Preview Activity 3.2.1

Consider the vectors

\begin{equation*} \vvec_1 = \twovec{2}{1}, \vvec_2 = \twovec{1}{2} \end{equation*}

in \(\real^2\text{.}\)

  1. Indicate the linear combination \(\vvec_1 - 2\vvec_2\) on FigureĀ 3.2.2.

    Figure 3.2.2 Linear combinations of \(\vvec_1\) and \(\vvec_2\text{.}\)

  2. Express the vector \(\twovec{-3}{0}\) as a linear combination of \(\vvec_1\) and \(\vvec_2\text{.}\)

  3. Find the linear combination \(10\vvec_1 - 13\vvec_2\text{.}\)

  4. Express the vector \(\twovec{16}{-4}\) as a linear combination of \(\vvec_1\) and \(\vvec_2\text{.}\)

  5. Explain why every vector in \(\real^2\) can be written as a linear combination of \(\vvec_1\) and \(\vvec_2\) in exactly one way.

in-context