##### Preview Activity3.3.1

Since we will be using various bases and the coordinate systems they define, let's review how we translate between coordinate systems.

1. Suppose that we have a basis $$\bcal=\{\vvec_1,\vvec_2,\ldots,\vvec_m\}$$ for $$\real^m\text{.}$$ Explain what we mean by the representation $$\coords{\xvec}{\bcal}$$ of a vector $$\xvec$$ in the coordinate system defined by $$\bcal\text{.}$$

2. If we are given the representation $$\coords{\xvec}{\bcal}\text{,}$$ how can we recover the vector $$\xvec\text{?}$$

3. If we are given the vector $$\xvec\text{,}$$ how can we find $$\coords{\xvec}{\bcal}\text{?}$$

4. Suppose that

\begin{equation*} \bcal=\left\{\twovec{1}{3},\twovec{1}{1}\right\} \end{equation*}

is a basis for $$\real^2\text{.}$$ If $$\coords{\xvec}{\bcal} = \twovec{1}{-2}\text{,}$$ find the vector $$\xvec\text{.}$$

5. If $$\xvec=\twovec{2}{-4}\text{,}$$ find $$\coords{\xvec}{\bcal}\text{.}$$

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