math115: up to date

docs/ce1/math115.md

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 # MATH 115: Linear Algebra
+
+## Set theory
+
+!!! definition
+    - Natural numbers ($\mathbb N$) are all **integers** greater than zero.
+    - Integers ($\mathbb Z$) are all non-decimal numbers.
+    - Rational numbers ($\mathbb Q$) are all numbers representable as a fraction.
+    - Irrational numbers are all **real** numbers not representable as a fraction.
+    - Real numbers ($\mathbb R$) are all rational or irrational numbers.
+
+The **subset sign** ($\subseteq$) indicates that one **set** is strictly within another. The **not subset sign** ($\not\subseteq$) indicates that at least one element in the first set is not in the second.
+
+!!! example
+    - Natural numbers are a subset of integers, or $\mathbb N \subseteq \mathbb Z$.
+    - Integers are not a subset of natural numbers, or $\mathbb Z \not\subseteq \mathbb N$.
+
+!!! warning
+    The subset sign is not to be confused with the **element of** sign ($\in$), as the former only applies to sets while the latter only applies to elements.
+
+Sets can be subtracted with a **backslash** (\\), returning a set with all elements in the first set not in the second.
+
+!!! example
+    The set of irrational numbers can be represented as the difference between the real and rational number sets, or:
+    $$\mathbb R \backslash \mathbb Q$$
+
+## Complex numbers
+
+A complex number can be represented in the form:
+$$x+yj$$
+
+where $x$ and $y$ are real numbers, and $j$ is the imaginary $\sqrt{-1}$ (also known as $i$ outside of engineering). This implies that every real number is also in the set of complex numbers as $y$ can be set to zero.
+
+!!! definition
+    - $Re(z)$ is the **real component** of complex number $z$.
+    - $Im(z)$ is the **imaginary component** of complex number $z$.
+
+These numbers can be treated effectively like any other number.
+
+### Properties of complex numbers
+
+All of these properties can be derived from expanding the standard forms.
+
+Where $z=x+yj$ and $w=a+bj$:
+
+$$
+\begin{align*}
+zw&=(ax-by)+(bx+ay)j \\
+\frac{1}{z} &= \frac{x}{x^2+y^2} - \frac{y}{x^2+y^2}j \\
+z^0 &= 1
+\end{align*}
+$$
+
+??? example
+    If $z=2+5j$ and $w=1+3j$:
+    $$
+    \begin{align*}
+    \frac{z}{w} &= (2+5j)(\frac{1}{1+9}-\frac{3}{1+9}j) \\
+    &= (2+5j)(\frac{1}{10}-\frac{3}{10j}) \\
+    &= \frac{1}{5}-\frac{3}{5}j+\frac{1}{2}j+\frac{3}{2} \\
+    &= \frac{17}{10}-\frac{1}{10}j
+    \end{align*}
+    $$
+
+??? example
+    To solve for $z$ in $z^2+4=0$:
+    $$
+    \begin{align*}
+    (x+yj)^2&=-4 \\
+    x^2+2xyj - y^2 &= -4 + 0j \\
+    (x^2-y^2) + 2xyj &= -4+0j \\
+    \\
+    ∵ x, y \in \mathbb R: 2xyj &= 0j \\
+    ∴ \begin{cases}
+    x^2-y^2=-4 \\
+    2xy = 0
+    \end{cases} \\
+    \\
+    x=0 &\text{ or } y=0 \\
+    \pu{if } x=0&: y =\pm 2 \\
+    \pu{if } y=0&: \text{no real solutions} \\
+    \\
+    ∴ x&=0, y=\pm 2 \\
+    z&=\pm 2j
+    \end{align*}
+    $$
+
+??? example
+    To solve for $z$ in $z^2=5+12j$:
+    $$
+    \begin{align*}
+    (x+yj)^2&=5+12j \\
+    (x^2-j^2)+2xyj = 5+12j \\
+    \\
+    \begin{cases}
+    x^2-y^2=5 \\
+    2xy = 12
+    \end{cases} \\
+    \\
+    y &= \frac{6}{x} \\
+    x^2 - \frac{6}{x}^2 &= 5\\
+    x^4 - 36 - 5x^2 &= 0 \\
+    x^2 &= 9, -4, x\in \mathbb R \\
+    x &= 3, -3 \\
+    y &= 2, -2 \\
+    z &= 3+2j, -3-2j
+    \end{align*}
+    $$
+
+### Conjugates
+
+The conjugate of any number can be written with a bar above it.
+$$\overline{x+yj} = x-yj$$
+
+The conjugate of a conjugate is the original number.
+$$\overline{\overline{ z}} = z$$
+
+$z$ is a real number **if and only if** its conjugate is itself.
+$$z\in\mathbb R \iff \overline{z}=z$$
+
+$z$ is purely imaginary **if and only if** its conjugate is the negative version of itself.
+$$z\in\text{only imaginary} \iff \overline{z}=-z$$
+
+Conjugates are flexible and can almost be treated as just another factor.
+$$
+\begin{align*}
+\overline{z+w}&=\overline{z}+\overline{w} \\
+\overline{zw}&=(\overline{z})(\overline{w}) \\
+\overline{z^k}&=\overline{z}^k \\
+\overline{\biggr(\frac{z}{w}\biggr)} &= \frac{\overline{z}}{\overline{w}}, w\neq 0
+\end{align*}
+$$
+
+### Modulus
+
+The modulus of a number is represented by the absolute value sign. It is equal to its magnitude if the complex number were a vector.
+$$|z| = \sqrt{x^2+y^2}$$
+
+!!! example
+    The modulus of complex number $2-j$ is:
+    $$
+    \begin{align*}
+    |2-j|&=\sqrt{2^2+(-1)^2} \\
+    &= -5
+    \end{align*}
+    $$
+
+If there is no imaginary component, a complex number's modulus is its absolute value.
+$$z\in\mathbb R: |z|=|Re(z)|$$
+
+Complex numbers cannot be directly compared because imaginary numbers have no inequalities, but their moduli can — the modulus of one complex number can be greater than another's.
+
+#### Properties of moduli
+
+These can be also be manually derived.
+
+If the modulus is zero, the complex number is zero.
+$$|z|=0 \iff z=0$$
+
+The modulus of the conjugate is equal to the modulus of the original.
+$$|\overline{z}| = |z|$$
+
+The number multiplied by the conjugate modulus is the square of the modulus.
+$$z|\overline{z}|=|z|^2$$
+
+Moduli are also almost just a factor:
+$$
+\begin{align*}
+\biggr|\frac{z}{w}\biggr| &= \frac{|z|}{|w|}, w \neq 0 \\
+|zw| &= |z||w|
+\end{align*}
+$$
+
+The moduli of the sum is always less than the sum of the moduli of the individual numbers — this is also known as the triangle inequality theorem.
+
+$$|z+w| \leq |z|+|w|$$
+
+### Geometry
+
+In setting the x- and y-axes to the imaginary and real components of a complex number, complex numbers can be represented almost as vectors.
+
+<img src="https://upload.wikimedia.org/wikipedia/commons/6/69/Complex_conjugate_picture.svg">(Source: Wikimedia Commons, GNU FGL 1.2 or later)</img>
+
+The complex number $x+yj$ will be on the point $(x, y)$, and the modulus is the magnitude of the vector. Complex number moduli can be compared graphically if their points lie within a drawn circle centred on the origin with a point on another vector.
+
+### Polar form
+
+The variable $r$ is equal to the modulus of a complex number $|z|$.
+
+From the Pythagorean theorem, the polar form of a complex number can be expressed using the angle of the modulus to the real axis. Where $\theta$ is the angle of the modulus to the real axis:
+$$z=r(\cos\theta + j\sin\theta)$$
+
+Trigonometry can be used to calculate $\cos\theta$ and $\sin\theta$ as $\cos\theta = \frac{x}{r}$ and $\sin\theta = \frac{y}{r}$.
+
+!!! example
+    $1+\sqrt{3}j=2\big(\cos\frac{\pi}{3} + j\sin\frac{\pi}{3}\big)$
+
+!!! warning
+    The polar form is not unique because going around 360° results in the same vector. Where $k$ is any integer.
+    $$r(\cos\theta + j\sin\theta) = r(\cos(\theta+2k\pi) + j\sin(\theta+2k\pi))$$
+
+The polar form is useful for the multiplication of complex numbers.
+
+Because of the angle sum identities:
+$$z_1z_2=r_1r_2(\cos(\theta_1+\theta_2) + j\sin(\theta_1+\theta_2))$$
+
+This can be extrapolated into Moivre's theorem:
+$$z^n=r^n(\cos(n\theta) + j\sin(n\theta))$$
+
+To determine the roots of a complex number, Moivre's theorem can be used again:
+$$\sqrt[n]{z} = \sqrt[n]{r}\big(\cos\big(\frac{\theta + 2k\pi}{n}\big) + j\sin\big(\frac{\theta + 2k\pi}{n}\big)\big)$$
+
+where $k$ is every number in the range $[0, n-1], k\in\mathbb Z$.
+
+!!! example
+    To find all answers for $w^5=-32$:
+    $$
+    \begin{align*}
+    w^5 &= 32(\cos\theta + \sin\theta) \\
+    w_k &= \sqrt[5]{32}\biggr[\cos\biggr(\frac{\pi + 2k\pi}{5}\biggr) + j\sin\biggr(\frac{\pi+2k\pi}{5}\biggr)\biggr]
+    w_0 &= 2\biggr(\cos\frac{\pi}{5} + j\sin\frac{\pi}{5}\biggr) = 2e^{j\frac{\pi}{5}} \\
+    w_1 &= 2\biggr(\cos\frac{3\pi}{5} + j\sin\frac{3\pi}{5}\biggr) = 2e^{j\frac{3\pi}{5}} \\
+    w_2 &= 2(\cos\pi + j\sin\pi = 2e^{j\pi} \\
+    \\
+    \text{etc.}
+    \end{align*}
+    $$
+    
+The **exponential** form of a complex number employs **Euler's identity**:
+$$
+\begin{align*}
+e^{j\pi} &= -1 \\
+e^{j\pi} &= \cos\theta + j\sin\theta \\
+z &= re^{j\pi}
+\end{align*}
+$$
+
+### Proofs
+
+!!! example
+
+## Vectors
+
+The column vector shows a vector of the form $(x, y, ...)$ from top to bottom as $(x_1, x_2, ...)$ as the number of dimensions increases.
+
+$$
+\newcommand\colv[1]{\begin{bmatrix}#1\end{bmatrix}}
+\colv{x_1 \\ x_2 \\ x_3}
+$$
+
+The zero vector is full of zeroes.
+$$
+\colv{0 \\ 0 \\ 0}
+$$
+
+!!! warning
+    Vectors of different dimensions cannot be compared — the missing dimensions cannot be treated as 0.
+
+The standard form of a vector is written as the difference between two points: $\vec{OA}$ where $O$ is the origin and $A$ is any point. $\vec{AB}$ is the vector as a difference between two points.