Archive for 八月, 2009

Challenging problem in Linear Algebra

八月 19, 2009

Let $\mathcal{M} = \left\{ \begin{pmatrix} a & -b \\b & a \end{pmatrix} \, | \, a, b, \in \mathbb{R} \right\}.$

Solve the following matrix equation $X^4 + X^2 + I = 0$ where $X \in \mathcal{M}.$

Posted in HKAL, Linear Algebra, Pure Mathematics, Teaching | 1 Comment »

Answers to Linear Algebra Concept Check

八月 19, 2009

Let (E) be the following system:

$a_{11}x + a_{12}y + a_{13}z = b_1\\a_{21}x + a_{22}y + a_{23}z = b_2 \\a_{31}x + a_{32}y + a_{33}z = b_3$

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Mental exercises for my Pure Complex Class (1)

八月 19, 2009

Dear Students,

Please avoid as much calculations as possible. In ideal situation, you should not do any calculation at all.

Problem 1
Let $z = 3 + 4i$ be a complex number.
(a) Find $Re(z).$
(b) Find $Im(z).$
(c) Find $\overline{z}.$
(d) Find $|z|^2.$
(e) Find $Re(1/z).$
(f) What is $z \overline{z}?$
(g) Suppose $z = re^{i \theta}$ , write $r$ in terms of $z$

Problem 2
(a) Write $z = i$ in the form $re^{i \theta}.$
(b) Find the two square roots of $i$ and represent the two square roots on the Gaussian Plane.

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Posted in HKAL, Pure Mathematics, Teaching | 11 意見»

Selected Solutions to My Pure Complex Class (1)

八月 19, 2009

Lesson 1:
Page 26
$\begin{aligned} \sum_{k = 1}^{n} \cos^2(k \theta) &= \sum_{k = 1}^{n} \frac{ 1 + \cos(2 k \theta) }{2} \\ &= \frac{n}{2} + \frac{1}{2}\sum_{k = 1}^{n} \cos(2 k \theta) \\ &= \frac{n}{2} + \frac{\sin(n \theta) \cos((n + 1) \theta)}{2\sin(\theta)}, \end{aligned}$ where the last line follows from part a(ii).