###### 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*}
in-context