###### 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.

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.

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