diff --git a/docs/ce1/math115.md b/docs/ce1/math115.md index e87f384..e90f036 100644 --- a/docs/ce1/math115.md +++ b/docs/ce1/math115.md @@ -1 +1,259 @@ # 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. + +(Source: Wikimedia Commons, GNU FGL 1.2 or later) + +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.