###### Preview Activity4.3.1

Let's recall how a vector in $$\real^2$$ can be represented in a coordinate system defined by a basis $$\bcal=\{\vvec_1, \vvec_2\}\text{.}$$

1. Suppose that we consider the basis $$\bcal$$ defined by

\begin{equation*} \vvec_1 = \twovec{1}{1},\qquad \vvec_2 = \twovec{-1}{0}\text{.} \end{equation*}

Find the vector $$\xvec$$ whose representation in the coordinate system defined by $$\bcal$$ is $$\coords{\xvec}{\bcal} = \twovec{-3}{2}\text{.}$$

2. Consider the vector $$\xvec=\twovec{4}{5}$$ and find its representation $$\coords{\xvec}{\bcal}$$ in the coordinate system defined by $$\bcal\text{.}$$

3. How do we use the matrix $$C_{\bcal} = \left[\begin{array}{rr} \vvec_1 \amp \vvec_2 \end{array}\right]$$ to convert a vector's representation $$\coords{\xvec}{\bcal}$$ in the coordinate system defined by $$\bcal$$ into its standard representation $$\xvec\text{?}$$ How do we use this matrix to convert $$\xvec$$ into $$\coords{\xvec}{\bcal}\text{?}$$

4. Suppose that we have a matrix $$A$$ whose eigenvectors are $$\vvec_1$$ and $$\vvec_2$$ and associated eigenvalues are $$\lambda_1=4$$ and $$\lambda_2 = 2\text{.}$$ Express the vector $$A(-3\vvec_1 +5\vvec_2)$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

5. If $$\coords{\xvec}{\bcal} = \twovec{-3}{5}\text{,}$$ find $$\coords{A\xvec}{\bcal}\text{.}$$

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