Linear Algebra, 5th Edition

2.5 The Change of Coordinate Matrix

In many areas of mathematics, a change of variable is used to simplify the appearance of an expression. For example, in calculus an antiderivative of $2x{e}^{x^2}$ can be found by making the change of variable $u=x^2$. The resulting expression is of such a simple form that an antiderivative is easily recognized:

$$\int 2x{e}^{x^2}dx=\int e^{u}du=e^u+c=e^{x^2}+c.$$

Similarly, in geometry the change of variable

$$\begin{array}{l}x={\displaystyle \frac{2}{\sqrt{5}}}{x}^{\prime }-{\displaystyle \frac{1}{\sqrt{5}}}{y}^{\prime }\\ y={\displaystyle \frac{1}{\sqrt{5}}}{x}^{\prime }+{\displaystyle \frac{2}{\sqrt{5}}}{y}^{\prime }\end{array}$$

can be used to transform the equation $2x^2-4xy+5y^2=1$ into the simpler equation ${(x^{\prime })}^2+6{(y^{\prime })}^2=1$, in which form it is easily seen to be the equation of an ellipse. (See Figure 2.4.) We will see how this change of variable is determined in Section 6.5. Geometrically, the change of variable

$$\left(\begin{array}{c}x\\ y\end{array}\right)\to \left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)$$

Figure 2.4

Figure ...