Exercise8
A theme that will later unfold concerns the use of coordinate systems. We can identify the point \((x,y)\) with the tip of the vector \(\left[\begin{array}{r}x\\y\end{array}\right]\text{,}\) drawn emanating from the origin. We can then think of the usual Cartesian coordinate system in terms of linear combinations of the vectors
\begin{equation*} \evec_1 = \left[\begin{array}{r} 1 \\ 0 \end{array}\right], \evec_2 = \left[\begin{array}{r} 0 \\ 1 \end{array}\right] \text{.} \end{equation*}The point \((2,3)\) is identified with the vector
\begin{equation*} \left[\begin{array}{r} 2 \\ 3 \end{array}\right] = 2\evec_1  3\evec_2 \text{.} \end{equation*}If we have vectors
\begin{equation*} \vvec_1 = \left[\begin{array}{r} 2 \\ 1 \end{array}\right], \vvec_2 = \left[\begin{array}{r} 1 \\ 2 \end{array}\right] \text{,} \end{equation*}we may define a new coordinate system, such that a point \(\{x,y\}\) will correspond to the vector
\begin{equation*} x\vvec_1 + y\vvec_2 \text{.} \end{equation*}For instance, the point \(\{2,3\}\) is shown on the right side of FigureĀ 8

Write the point \(\{2,3\}\) in standard coordinates; that is, find \(x\) and \(y\) such that
\begin{equation*} (x,y) = \{2,3\} \text{.} \end{equation*} 
Write the point \((2,3)\) in the new coordinate system; that is, find \(a\) and \(b\) such that
\begin{equation*} \{a,b\} = (2,3) \text{.} \end{equation*} Convert a general point \(\{a,b\}\text{,}\) expressed in the new coordinate system, into standard Cartesian coordinates \((x,y)\text{.}\)
What is the general strategy for converting a point from standard Cartesian coordinates \((x,y)\) to the new coordinates \(\{a,b\}\text{?}\) Actually implementing this strategy in general may take a bit of work so just describe the strategy. We will study this in more detail later.