Preview Activity 4.1.1

Before we introduce the definition of eigenvectors and eigenvalues, it will be helpful to remember some ideas we have seen previously.

  1. Suppose that \(\vvec\) is the vector shown in the figure. Sketch the vector \(2\vvec\) and the vector \(-\vvec\text{.}\)

  2. State the geometric effect that scalar multiplication has on the vector \(\vvec\text{.}\) Then sketch all the vectors of the form \(\lambda \vvec\) where \(\lambda\) is a scalar.

  3. State the geometric effect of the matrix transformation defined by

    \begin{equation*} \left[\begin{array}{rr} 3 \amp 0 \\ 0 \amp -1 \\ \end{array}\right]\text{.} \end{equation*}
  4. Suppose that \(A\) is a \(2\times 2\) matrix and that \(\vvec_1\) and \(\vvec_2\) are vectors such that

    \begin{equation*} A\vvec_1 = 3 \vvec_1, \qquad A\vvec_2 = -\vvec_2\text{.} \end{equation*}

    Use the linearity of matrix multiplication to express the following vectors in terms of \(\vvec_1\) and \(\vvec_2\text{.}\)

    1. \(A(4\vvec_1)\text{.}\)

    2. \(A(\vvec_1 + \vvec_2)\text{.}\)

    3. \(A(4\vvec_1 -3\vvec_2)\text{.}\)

    4. \(A^2\vvec_1\text{.}\)

    5. \(A^2(4\vvec_1 - 3\vvec_2)\text{.}\)

    6. \(A^4\vvec_1\text{.}\)