##### Preview Activity4.1.1

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. $$A^4\vvec_1\text{.}$$

in-context