Activity2.3.3

In this activity, we will look at the span of sets of vectors in \(\real^3\text{.}\)

  1. Suppose \(v=\threevec{1}{2}{1}\text{.}\) Give a written description of \(\span{v}\) and a rough sketch of it below.

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  2. Consider now the two vectors

    \begin{equation*} \evec_1 = \threevec{1}{0}{0}, \evec_2 = \threevec{0}{1}{0} \text{.} \end{equation*}

    Sketch the vectors below. Then give a written description of \(\span{\evec_1,\evec_2}\) and a rough sketch of it below.

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    Let's now look at this algebraically by writing write \(\bvec = \threevec{b_1}{b_2}{b_3}\text{.}\) Determine the conditions on \(b_1\text{,}\) \(b_2\text{,}\) and \(b_3\) so that \(\bvec\) is in \(\span{\evec_1,\evec_2}\) by considering the linear system

    \begin{equation*} \left[\begin{array}{rr} \evec_1 \amp \evec_2 \\ \end{array}\right] \xvec = \bvec \end{equation*}

    or

    \begin{equation*} \left[\begin{array}{rr} 1 \amp 0 \\ 0 \amp 1 \\ 0 \amp 0 \\ \end{array}\right] \xvec = \threevec{b_1}{b_2}{b_3} \text{.} \end{equation*}

    Explain how this relates to your sketch of \(\span{\evec_1,\evec_2}\text{.}\)

  3. Consider the vectors

    \begin{equation*} \vvec_1 = \threevec{1}{1}{-1}, \vvec_2 = \threevec{0}{2}{1} \text{.} \end{equation*}
    1. Is the vector \(\bvec=\threevec{1}{-2}{4}\) in \(\span{\vvec_1,\vvec_2}\text{?}\)

    2. Is the vector \(\bvec=\threevec{-2}{0}{3}\) in \(\span{\vvec_1,\vvec_2}\text{?}\)

    3. Give a written description of \(\span{\vvec_1,\vvec_2}\text{.}\)

  4. Consider the vectors

    \begin{equation*} \vvec_1 = \threevec{1}{1}{-1}, \vvec_2 = \threevec{0}{2}{1}, \vvec_3 = \threevec{1}{-2}{4} \text{.} \end{equation*}

    Form the matrix \(\left[\begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \vvec_3 \end{array}\right]\) and find its reduced row echelon form.

    What does this tell you about \(\span{\vvec_1,\vvec_2,\vvec_3}\text{?}\)

  5. If a set of vectors \(\vvec_1,\vvec_2,\ldots,\vvec_n\) spans \(\real^3\text{,}\) what can you say about the pivots of the matrix \(\left[\begin{array}{rrrr} \vvec_1\amp\vvec_2\amp\ldots\amp\vvec_n \end{array}\right]\text{?}\)

  6. What is the smallest number of vectors such that \(\span{\vvec_1,\vvec_2,\ldots,\vvec_n} = \real^3\text{?}\)

in-context