Exercise 8
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
The point \((2,3)\) is identified with the vector
If we have vectors
we may define a new coordinate system, such that a point \(\{x,y\}\) will correspond to the vector
For instance, the point \(\{2,3\}\) is shown on the right side of FigureĀ 2.1.8

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.