##### 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