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