COMP0011 but wtf

1. Complex Numbers

The Imaginary Unit

$i = \sqrt{-1}$
I.e., $i^2 = -1$.

Complex Numbers

$\mathbb{C} = \big\{ a + bi \;\big\vert\; a, b \in \R \big\}$
I.e., a linear combination of $1$ and $i$.

We write $z = a + bi$ to illustrate the components of a $z \in \mathbb{C}$.

Remember that $\R \subset \mathbb{C}$.

Forms of Representation

Complex numbers are commonly treated as points on a 2-d plane; $1$ is the horizontal unit vector and $i$ is the vertical unit vector.
In this case, rectangular form is equivalent to cartesian coordinates; polar form is equivalent to, well, polar coordinates.

Rectangular

the 2-term linear combination described above
For a $z = a + bi$, $a$ is the real component and $b$ is the imaginary component.

Aka. Cartesian form (for obvious reasons).

Polar

whirry form.
For a $z = a + bi = r(\cos \theta + i \sin \theta)$,
$r = |z| = \sqrt{a^2 + b^2}$ is the modulus and $\theta = \operatorname{arg}(z) = \operatorname{atan2}(b, a)$ is the argument.
Also, $z = re^{i\theta}$ (ask euler).
$r, \theta \in \R$, $r \ge 0$.

$r \operatorname{cis}\theta$ is short for $r(\cos \theta + i \sin \theta)$

Arithmetic

In rectangular form, everything actually just works under standard “algebraic/symbolic” arithmetic, treating $i$ as an atomic symbol, and simplifying $i^2$ to $-1$ wherever possible.

Addition

$(a_1 + b_1 i) + (a_2 + b_2 i) = (a_1 + a_2) + (b_1 + b_2) i$

$|z_1 + z_2| \le |z_1| + |z_2|$

Multiplication

$(a_1 + b_1 i) \times (a_2 + b_2 i) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + b_1 a_2) i$
$(r_1 \operatorname{cis} \theta_1) \times (r_2 \operatorname{cis} \theta_2) = r_1 r_2 \operatorname{cis}(\theta_1 + \theta_2)$

Conjugation

The conjugate of $x$ is denoted $\bar x$.
$\overline{a + bi} = a - bi$
$\overline{r \operatorname{cis} \theta} = r \operatorname{cis} -\theta$

Observe that $x \times \bar x \in \R$

Division

$\displaystyle \frac{z_1}{z_2} = \frac{a_1 + b_1 i}{a_2 + b_2 i} = \frac{a_1 a_2 + b_1 b_2}{z_2 \bar z_2} + \frac{b_1 a_2 - a_1 b_2}{z_2 \bar z_2}i \\[-0.3em]$
$\displaystyle \frac{z_1}{z_2} = \frac{r_1 \operatorname{cis} \theta_1}{r_2 \operatorname{cis} \theta_2} = \frac{r_1}{r_2} \operatorname{cis}(\theta_1 - \theta_2) \\[1em]$

Exponentiation

$(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(r \theta)$

This is De Moivre’s Theorem.

Standard Angles

know them stuffs

Roots of Unity

I.e., complex $n$-th roots of 1, of the form $z^n = 1$ for some $n \in \N$

$\displaystyle z^n = (re^{i\theta})^n = r^n e^{in\theta} = 1$
Clearly, $r = 1$;
$\operatorname{arg}(z^n) = \operatorname{arg}(1 + 0i) = \arctan(0) = 2k\pi$
$\displaystyle \!\!\implies n\theta = 2k\pi \iff \theta = \frac{2k\pi}{n} \\[-0.5em]$
I.e., $\displaystyle z^n = 1 \implies z = \operatorname{cis}\Big( k \frac{2\pi}{n} \Big)$

The $n$ unique solutions are given by the first $n$ values of $k$ (from $0$ to $n-1$).

Geolaulihafhiafahu

yep

2. Continuous Functions

Function

$A \mapsto B$, for sets $A$ ad $B$
Maps every $a \in A$ to some $b \in B$

Exp & Log

Exponential Function

$\operatorname{exp}: A \mapsto A$ for a ring $A$ where
$\operatorname{exp}(a + b) = \operatorname{exp}(a) \ast \operatorname{exp}(b)$

Logarithm Function

Inverse of the above
$\log_a (a^x) = x$

$\ln(x \times y) = \ln(x) + \ln(y)$
$\ln(x / y) = \ln(x) - \ln(y)$
$\ln(x^y) = y \ln(x)$
$\log_b(x) = 1 / \log_x(b)$
$\log_b(x) = \ln(x) / \ln(b)$
($\ln$ denotes a logarithm of arbitrary (but same within the same line) base.)

Limits & Continuity

$(\epsilon, \delta)$-definition of a limit:
$$\lim_{x \to a} f(x) = L \impliedby \big( \forall \varepsilon \gt 0 \ \exists \delta \gt 0 : 0 \lt |x - a| \lt \delta \implies |f(x) - L| \lt \varepsilon \big)$$

Observe that due to $\bm{0 \lt}\; |x - a|$, $f$ does not have to be continuous at $a$.

$\lim_{x \to +\infty} f(x) = L \iff \lim_{x' \to 0^+} f(1 / x') = L$
$\lim_{x \to a} f(x) = \infty \iff \lim_{x \to c}1 / f(x) = 0$

Little $o$ notation

$f = o(g)$ near $b$ $\iff$ $\displaystyle \lim_{x \to b} \frac{f(x)}{g(x)} = 0$

Big $O$ notation

$f = O(g)$ near $b$ $\iff$ $\displaystyle \limsup_{x \to b} \left| \frac{f(x)}{g(x)} \right| < \infty$

Indeterminate Forms

$0 / 0$
$\infty / \infty$
$0 \times \infty$
$\infty - \infty$
$0^0$
$1^\infty$¹
$\infty^0$

Limit-definition of continuity:
$f : D \mapsto \R$ is continuous at $a$ $\impliedby$ $\lim_{x \to a} f(x) = f(a)$

A function is continuous on an open interval $(l, r)$ if it is continuous at all values between $l$ and $r$.
A function is continuous (everywhere) if it is continuous on $(-\infty, +\infty)$.

Combination Theorem (for continuous functions)

For continuous (real) functions $f$ and $g$,
$f + g$, $f - g$, $f \times g$, $f \circ g$, and $f / g$ are all continuous.
(In the division case, discontinuous at $g(x) = 0$)

I.e., the arithmetic operations “preserve” continuity (except div by 0).

Intermediate Value Theorem

For (real) function $f$ continuous on $[l, r]$,
let $L = f(l)$, $R = f(r), L \lt R$, then
$\forall k \in [L, R] \ \exists a \in [l, r] : f(a) = k$

I.e., the image of $f$ over $[l, r]$ is $[f(l), f(r)]$.

3. Vector Spaces

See notes on Algebraic Structures which formally defines this towards the bottom.

Collinearity

$\vec{u} = a\vec{v}$ $\iff$ $\vec{u}$ and $\vec{v}$ are colinear (for $a \in F$).

Linear Combination

some $a\vec{u} + b\vec{v}$

Span

the set of all linear combinations of $\vec{u} \in S$ for some family of vectors $S$.

Linear Independence

not colinear
For a set of vectors (more than 2), it means that no vector is in the span of any other(s).

I.e., no $a_n \ne 0$ s.t. $a_1\vec{u}_1 + \cdots + a_n\vec{u}_n = \vec{0}$

Dimensionality

yes
$\R^n$ denotes an $n$-dimensional vector space over $\R$

Basis

A set $S$ of vectors where the span of $S$ is $V$ (its vector space)
(and $S$ are linearly independent)
$|S|$ is the dimension of $V$.

A canonical basis is aslkdfjasl;dfjslkjaf yeah the obvious one

Linear Maps

A mapping/transformation (between vector spaces) $f : V \mapsto W$ is linear if
$\forall u, v \in V \, \exists a \in F : \quad f(u + v) = f(u) + f(v) \ \land\ f(a \cdot u) = a \cdot f(u) \space$.

Remember that linear maps can be composed.
A linear map $f$ is an endomorphism iff $V = W$ (in $f : V \mapsto W$).l

Kernel

$\operatorname{Ker}(f) = \{ v \in V \mid f(v) = \vec{0} \} \space$ (for a linear map $f: V \mapsto W$ on vector space $V$).

This is aka. the Null Space (of the equivalent matrix representation).

Image

usual definition as per any other function

Polynomials (over $F$, in some indeterminant) is a vector space (over $F$).
The canonical basis of $\R_2[x]$ is $\{ 1, x, x^2 \}$.

4. Matrices

A linear map $f : V \mapsto W$ where $V$ has dimension $m$ and $W$ $n$ can be represented as a $m \times n$ matrix.
Conversely, all matrices (of a field $F$) are representative of some linear map across the vector spaces of $F$.

Matrix multiplication is the equivalent of function composition, and as such shares all the properties of composition.
$\mathcal{M}(m, n) \times \mathcal{M}(n, k) \in \mathcal{M}(m, k)$.
Remember that matrix multiplication is notationally right-associative.

$M$ has an inverse
$\iff$ $M$ as column vectors are linearly independent
$\iff$ $M$ as row vectors are linearly independent
$\iff$ $\operatorname{Ker}(M) = 0$
$\iff$ its equivalent map $f$ is bijective

I.e., $M$ does not reduce (change) dimensionality – $M$ must be a square matrix.

Determinant

$ad - bc$, otherwise eeeeeeeeee

5. Sequences & Series

Monotonic

non-decreasing or non-increasing sequence

Bounded

sequence that does not reach infinity

Convergence

the existence of a limit at infinity for a sequence

Theorem of Monotone Convergence

a monotone sequence converges iff it is bounded

Squeeze Theorem

yeah

Arithmetic Series

$\displaystyle S_n = n \cdot \frac{a_1 + a_n}{2} = n \cdot \frac{2 a_1 + (n - 1) d}{2}$

Geometric

$\displaystyle S_n = a \cdot \frac{1 - r^n}{1 - r} \\[-0.5em]$
$\displaystyle S_\infty = a \cdot \frac{1}{1 - r}$ (converges when $|r| < 1$) $\\[0.8em]$

Absolute Convergence

when the absolute series $\sum |a_n|$ converges

Note that absolute convergence is stronger than (implies) convergence.

D’Alembert Criterion

For ratio $r = \left| \frac{a_{n + 1}}{a_n} \right|$ of a series,
$\liminf_{n \to \infty} r > 1$ $\implies$ series diverges;
$\limsup_{n \to \infty} r < 1$ $\implies$ series absolutely converges;
otherwise :shrug:

6. Differential Calculus

$\displaystyle \frac{\mathop{\mathrm{d}}}{\mathop{\mathrm{d}x}} f(x) = f'(x)$

$$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$

Elementary Differentiation Rules

Linearity

$\big( a f(x) \big)' = a f'(x)$
$\big( f(x) + g(x) \big)' = f'(x) + g'(x)$

Product Rule

$\displaystyle \big( f(x)g(x) \big)' = f'(x)g(x) + f(x)g'(x)$

Chain Rule

$\displaystyle \big( (f \circ g)(x) \big)' = (f' \circ g)(x)g'(x)$

Inverse Function Rule

$\displaystyle \big( f^{-1}(x) \big)' = 1/(f' \circ f^{-1})(x)$

Power Rule

$\displaystyle \Big( f(x)^{g(x)} \Big)' = \exp \!\big(g(x) \ln f(x) \big)' \\ = \Big( g(x) \frac{f'(x)}{f(x)} + g'(x) \ln f(x) \Big) f(x)^{g(x)}$

Derived rules:

Poorman’s Power Rule

$\displaystyle \frac{\mathop{\mathrm{d}}}{\mathop{\mathrm{d}x}} x^n = nx^{n-1}$

Reciprocal Rule

$\displaystyle \big( 1/f(x) \big)' = f'(x) / f(x)^2$

Quotient Rule

$\displaystyle \bigg( \frac{f(x)}{g(x)} \bigg)' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$

Exp & Log rules:

Log Rule

$\displaystyle \big( \log_c x \big)' = \frac{1}{x \ln c} \qquad c > 1 \\[-0.5em]$
$\displaystyle \big( \ln |x| \big)' = \frac{1}{x} \qquad x \ne 0$

Exp Rule

$\displaystyle ( c^{ax} )' = ac^{ax} \ln c \qquad c > 0$
$\displaystyle (e^{ax})' = ae^{ax}$

Standard Derivatives (Trig)

$$\begin{align*} \sin x &\to \cos x \\ \cos x &\to -\sin x \\ \tan x &\to \sec^2 x = 1 + \tan^2 x \\ \csc x &\to -\csc x \cot x \\ \sec x &\to \sec x \tan x \\ \cot x &\to -\csc^2 x = -1 - \cot^2 x \\ \arcsin x &\to \frac{1}{\sqrt{1 - x^2}} \\ \arccos x &\to -\frac{1}{\sqrt{1 - x^2}} \\ \arctan x &\to \frac{1}{1 + x^2} \\ \end{align*}$$

7. Integral Calculus

8. Polynomials

$\operatorname{deg}(P + Q) \le \max\!\big( \!\operatorname{deg}(P), \operatorname{deg}(Q) \big)$
$\operatorname{deg}(P \times Q) = \operatorname{deg}(P) + \operatorname{deg}(Q)$

Irreducible Polynomial

a polynomial with no factors of degree ≥ 1

A polynomial splits if it can be factored into just degree-1 (linear) factors.

Euclidean Division

For polynomials $A$ and $B$, there exists $Q$ and $R$ s.t.
$A = Q \cdot B + R, \quad \operatorname{deg}(R) \lt \operatorname{deg}(B)$

In $\R$, all irreducible polynomials are of degree 2. (when discriminant is negative)
Every polynomial in $\R$ can be factorized into a combination of linears and degree-2’s.

In $\mathbb{C}$, all polynomials split into linear factors.

The multiplicity of a root is its, well, multiplicity.

9. Probabilities

Universe / sample space $\Omega$
An event is a set of “accepting” outcomes; a subset of the universe.
As such, set operations naturally correspond to logical operations.

$A \cup A' = \Omega$, obviously,

e:
$$\begin{align*} P(A) &= P(A \cap B) + P(A \cap B') \\[1em] P(A \cap B) &= P(A) \cdot P(B | A) = P(B) \cdot P(A | B) \\[1em] P(A | B) &= \frac{P(A \cap B)}{P(B)} \\[1em] P(A | B) &= \frac{P(A) \cdot P(B | A)}{P(B)} \\[1em] P(A | B) &= \frac{P(A) \cdot P(B | A)}{P(A) \cdot P(B | A) + P(A') \cdot P(B | A')} \end{align*}$$

$A$ and $B$ are independent if $P(A \cap B) = P(A) \cdot P(B)$ (or, equivalently,) $P(A | B) = P(A)$; $P(B | A) = P(B)$.

Expected Value:

For a numeric sample space,
$$\operatorname{E}[X] = \sum_{x \in \Omega} x \cdot P(X = x)$$

$E$ is linear; $\operatorname{E}[X + Y] = \operatorname{E}[X] + \operatorname{E}[Y]$ and $\operatorname{E}[a \cdot X] = a \cdot \operatorname{E}[X]$.

For Binomial $X$, $\operatorname{E}[X] = np$.
For Normal $X$, $\operatorname{E}[X] = \mu$

Variance:

$$\begin{align*} \operatorname{Var}(X) &= \operatorname{E}\!\big[ (X - \operatorname{E}[X])^2 \big] \\ &= \operatorname{E}\!\big[X^2 - 2X\operatorname{E}[X] + \operatorname{E}[X]^2 \big] \\ &= \operatorname{E}[X^2] - 2\operatorname{E}[X]\operatorname{E}[X] + \operatorname{E}[X]^2 \\ &= \operatorname{E}[X^2] - \operatorname{E}[X]^2 \end{align*}$$

For Bernoulli $X$, $\operatorname{Var}(X) = p - p^2 = p(1 - p)$.

10. Statistics

Standard deviation, $\displaystyle \sigma = \sqrt{\operatorname{Var}} = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \mu)^2}$

Confidence Interval
$\displaystyle \mu \pm z \left( \frac{\sigma}{\sqrt{N}} \right)$

%	$z$-score
80%	1.28
85%	1.44
90%	1.64
95%	1.96
98%	2.33
99%	2.58

Hypothesis Testing

H0, null hypo, and H1, alternative hypo

Assume H0, calculate probability of observed outcome. H0 is rejectable if probability below a chosen significance threshold.

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1. not literally indeterminate (=1), but yes when your limit goes like this (go google) ↩︎