I am studying an intro to complex analysis and geometry book in order to become more adept with complex numbers and hopefully eventually the basics of complex analysis.
I love the explanations but I am having a lot of trouble with the exercises. The following question is one of them.
Suppose one has, $L$ , the $ 2 \times 2$ matrix
\begin{array}{cc} a & -b \\ b & a \end{array}
A) suppose that $a^2 + b^2 = 1$. What is the geometric meaning of multiplication by $L$.
B) suppose that $b = 0$. What is the geometric meaning of multiplication by $L$.
This is the part of the book where he talks about viewing complex numbers as a general linear transformation from $R^2$ to $R^2$, namely $(x,y) \longrightarrow (ax - by, bx + ay)$. Thank you for any help or relevant references.
Hint: A) $1 = a^2 + b^2 $ so there is a $\theta$ with $a = \cos\theta$ and $b=-\sin\theta$, then this is a rotation of an angle $\theta$ around the origin.
B) the transformation is an enlargement of $x,y$ with the same ratio.