##### Activity2.1.2

In this activity, we will look at linear combinations of a pair of vectors,

\begin{equation*} \vvec = \left[\begin{array}{r} 2 \\ 1 \end{array}\right], \wvec = \left[\begin{array}{r} 1 \\ 2 \end{array}\right] \end{equation*}

with weights $$a$$ and $$b\text{.}$$

LINEAR-COMBINATIONS

1. The weight $$b$$ is initially set to 0. Explain what happens as you vary $$a$$ with $$b=0\text{?}$$ How is this related to scalar multiplication?

2. What is the linear combination of $$\vvec$$ and $$\wvec$$ when $$a = 1$$ and $$b=-2\text{?}$$ You may find this result using the diagram, but you should also verify it by computing the linear combination.

3. Describe the vectors that arise when the weight $$b$$ is set to 1 and $$a$$ is varied. How is this related to our investigations in the preview activity?

4. Can the vector $$\left[\begin{array}{r} 0 \\ 0 \end{array} \right]$$ be expressed as a linear combination of $$\vvec$$ and $$\wvec\text{?}$$ If so, what are weights $$a$$ and $$b\text{?}$$

5. Can the vector $$\left[\begin{array}{r} 2 \\ -2 \end{array} \right]$$ be expressed as a linear combination of $$\vvec$$ and $$\wvec\text{?}$$ If so, what are weights $$a$$ and $$b\text{?}$$

6. Verify the result from the previous part by algebraically finding the weights $$a$$ and $$b$$ that form the linear combination $$\left[\begin{array}{r} 2 \\ -2 \end{array} \right]\text{.}$$

7. Can the vector $$\left[\begin{array}{r} 1.3 \\ -1.7 \end{array} \right]$$ be expressed as a linear combination of $$\vvec$$ and $$\wvec\text{?}$$ What about the vector $$\left[\begin{array}{r} 15.2 \\ 7.1 \end{array} \right]\text{?}$$

8. Are there any two-dimensional vectors that cannot be expressed as linear combinations of $$\vvec$$ and $$\wvec\text{?}$$

in-context