Chapter 6.3, Problem 18E

If $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ is an invertible linear transformation from ${ℝ}^2$ to ${ℝ}^2$ , then the image $T\left(\Omega \right)$ of the unit circle $\Omega$ is an ellipse. See Exercise 2.2.54.
a. Sketch this ellipse when $A=\left[\begin{array}{cc}p& 0\\ 0& q\end{array}\right]$ where p and q are positive. What is its area?
b. For an arbitrary invertible transformation $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ denote the lengths of the semimajor and semi minor axes of $T\left(\Omega \right)$ by a and b, respectively. What is the relationship among a, b, and $\det \left(A\right)$ ?

c. For the transformation $T\left(\stackrel{\to }{x}\right)=\left[\begin{array}{cc}3& 1\\ 1& 3\end{array}\right]\stackrel{\to }{x}$ sketch this ellipse and determine its axes. Hint: Consider $T\left[\begin{array}{l}1\\ 1\end{array}\right]$ and $T\left[\begin{array}{l}1\\ -1\end{array}\right]$ .