Preview Activity 3.3.1
Since we will be using various bases and the coordinate systems they define, let's review how we translate between coordinate systems.
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 \(\bcal\text{.}\)
If we are given the representation \(\coords{\xvec}{\bcal}\text{,}\) how can we recover the vector \(\xvec\text{?}\)
If we are given the vector \(\xvec\text{,}\) how can we find \(\coords{\xvec}{\bcal}\text{?}\)

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 \(\xvec\text{.}\)
If \(\xvec=\twovec{2}{4}\text{,}\) find \(\coords{\xvec}{\bcal}\text{.}\)