Comprehensive Study Guide for Higher Mathematics: Relations, Vector Spaces, Complex Numbers, Matrices, and Diagonalization

General Concepts on Relations and Algebraic Structures

• Relations Between Elements

  • Definition of Relation:
  • A relation between two sets is a triplet $\mathcal{R} = (E, F, \Gamma)$ consisting of a starting set $E$ (domain of departure), an arrival set $F$ (target domain), and a subset $\Gamma$ of the Cartesian product $E \times F$, called the graph of the relation $\mathcal{R}$.
  • Two relations $\mathcal{R}_1 = (E_1, F_1, \Gamma_1)$ and $\mathcal{R}_2 = (E_2, F_2, \Gamma_2)$ are equal if and only if $E_1 = E_2$, $F_1 = F_2$, and $\Gamma_1 = \Gamma_2$
  • Example:
  • Let $E = \{2, 3, 5\}$ and $F = \{4, 6, 8\}$. The relation associating $x \in E$ with its double in $F$ is given by $\mathcal{R} = \{(2, 4), (3, 6)\}$.
  • Properties of a Binary Relation:
  • A binary relation $\mathcal{R}$ defined on a set $E$ is said to be:
    • Reflexive: if $x\mathcal{R}x$ for all $x \in E$
    • Symmetric: if $x\mathcal{R}y \implies y\mathcal{R}x$ for all $x, y \in E$
    • Anti-symmetric: if $x\mathcal{R}y$ and $y\mathcal{R}x \implies x = y$ for all $x, y \in E$
    • Transitive: if $x\mathcal{R}y$ and $y\mathcal{R}z \implies x\mathcal{R}z$ for all $x, y, z \in E$

• Equivalence Relation and Quotient Set

  • Definition:
  • A binary relation $\mathcal{R}$ on a set $E$ is an equivalence relation if and only if it is reflexive, symmetric, and transitive.
  • Quotient Set $E/\mathcal{R}$:
  • Given an equivalence relation $\mathcal{R}$ on $E$:
    • The equivalence class of $x \in E$, denoted $\text{cl}(x)$ or $\dot{x}$, is defined as $\text{cl}(x) = \dot{x} = \{y \in E : y\mathcal{R}x\}$.
    • The quotient set relative to $\mathcal{R}$, denoted $E/\mathcal{R}$, is the set of all equivalence classes: $E/\mathcal{R} = \{\text{cl}(x) : x \in E\}$.
  • Example: Congruence Modulo $n$:
  • Let $n$ be a fixed positive integer. An integer $x$ is congruent to an integer $y$ modulo $n$ (written $x \equiv y \pmod n$) if and only if there exists an integer $k \in \mathbb{Z}$ such that $x - y = k n$.
  • Congruence modulo $n$ is an equivalence relation on $\mathbb{Z}$:
    • Reflexivity: $\forall x \in \mathbb{Z}$, $x - x = 0 = 0 \times n$ with $k = 0 \in \mathbb{Z}$, so $x\mathcal{R}x$
    • Symmetry: $\forall x, y \in \mathbb{Z}$, if $x\mathcal{R}y \implies \exists k \in \mathbb{Z}$ such that $x - y = kn \implies y - x = (-k)n \implies y\mathcal{R}x$
    • Transitivity: $\forall x, y, z \in \mathbb{Z}$, if $x\mathcal{R}y$ and $y\mathcal{R}z \implies \exists k, k' \in \mathbb{Z}$ such that $x - y = kn$ and $y - z = k'n$. Adding both equations yields $x - z = (k + k')n \implies x\mathcal{R}z$
  • The equivalence class of $x \in \mathbb{Z}$ is $\dot{x} = \{x + kn : k \in \mathbb{Z}\}$.
  • The quotient set is $\mathbb{Z}_n = \{\dot{0}, \dot{1}, \dots, \dot{n-1}\} = \{0, 1, \dots, n-1\}$.
  • Detailed evaluation for $n = 7$:
    • $\dot{0} = \{7k : k \in \mathbb{Z}\} = \{\dots, -21, -14, -7, 0, 7, 14, 21, \dots\}$
    • $\dot{1} = \{1 + 7k : k \in \mathbb{Z}\} = \{\dots, -20, -13, -6, 1, 8, 15, 22, \dots\}$
    • $\dot{2} = \{2 + 7k : k \in \mathbb{Z}\} = \{\dots, -19, -12, -5, 2, 9, 16, 23, \dots\}$
    • $\dot{3} = \{3 + 7k : k \in \mathbb{Z}\} = \{\dots, -18, -11, -4, 3, 10, 17, 24, \dots\}$
    • $\dot{4} = \{\dots, -17, -10, -3, 4, 11, 18, 25, \dots\}$
    • $\dot{5} = \{\dots, -16, -9, -2, 5, 12, 19, 26, \dots\}$
    • $\dot{6} = \{\dots, -15, -8, -1, 6, 13, 20, 27, \dots\}$
    • $\dot{7} = \dot{0}$, $\dot{8} = \dot{1}$, $\dot{9} = \dot{2}$
    • $\mathbb{Z}_7 = \{\dot{0}, \dot{1}, \dot{2}, \dot{3}, \dot{4}, \dot{5}, \dot{6}\}$
  • Example: Consumer Indifference Curves:
  • A consumer holds two goods $A$ and $B$ in non-negative quantities $x, y \ge 0$. A state $(x, y)$ represents a point $M(x, y) \in \mathbb{R}^2$
  • The relation $\mathcal{R}$ between states $(x_1, y_1)$ and $(x_2, y_2)$ is defined by $(x_1, y_1)\mathcal{R}(x_2, y_2) \iff (2x_1 + 1)(3y_1 + 2) = (2x_2 + 1)(3y_2 + 2)$.
  • Equivalence classes under $\mathcal{R}$ are called indifference curves.
  • For state $(3, 4)$: $(2 \times 3 + 1)(3 \times 4 + 2) = 7 \times 14 = 98$
  • Equation of the indifference curve: $(2x + 1)(3y + 2) = 98 \implies y = f(x) = \frac{98}{3(2x + 1)} - \frac{2}{3} = \frac{96 - 4x}{3(2x + 1)}$.

• Order Relation

  • Definition:
  • A binary relation $\mathcal{R}$ on $E$ is an order relation if it is reflexive, anti-symmetric, and transitive.
  • An order relation is generally denoted by $\le$. The pair $(E, \le)$ is called an ordered set.
  • Two elements $x, y \in E$ are comparable if $x \le y$ or $y \le x$
  • An ordered set $(E, \le)$ is totally ordered if all pairs of elements are comparable.
  • Examples:
  • $(\mathbb{N}, \le)$, $(\mathbb{Z}, \le)$, $(\mathbb{Q}, \le)$, and $(\mathbb{R}, \le)$ are totally ordered sets.
  • For $E = \{1, 2\}$, the power set $( \mathcal{P}(E), \subseteq )$ is partially ordered, because singletons ({1}) and ({2}) are not comparable.

• Review of Number Sets

  • Natural Numbers $\mathbb{N}$:
  • $\mathbb{N} = \{0, 1, 2, \dots, n, \dots\}$. Contains a smallest element $0$, and every natural number $n$ has a successor.
  • The equation $x + 5 = 0$ has no solution in $\mathbb{N}$ ($x = -5 \notin \mathbb{N}$).
  • Integers $\mathbb{Z}$:
  • $\mathbb{Z} = \{\dots, -n, \dots, -1, 0, 1, \dots, n, \dots\}$.
  • Every element has a predecessor and a successor.
  • $(\mathbb{Z}, +)$ forms an Abelian (commutative) group:
    • Associativity: $(x + y) + z = x + (y + z)$
    • Neutral element $0$: $x + 0 = 0 + x = x$
    • Inverse $-x$: $x + (-x) = (-x) + x = 0$
    • Commutativity: $x + y = y + x$
  • $(\mathbb{Z}, +, \times)$ forms a commutative unitary ring:
    • Multiplication is associative, distributive over addition, and has a multiplicative identity $1$
  • Incomplete because $2x - 3 = 0$ has no solution in $\mathbb{Z}$.
  • Rational Numbers $\mathbb{Q}$:
  • $\mathbb{Q} = \{\frac{a}{b} : a \in \mathbb{Z}, b \in \mathbb{Z}^*\}$.
  • $(\mathbb{Q}, +, \times)$ forms a field (corps commutatif): every non-zero element $x \in \mathbb{Q}$ has a multiplicative inverse $\frac{1}{x}$ such that $x \times \frac{1}{x} = 1$
  • Real Numbers $\mathbb{R}$:
  • $x^2 - 2 = 0$ has no solution in $\mathbb{Q}$ ($x = \pm \sqrt{2} \notin \mathbb{Q}$).
  • Algebraic irrationals: numbers that are roots of polynomial equations with integer coefficients (e.g., $\sqrt{2}, -\sqrt{2}$).
  • Transcendental irrationals: non-algebraic real numbers (e.g., $\pi, e$).
  • Inclusion chain: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$.
  • Intervals in $\mathbb{R}$:
    • Open interval: $]a, b[ = \{x \in \mathbb{R} : a < x < b\}$
    • Closed interval: $[a, b] = \{x \in \mathbb{R} : a \le x \le b\}$
    • Semi-open intervals: $[a, b[$ and $]a, b]$
    • Unbounded intervals: $[a, +\infty[$, $]a, +\infty[$, $]-\infty, a[$, $]-\infty, a]$

• Combinatorics (Combinatorial Analysis)

  • Types of Mappings:
  • Injective: for every distinct pair $(x_1, x_2) \in E \times E$, $f(x_1) \neq f(x_2)$.
  • Surjective: for every $y \in F$, there exists $x \in E$ such that $y = f(x)$.
  • Bijective: both injective and surjective (unique $x \in E$ for every $y \in F$).
  • Cardinality Properties:
  • For an application $f: A_n \to B_p$ between finite sets of sizes $n$ and $p$:
    • Injection $\implies n \le p$
    • Surjection $\implies p \ge n$
    • Bijection $\implies p = n$
  • Fundamental Counting Principle:
  • If task $A_1$ can be performed in $n_1$ ways, and task $A_2$ in $n_2$ ways, both tasks together can be performed in $n_1 \times n_2$ ways.
  • Example: Rolling a green die and a red die yields $6 \times 6 = 36$ outcomes.
  • Cartesian Product:
  • $\#(E_n \times E_p) = n \times p$
  • Example: Forming couples (man, woman) from 6 women and 4 men yields $\#(H_4 \times F_6) = 4 \times 6 = 24$ possibilities.
  • Arrangements with Repetition:
  • Selecting $p$ elements from $n$ elements with repetition allowed: $B_n^p = n^p$
  • Example: Vehicle license plates consisting of 4 non-zero digits from $\{1, 2, \dots, 9\}$ give $9^4 = 6561$ possibilities.
  • Arrangements without Repetition:
  • Injections from $A_p$ to $A_n$ ($p \le n$):       $A_n^p = n(n - 1)(n - 2)\dots(n - p + 1) = \frac{n!}{(n - p)!}$
  • Example 1: Parking 3 cars in a 4-spot parking lot yields $A_4^3 = \frac{4!}{(4-3)!} = 24$ ways.
  • Example 2: Forming 4-digit distinct non-zero numbers from $\{1, \dots, 9\}$ yields $A_9^4 = \frac{9!}{(9-4)!} = \frac{9!}{5!} = 3024$
  • Permutations:
  • Arrangement of $n$ elements taken $n$ at a time: $P_n = A_n^n = n!$
  • Example: Parking 4 cars in 4 parking spots yields $P_4 = 4! = 24$ ways.
  • Combinations (Subsets of size $p$):
  • Number of subsets of size $p$ from an $n$ element set: $C_n^p = \binom{n}{p} = \frac{n!}{(n - p)! p!}$
  • Example: Selecting a 5-member delegation from 15 students yields $C_{15}^5 = \frac{15!}{10! 5!} = 3003$ ways.
  • Binomial Theorem (Newton's Formula):
  • $(a + b)^n = \sum_{i=0}^n C_n^i a^{n-i} b^i$
  • Example: $(a + b)^3 = C_3^0 a^3 b^0 + C_3^1 a^2 b + C_3^2 a b^2 + C_3^3 a^0 b^3 = a^3 + 3a^2b + 3ab^2 + b^3$
  • Pascal's Triangle:
  • The coefficients of the binomial expansion are arranged in Pascal's triangle:

• Introduction to Discrete Probabilities
  • Definitions:
  • A random experiment is an experiment with known possible outcomes, but whose specific outcome cannot be predicted with certainty.
  • An event is any subset of the sample space of outcomes.
  • Complementary events: $A$ and $B$ such that $B = A^c$
  • Certain event: the set of all possible outcomes ($P = 1$).
  • Impossible event: empty set ($P = 0$).
  • Probability Formula:
  • For equiprobable outcomes: $P(A) = \frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}$
  • Property: $P(A^c) = 1 - P(A)$
  • Examples:
  • Drawing 1 card from a standard 52-card deck:
    • Red card: $P = \frac{26}{52} = \frac{1}{2}$
    • Club card: $P = \frac{13}{52} = \frac{1}{4}$
    • King: $P = \frac{4}{52} = \frac{1}{13}$
    • Not the Ace of Hearts: $P = \frac{51}{52}$
  • Drawing 3 cards simultaneously from 52 cards:
    • Total possible outcomes: $C_{52}^3 = 22100$
    • All 3 red cards: $P(A) = \frac{C_{26}^3}{C_{52}^3} = \frac{26 \times 25 \times 24}{52 \times 51 \times 50} = \frac{2}{17}$
    • All 3 same suit: $P(B) = \frac{4 \times C_{13}^3}{C_{52}^3} = \frac{22}{425}$
    • All 3 aces: $P(C) = \frac{C_4^3}{C_{52}^3} = \frac{4}{22100} = \frac{1}{5525}$

Vector Spaces and Linear Maps

• Generalities on Vector Spaces

  • Definition:
  • A real vector space (or vector space over $\mathbb{R}$) is a triplet $(V, +, \cdot)$ consisting of a non-empty set $V$, an internal operation $+: V \times V \to V$, and an external operation $\cdot : \mathbb{R} \times V \to V$ such that:
    • $(V, +)$ is a commutative group.
    • For all $x, y \in V$ and $\lambda, \mu \in \mathbb{R}$:
    • $\lambda(x + y) = \lambda x + \lambda y$
    • $(\lambda + \mu)x = \lambda x + \mu x$
    • $\lambda(\mu x) = (\lambda \mu)x$
    • $1 \cdot x = x$
  • Elements of $V$ are called vectors, elements of the field $K$ are called scalars.
  • Standard Examples:
  • $\mathbb{R}$ over $\mathbb{R}$.
  • $\mathbb{R}^n = \{(x_1, \dots, x_n) : x_i \in \mathbb{R}\}$ with componentwise addition and scalar multiplication.
  • $\mathcal{P}_2(x) = \{a_0 + a_1 x + a_2 x^2 : a_0, a_1, a_2 \in \mathbb{R}\}$: polynomials of degree $\le 2$
  • Subspaces:
  • A subset $F \subseteq E$ is a subspace of $E$ if and only if:
    • $0 \in F$
    • $\forall x, y \in F$, $x + y \in F$
    • $\forall a \in K, \forall x \in F$, $a \cdot x \in F$
  • Examples:
    • Subspace $\{0\}$ and $E$ itself.
    • $F = \{(x, y, 0) : x, y \in \mathbb{R}\} \subset \mathbb{R}^3$
    • Set $W$ of symmetric matrices $A = (a_{ij})$ where $a_{ij} = a_{ji}$.
  • Linear Combination and Span:
  • Expression $\sum_{i=1}^n \lambda_i x_i$ is a linear combination of $x_1, \dots, x_n$
  • Subspace generated by $x_1, \dots, x_n$: denoted $\text{span}\{x_1, \dots, x_n\}$ or $\text{vect}\{x_1, \dots, x_n\}$.
  • Example:
    • $W = \left\{ \begin{pmatrix} a \\ b \\ c \end{pmatrix} : a + b + c = 0, a, b, c \in \mathbb{R} \right\} = \text{span}\left\{ \begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix} \right\}$.
  • Linear Independence and Basis:
  • Vectors $x_1, \dots, x_n$ are linearly independent if $\sum_{i=1}^n \lambda_i x_i = 0 \implies \lambda_i = 0, \forall i$
  • A family $(x_1, \dots, x_n)$ forms a basis of $E$ if it is linearly independent and spans $E$
  • Canonical basis of $\mathbb{R}^3$: $e_1 = (1, 0, 0)$, $e_2 = (0, 1, 0)$, $e_3 = (0, 0, 1)$.
  • Theorem: All bases of a finite-dimensional vector space $E$ have the same number of elements $n = m$
  • Dimension: $\dim E = n$ represents the number of elements in any basis. $\dim \{0\} = 0$
  • Direct Sum:
  • $E = U \oplus V \iff \forall x \in E$, there exists a unique $u \in U$ and unique $v \in V$ such that $x = u + v$
  • Theorem: Every subspace of a finite-dimensional vector space has a complementary subspace.

• Linear Mappings (Linear Applications)

  • Definition:
  • A mapping $f: E \to F$ between vector spaces over $K$ is linear if for all $x, y \in E$ and $\lambda \in K$:
    • $f(x + y) = f(x) + f(y)$
    • $f(\lambda x) = \lambda f(x)$
  • If $E = F$, $f$ is an endomorphism.
  • If $F = \mathbb{R}$, $f$ is a linear form.
  • The space of linear maps from $E$ to $F$ is denoted $\text{Hom}(E, F)$.
  • Image and Kernel:
  • Image: $\text{Im}(f) = \{y \in F : \exists x \in E, f(x) = y\}$, subspace of $F$
  • Kernel: $\text{Ker}(f) = \{x \in E : f(x) = 0_F\}$, subspace of $E$
  • Examples:
  • $f: \mathbb{R} \to \mathbb{R}, x \mapsto 2x$ (linear).
  • $f: \mathbb{R}^2 \to \mathbb{R}^2, (x, y) \mapsto (2x + y, y + x)$ (linear).
  • $f: \mathbb{R} \to \mathbb{R}, x \mapsto 2x + 1$ (not linear, due to constant term $1$).
  • Isomorphisms & Rank-Nullity Theorem:
  • An isomorphism is a bijective linear mapping.
  • $f$ is injective $\iff \text{Ker}(f) = \{0_E\}$.
  • $f$ is surjective $\iff \text{Im}(f) = F$
  • Rank-Nullity Theorem: $\dim \text{Im}(f) + \dim \text{Ker}(f) = \dim E$

• Usual Linear Mappings

  • Vector Homotheties:
  • $h_k(x) = k x$. Represented by scalar matrix $\text{diag}(k, \dots, k)$.
  • $k = 1$: identity map; $k = 0$: zero map; $k = -1$: central symmetry.
  • Dilations, Projections, and Symmetries:
  • Given $E = S \oplus T$, a dilation of ratio $k$ on $S$ parallel to $T$ is defined as $d_k(s + t) = s + k t$
  • $k = -1$: symmetry with respect to $S$ parallel to $T$ ($d_{-1} \circ d_{-1} = \text{Id}$ - involutive).
  • $k = 0$: projection onto $S$ parallel to $T$ ($d_0 \circ d_0 = d_0$ - idempotent).
  • Worked Example 2.3.1:
  • Line $D = \text{span}\left\{ \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} \right\}$, Plane $P = \text{span}\left\{ \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ -2 \\ 1 \end{pmatrix} \right\}$ in $\mathbb{R}^3$
  • Determinant of basis matrix:       $\begin{vmatrix} 1 & 1 & 0 \\ 2 & 2 & -2 \\ -1 & 0 & 1 \end{vmatrix} = 2 \neq 0 \implies D \oplus P = \mathbb{R}^3$
  • Decomposition system for $v = (x, y, z)^t = \alpha v_1 + \beta v_2 + \gamma v_3$:       $\alpha = x - \frac{1}{2}y - z$, $\beta = \frac{1}{2}y + z$, $\gamma = x - \frac{1}{2}y$
  • Projection onto $D$ parallel to $P$ ($P_D$):       $P_D(x, y, z) = \left( x - \frac{1}{2}y - z, 2x - y - 2z, -x + \frac{1}{2}y + z \right)$
  • Projection onto $P$ parallel to $D$ ($P_P$):       $P_P(x, y, z) = \left( \frac{1}{2}y + z, -2x + 2y + 2z, x - \frac{1}{2}y \right)$
  • Dilation of ratio $k$ on $P$ parallel to $D$:       $d_k(x, y, z) = k \left( x - \frac{1}{2}y - z \right) \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} + \left( \frac{1}{2}y + z \right) \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + \left( x - \frac{1}{2}y \right) \begin{pmatrix} 0 \\ -2 \\ 1 \end{pmatrix}$

Complex Numbers and Polynomials

• Complex Numbers
  • Algebraic Form:
  • $z = a + bi$, where $a = \text{Re}(z) \in \mathbb{R}$ is the real part, $b = \text{Im}(z) \in \mathbb{R}$ is the imaginary part, and $i^2 = -1$
  • Complex conjugate: $\bar{z} = a - bi$
  • Algebraic Operations:
  • Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
  • Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$
  • Multiplicative Inverse: $\frac{1}{a + bi} = \frac{a - bi}{a^2 + b^2}$
  • Conjugate Identities: $z + \bar{z} = 2\text{Re}(z)$, $z - \bar{z} = 2i\text{Im}(z)$.
  • Polar / Trigonometric Form:

- Modulus: $R = |z| = \sqrt{a^2 + b^2}$
- Argument: $\theta$ defined modulo $2\pi$ such that $\cos \theta = \frac{a}{|z|}$ and $\sin \theta = \frac{b}{|z|}$.
- Trigonometric form: $z = R(\cos \theta + i \sin \theta)$.
- Examples:
  - $z = 1 + i = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)$
  - $z = -1 - i = \sqrt{2} \left( \cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4} \right)$
- Operations in Polar Form:
  - $z_1 z_2 = R_1 R_2 [\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$
  - $\frac{z_1}{z_2} = \frac{R_1}{R_2} [\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$
  - **De Moivre's Formula**: $z^n = R^n (\cos n\theta + i \sin n\theta)$.
  - Powers: z_1^{20} = (\sqrt{2})^{20} \left( \cos \frac{20\pi}{4} + i \sin \frac{20\pi}{4} \right) = 2^{10} (\cos 5\pi + i \sin 5\pi) = -1024$.\n      - Trigonometric identities via De Moivre:\n        (\cos \theta + i \sin \theta)^2 = \cos 2\theta + i \sin 2\theta = \cos^2 \theta - \sin^2 \theta + 2i \cos \theta \sin \theta\n        \implies \cos 2\theta = \cos^2 \theta - \sin^2 \theta$,$\sin 2\theta = 2 \sin \theta \cos \theta\n  - **Square Roots of Complex Numbers**:\n    - Solving (u + iv)^2 = a + bi leads to:\n      - u^2 - v^2 = a\n      - 2uv = b\n      - u^2 + v^2 = \sqrt{a^2 + b^2}\n      - Solutions: u^2 = \frac{1}{2}(a + \sqrt{a^2 + b^2})$,$v^2 = \frac{1}{2}(-a + \sqrt{a^2 + b^2})$, where sign of$u v$matches sign of$b$.
- Example 1: Roots of $21 - 20i$: $\sqrt{21^2 + 20^2} = 29 \implies u^2 = 25, v^2 = 4 \implies \sqrt{21 - 20i} = \pm(5 - 2i)$.
- Example 2: Roots of $-a^2 < 0$: $\pm |a| i$
- Example 3: Solving $z^2 + 2z + 5 = 0 \implies \Delta = -16 = (4i)^2 \implies z = -1 \pm 2i$
- Example 4: Solving $4z^2 + 8iz - 25 + 20i = 0 \implies \Delta = 16(21 - 20i) = [4(5 - 2i)]^2 \implies z_1 = \frac{5}{2} - 2i, z_2 = -\frac{5}{2}$.

• $n$-th Roots of Complex Numbers:

  • $\sqrt[n]{z} = \sqrt[n]{R} \left( \cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} \right)$, for $k = 0, 1, \dots, n-1$

  • $n$-th roots of unity: $\omega_k = \cos \frac{2k\pi}{n} + i \sin \frac{2k\pi}{n} = e^{i \frac{2k\pi}{n}}$.

  • The set of $n$-th roots of unity $U_n$ forms a group under multiplication isomorphic to $(\mathbb{Z}_n, +)$.

  • Primitive roots of unity: $\varepsilon$ is primitive if $\{\varepsilon^k : k = 0, \dots, n-1\} = U_n$

  • Cube roots of 1: $z_0 = 1$, $z_1 = -\frac{1}{2} + i \frac{\sqrt{3}}{2} = j$, $z_2 = -\frac{1}{2} - i \frac{\sqrt{3}}{2} = j^2$

  • Polynomial Functions

• Definition:

  • $P(z) = p_0 + p_1 z + p_2 z^2 + \dots + p_n z^n$, where $p_n \neq 0$, degree $\deg P = n$

• Derivatives and Taylor Formula:

  • $P^{(k)}(0) = k! p_k \implies p_k = \frac{P^{(k)}(0)}{k!}$
  • Taylor's Formula for Polynomials:       $P(z) = P(a) + \frac{P'(a)}{1!}(z - a) + \frac{P''(a)}{2!}(z - a)^2 + \dots + \frac{P^{(n)}(a)}{n!}(z - a)^n$

• Roots and Factorization:

  • $a$ is a root of $P$ if $P(a) = 0 \iff P(z) = (z - a)Q(z)$
  • $a$ is a multiple root if $P(a) = P'(a) = 0 \iff P(z) = (z - a)^2 R(z)$
  • Complex conjugate root theorem: If $P$ has real coefficients and $P(a) = 0$, then $P(\bar{a}) = 0$
  • Fundamental Theorem of Algebra (D'Alembert-Gauss):       Every polynomial $P$ of degree $n \ge 1$ over $\mathbb{C}$ splits completely into linear factors:       $P(z) = a(z - u_1)^{n_1}(z - u_2)^{n_2}\dots(z - u_k)^{n_k}$, where $\sum n_i = n$
  • Geometric series identity: $1 + z + z^2 + \dots + z^{n-1} = \frac{1 - z^n}{1 - z}$ for $z \neq 1$

Matrices and Systems of Linear Equations

• Generalities on Matrices
  • Definition:
  • A real matrix $A$ of dimension $n \times p$ is a rectangular array of real numbers with $n$ rows and $p$ columns: $A = (a_{ij})_{1 \le i \le n, 1 \le j \le p}$.
  • Stock Matrix Example:

• Special Types of Matrices:

  • Transpose: $A^t = (a_{ji})$, dimension $p \times n$

  • Square Matrix: $n = p$ (order $n$).

  • Row / Column Vectors: dimension $1 \times p$ and $n \times 1$ respectively.

  • Symmetric Matrix: $A^t = A$

  • Lower Triangular: $a_{ij} = 0$ for all $i < j$

  • Upper Triangular: $a_{ij} = 0$ for all $i > j$

  • Diagonal Matrix: $a_{ij} = 0$ for all $i \neq j$

  • Scalar Matrix: diagonal matrix with equal diagonal elements $\lambda I_n$

  • Identity Matrix $I_n$: scalar matrix with diagonal elements equal to 1.

  • Zero Matrix $0$: all entries equal to 0.

  • Matrix Operations

• Addition: $A + B = (a_{ij} + b_{ij})$, defined only for matrices of identical dimensions $n \times p$. Commutative group $(M_{n \times p}(\mathbb{R}), +)$.

• Scalar Multiplication: $\lambda A = (\lambda a_{ij})$.

• Matrix Product:

  • For $A_{n \times m}$ and $B_{m \times q}$, product $C = A B$ is dimension $n \times q$ with entries $c_{ij} = \sum_{k=1}^m a_{ik} b_{kj}$.

  • Associative, non-commutative in general.

  • Determinant of a Square Matrix

• Order 2: $\det \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = a_{11}a_{22} - a_{21}a_{12}$.

• Order 3 (Sarrus Rule):     $\det A = (a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32}) - (a_{31}a_{22}a_{13} + a_{32}a_{23}a_{11} + a_{33}a_{21}a_{12})$.

• General Order $n \ge 3$:

  • Signature: $s_{ij} = (-1)^{i+j}$

  • Minor $m_{ij}$: determinant of submatrix $M_{ij}$ obtained by removing row $i$ and column $j$

  • Cofactor: $\text{cof}_{ij} = (-1)^{i+j} m_{ij}$

  • Laplace Expansion:       $\det A = \sum_{j=1}^n a_{ij} \text{cof}_{ij}$ (along row $i$) or $\det A = \sum_{i=1}^n a_{ij} \text{cof}_{ij}$ (along column $j$).

  • Invertible Square Matrices

• A matrix $A$ is invertible (regular) iff $\det A \neq 0$

• Formula for Inverse: $A^{-1} = \frac{1}{\det A} (\text{com}(A))^t$, where $\text{com}(A) = (\text{cof}_{ij})$ is the comatrix.

• IS-LM Model Application:

  • System: $\begin{cases} 0.3Y + 100i - 250 = 0 \\ 0.25Y - 200i - 176 = 0 \end{cases} \implies \begin{pmatrix} 0.3 & 100 \\ 0.25 & -200 \end{pmatrix} \begin{pmatrix} Y \\ i \end{pmatrix} = \begin{pmatrix} 250 \\ 176 \end{pmatrix}$.
  • $\det A = 0.3(-200) - 100(0.25) = -85$
  • $A^{-1} = -\frac{1}{85} \begin{pmatrix} -200 & -100 \\ -0.25 & 0.3 \end{pmatrix} = \begin{pmatrix} \frac{40}{17} & \frac{20}{17} \\ \frac{0.05}{17} & -\frac{0.06}{17} \end{pmatrix}$.
  • Equilibrium: $Y = 800$, $i = 0.12$

• Set of invertible $n \times n$ matrices forms the General Linear Group $GL_n(K)$.

  • Cramerian Systems & Applications

• Linear system $A X = B$ of $n$ equations and $n$ unknowns.

• If $\det A \neq 0$, system is Cramerian with unique solution $x_i = \frac{\det A_i}{\det A}$, or $X = A^{-1} B$

• Maternity Hospital Bed Allocation Problem: