viewing complex numbers as a linear transformation

Question

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.

Related: http://math.stackexchange.com/questions/886872/history-of-the-matrix-representation-of-complex-numbers http://math.stackexchange.com/questions/845907/usefulness-of-alternative-constructions-of-the-complex-numbers — Incnis Mrsi, Nov 05 '14 at 12:03

score 0 · Accepted Answer · answered Mar 14 '14 at 23:43

0

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.

answered Mar 14 '14 at 23:43

mookid

28,795

thanks. do you know of a book or notes that talk about geometric meanings of matrix transformations. – mark leeds Mar 14 '14 at 23:45
I have no idea. Almost every course covering the subject should contain such content. – mookid Mar 14 '14 at 23:48
This one looks nice. – mookid Mar 14 '14 at 23:52
thank you. I am going to check the question as closed because between your answer and the link, I should be able to understand. thanks again. – mark leeds Mar 14 '14 at 23:54
@mark leeds: thanks. – mookid Mar 14 '14 at 23:57

viewing complex numbers as a linear transformation

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