3\text{,}\)

    \begin{equation*} c_1\wvec_1 + c_2\wvec_2 + c_3\wvec_3 \end{equation*}

    can be written as a linear combination of \(\wvec_1\) and \(\wvec_2\text{.}\)

  5. Explain why

    \begin{equation*} \span{\wvec_1,\wvec_2,\wvec_3} = \span{\wvec_1,\wvec_2} \text{.} \end{equation*}
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