Applied Mathematics IB Course Notes

PART I: INTRODUCTION TO LINEAR ALGEBRA

Chapter 1: Vectors and Vector Spaces

Introduction

• Measurement Magnitudes vs. Vectors:
  • Quantities like mass, length, or time are determined solely by their magnitudes and require only a number and a unit of measure.
  • Quantities like force, displacement, or velocity require more information: they need direction in addition to magnitude.
  • Force: Requires magnitude and the direction in which it acts.
  • Displacement: Requires magnitude (how far) and direction (where it moves).
  • Velocity: Requires magnitude (speed) and direction (where the body is headed).
• Definition of Vectors:
  • Quantities possessing both magnitude and direction are represented by arrows called directed line segments or vectors.
  • The length of the arrow represents the magnitude of the action in terms of a chosen unit.
  • This chapter focuses on $n$-vectors or $n$-dimensional vectors in $\mathbb{R}^n$.
• Scope of Study:
  • Vectors provide geometric motivation for Linear Algebra.
  • Algebraic and geometric properties of points in $\mathbb{R}^n$ are discussed.
  • Scalar products (dot products) and vector products (cross products) are included.
  • Points in $\mathbb{R}^2$ and $\mathbb{R}^3$ are treated as special cases of $\mathbb{R}^n$.
  • The cross product is unique to 3-dimensional space (invalid in 2D, 4D, or other $n$ dimensions).

1.1 Definition of Points in n-space

• Practical Context: Vectors arise in the simultaneous study of multiple characteristics. For example, height ($a_1$), weight ($a_2$), age ($a_3$), and blood pressure ($a_4$) can be represented as an ordered 4-tuple $A = (a_1, a_2, a_3, a_4) \in \mathbb{R}^4$.
• Definition 1.1.1 (Operations): Let $A = (a_1, a_2, \dots, a_n)$ and $B = (b_1, b_2, \dots, b_n)$ be points in $n$-space, and $c \in \mathbb{R}$.
  • Addition: $A + B = (a_1 + b_1, a_2 + b_2, \dots, a_n + b_n)$.
  • Scalar Multiplication: $cA = (ca_1, ca_2, \dots, can)$.
• Example 1.1.1:
  • Let $A = (-1, 3, 6, 2)$, $B = (0, -5, -1, 4)$, $C = (2, -1, 3)$.
  • $A + B = (-1+0, 3-5, 6-1, 2+4) = (-1, -2, 5, 6)$.
  • $A + C$ is not defined because $A$ is in 4D space and $C$ is in 3D space.
  • $-2A = (-2(-1), -2(3), -2(6), -2(2)) = (2, -6, -12, -4)$.
• Definition 1.1.2 (Subtraction): $A - B = A + (-B)$.
• Theorem 1.1.1 (Properties): For points $A, B, C \in \mathbb{R}^n$ and $\alpha, \beta \in \mathbb{R}$:
  1. $A + B = B + A$ (Commutativity).
  2. $A + (B + C) = (A + B) + C$ (Associativity).
  3. $\alpha(A + B) = \alpha A + \alpha B$.
  4. $(\alpha + \beta)A = \alpha A + \beta A$.
  5. $\alpha(\beta A) = (\alpha\beta)A$.
  6. $A + 0 = 0 + A = A$.
  7. $1 \cdot A = A$ and $-1 \cdot A = -A$.
  8. $A + (-A) = 0$.

1.2 Vectors in n-space: Geometric Interpretation

• Vector Determination: A vector $\vec{AB}$ is determined by an initial point $A$ and a terminal point $B$.
• Relation between Points: If $A = (a_1, a_2)$ and $B = (b_1, b_2)$, then $B = A + (B - A)$.
• Definition 1.2.2 (Equivalence): $\vec{AB} \cong \vec{CD}$ if and only if $B - A = D - C$.
  • Origin Property: Every vector is equivalent to a vector whose initial point is the origin ($O$). If $\vec{AB} \cong \vec{OP}$, then $P = B - A$.
• Definition 1.2.3 (Parallelism): $\vec{AB} \parallel \vec{CD}$ if there exists $\alpha \in \mathbb{R}$ such that $B - A = \alpha(D - C)$.
  • Equivalent vectors are parallel where $\alpha = 1$.
  • If $\alpha > 0$, vectors have the same direction.
  • If $\alpha < 0$, vectors have opposite directions.

1.3 The Scalar Product

• Definition 1.3.1: For $A = (a_1, \dots, a_n)$ and $B = (b_1, \dots, b_n)$, the scalar product (dot product) is:
  • $A \cdot B = a_1b_1 + a_2b_2 + \dots + a_nb_n$.
• Theorem 1.3.1 (Properties):
  • $A \cdot B = B \cdot A$.
  • $A \cdot (B + C) = A \cdot B + A \cdot C$.
  • $(\alpha A) \cdot B = \alpha(A \cdot B) = A \cdot (\alpha B)$.
  • $A \cdot A \geq 0$; $A \cdot A = 0 \iff A = 0$.
• Definition 1.3.2 (Orthogonality): $A \perp B \iff A \cdot B = 0$.
• The Norm of a Vector (Definition 1.3.3):
  • The norm (length) is denoted $\Vert A \| = \sqrt{A \cdot A} = \sqrt{a_1^2 + a_2^2 + \dots + a_n^2}$.
  • Property: $\Vert \alpha A \| = |\alpha| \Vert A \|$.
• Unit Vectors: A vector $A$ is a unit vector if $\Vert A \| = 1$.
  • For any $A \neq 0$, the vector $u = \frac{A}{\Vert A \|}$ is a unit vector in the direction of $A$.
• Distance (Definition 1.3.5): $d(A, B) = \Vert A - B \| = \sqrt{(A-B) \cdot (A-B)}$.
• Angles (Theorem 1.3.5):
  • $A \cdot B = \Vert A \| \Vert B \| \cos(\theta)$ where $0 \leq \theta \leq \pi$.
  • $\cos(\theta) = \frac{A \cdot B}{\Vert A \| \Vert B \|}$.
• Directional Cosines: The angles $\alpha, \beta, \gamma$ a vector makes with the positive $X, Y, Z$ axes are the direction angles.
  • $\cos(\alpha) = \frac{a_1}{\Vert A \|}$, $\cos(\beta) = \frac{a_2}{\Vert A \|}$, $\cos(\gamma) = \frac{a_3}{\Vert A \|}$.
  • Property: $\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1$.
• Projection: The projection of vector $A$ onto $B$ is $P = \text{Proj}_B A = \frac{A \cdot B}{\Vert B \|^2} B$.
• Triangle Inequality (Theorem 1.3.6): $\Vert A + B \| \leq \Vert A \| + \Vert B \|$.

1.4 The Cross Product

• Definition 1.4.1 (3D space only): $\vec{A} \times \vec{B} = (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}$.
• Properties (Theorem 1.4.1):
  • $A \times B = -(B \times A)$.
  • If $A \parallel B$, then $A \times B = 0$.
  • $A \times B$ is perpendicular to both $A$ and $B$.
  • $\Vert A \times B \| = \Vert A \| \Vert B \| \sin(\theta)$. This equals the area of the parallelogram formed by $A$ and $B$.
  • Lagranges's Identity: $\Vert A \times B \|^2 = (A \cdot A)(B \cdot B) - (A \cdot B)^2$.
  • Unit Vector relations: $\mathbf{i} \times \mathbf{j} = \mathbf{k}$, $\mathbf{j} \times \mathbf{k} = \mathbf{i}$, $\mathbf{k} \times \mathbf{i} = \mathbf{j}$. Reversed products yield negative results (e.g., $\mathbf{j} \times \mathbf{i} = -\mathbf{k}$).

1.5 Lines and Planes

• Lines in Space:
  • A line $l$ is determined by a point $Q(x_0, y_0, z_0)$ and a vector $L = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ parallel to it.
  • Vector Equation: $r = r_0 + tL$.
  • Parametric Equations: $x = x_0 + at$, $y = y_0 + bt$, $z = z_0 + ct$.
  • Symmetric Equations: $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$.
• Distance from point $Q$ to line $l$: $D = \frac{\Vert L \times \vec{PQ} \|}{\Vert L \|}$ where $P$ is any point on $l$.
• Planes in Space:
  • A plane is determined by a point $Q(x_0, y_0, z_0)$ and a normal vector $N = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$.
  • Equation of Planet: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$.
• Distance from point $Q$ to plane $\pi$: $D = \frac{|N \cdot \vec{QP}|}{\Vert N \|}$ where $P$ is any point on $\pi$.

1.6 Applications

• Area of a Triangle: $Area = \frac{1}{2} \Vert A \times B \|$ where $A$ and $B$ are adjacent side vectors.
• Volume of a Parallelepiped: $Volume = |A \cdot (B \times C)|$. This is the absolute value of the scalar triple product.

Vector Spaces (Chapter 4)

4.1 Definition and Axioms

• Definition: A vector space $V$ over a field $F$ is a set of objects (vectors) satisfying:
  1. Closure under addition: $u + v \in V$.
  2. Closure under scalar multiplication: $\alpha v \in V$.
  3. Commutativity: $u + v = v + u$.
  4. Associativity of addition: $(u + v) + w = u + (v + w)$.
  5. Additive identity: There exists $0 \in V$ such that $0 + u = u$.
  6. Additive inverse: There exists $-u \in V$ such that $u + (-u) = 0$.
  7. Distributivity (scalar): $\alpha(u + v) = \alpha u + \alpha v$.
  8. Distributivity (field): $(\alpha + \beta)u = \alpha u + \beta u$.
  9. Associativity of scalar multiplication: $(\alpha\beta)u = \alpha(\beta u)$.
  10. Unitary Law: $1 \cdot u = u$.
• Theorem 4.1.2:
  • $0v = 0$ and $\alpha 0 = 0$.
  • $\alpha v = 0 \iff \alpha = 0$ or $v = 0$.
  • $(-1)v = -v$.

4.2 Subspaces

• Definition: A subset $W \subseteq V$ is a subspace if it satisfies closure under addition, closure under scalar multiplication, and contains the zero vector.
• Intersection Rule: The intersection of two subspaces $W_1 \cap W_2$ is always a subspace.
• Union Rule: The union $W_1 \cup W_2$ is generally not a subspace unless one is contained in the other.
• Sum of Subspaces: $W_1 + W_2 = \{u_1 + u_2 : u_1 \in W_1, u_2 \in W_2\}$ is a subspace.
• Linear Combination: A vector $v = \sum \alpha_i v_i$ is a linear combination.
• Generating Set: A set $S$ spans $V$ if every vector in $V$ can be written as a linear combination of elements in $S$.

4.3 Linear Independence

• Linear Dependence: Vectors $\vec{v_1}, \dots, \vec{v_n}$ are LD if $\sum \alpha_i \vec{v_i} = 0$ for some $\alpha_i \neq 0$.
• Linear Independence: Vectors are LI if $\sum \alpha_i \vec{v_i} = 0 \implies$ all $\alpha_i = 0$.
• Singleton Rule: $\left\{x\right\}$ is LI iff $x \neq 0$.

4.4 Basis and Dimension

• Basis: A set $B$ is a basis for $V$ if it spans $V$ and is linearly independent.
• Dimension ($dim V$): The number of vectors in any basis of $V$.
• Extension Theorem: Any linearly independent set can be extended to form a basis.
• Coordinate Vector: If $u = \sum a_i v_i$ relative to basis $\left\{v_k\right\}$, then $(a_1, \dots, a_n)$ is the coordinate vector.

Chapter 2: Matrices, Determinants, and Systems

2.1 Matrix Definitions

• Definition: An $m \times n$ matrix $A$ is a rectangular array of numbers with $m$ rows and $n$ columns.
• Equality: Matrices $A=B$ if they have the same size and all corresponding entries $a_{ij} = b_{ij}$ are equal.

2.2 Operations

• Addition: $(A+B)_{ij} = a_{ij} + b_{ij}$. Both must be same size.
• Scalar Multiplication: $(\alpha A)_{ij} = \alpha a_{ij}$.
• Multiplication: For $A_{m \times n}$ and $B_{n \times p}$, the product $C_{m \times p}$ is $c_{ik} = \sum_{j=1}^n a_{ij} b_{jk}$.
  • Note: Matrix multiplication is not commutative ($AB \neq BA$).
• Properties:
  • $A(BC) = (AB)C$ (Associative).
  • $A(B+C) = AB + AC$ (Distributive).

2.3 Types of Matrices

• Transpose ($A^t$): Interchanging rows and columns. $(A^t)_{ij} = a_{ji}$.
  • $(AB)^t = B^tA^t$.
• Identity ($I$): Square matrix with 1s on diagonal, 0s elsewhere.
• Diagonal: $a_{ij} = 0$ for $i \neq j$.
• Symmetric: $A^t = A \implies a_{ij} = a_{ji}$.
• Skew-Symmetric: $A^t = -A \implies a_{ij} = -a_{ji}$ (diagonal entries must be 0).
• Triangular: Upper triangular ($a_{ij} = 0$ for $i > j$) or Lower triangular ($a_{ij} = 0$ for $i < j$).

2.4 Inverses and Rank

• Elementary Row Operations:
  • Interchange two rows ($R_i \leftrightarrow R_j$).
  • Multiply a row by a non-zero constant ($R_i \to \alpha R_i$).
  • Add a multiple of one row to another ($R_i \to R_i + \alpha R_j$).
• Row Echelon Form (REF):
  • Zeros at bottom.
  • Leading non-zero entry is 1.
  • Leading 1 of a lower row is to the right of the leading 1 above it.
• Reduced Row Echelon Form (RREF): REF plus zeros above every leading 1.
• Inverse ($A^{-1}$): $AA^{-1} = A^{-1}A = I$. Only for non-singular matrices.
• Computation: Augment $A$ with $I$ ($[A | I]$) and use row operations to reach $[I | A^{-1}]$.
• Rank: Number of non-zero rows in REF.

2.6 Systems of Linear Equations

• Matrix Form: $Ax = b$.
• Gaussian Elimination: Reducing the augmented matrix to REF.
• Gauss-Jordan Elimination: Reducing the augmented matrix to RREF.
• Consistency:
  • If $rank(A) < rank(A|b)$, no solution (inconsistent).
  • If $rank(A) = rank(A|b) = n$, unique solution.
  • If $rank(A) = rank(A|b) < n$, infinitely many solutions (dependent on $n-r$ parameters).

Chapter 3: Determinants

• Definition 3.1.1:
  • $1 \times 1$: $|a_{11}| = a_{11}$.
  • $2 \times 2$: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$.
  • Expansion by Cofactors: $det A = \sum_{j=1}^n (-1)^{i+j} a_{ij} |A_{ij}|$ for any row $i$. Here $|A_{ij}|$ is the minor (determinant of submatrix after deleting row $i$ and col $j$).
• Properties:
  • $det(I) = 1$.
  • $det(AB) = det(A)det(B)$.
  • $det(A^t) = det(A)$.
  • If any row or col is all 0, $det(A) = 0$.
  • Swapping two rows changes the sign of the determinant.
  • Multiplying a row by $k$ results in $k \cdot det(A)$.
  • For triangular matrices, $det(A) = \prod_{i=1}^n a_{ii}$.
• Adjoint Inverse: $A^{-1} = \frac{1}{det A} adj A$, where $adj A$ is the transpose of the matrix of cofactors.
• Cramer’s Rule: For $Ax=b$, $x_i = \frac{det(B_i)}{det(A)}$, where $B_i$ replaces col $i$ of $A$ with $b$.

PART II: CALCULUS

Chapter 1: Limits, Continuity, and Differentiation

1.1 Limits

• Definition: $\lim_{x \to a} f(x) = L$ if $f(x)$ approach $L$ as $x$ approaches $a$.
• One-Sided Limits: $\lim_{x \to a^-} f(x)$ (left-hand) and $\lim_{x \to a^+} f(x)$ (right-hand).
• Theorems: Limits of sums, differences, products, and quotients are the operations of the individual limits (assuming denominators are non-zero).
• Squeeze Theorem: If $h(x) \leq f(x) \leq g(x)$ and $\lim h(x) = \lim g(x) = L$, then $\lim f(x) = L$.
• Trigonometric Limits: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ and $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$.

Continuity

• At a point: continuous at $a$ if $f(a)$ is defined, the limit exists, and $\lim_{x \to a} f(x) = f(a)$.
• Discontinuities: Removable (limit exists but $\neq f(a)$) or Essential (limit does not exist, e.g., jump or infinite).
• Intermediate Value Theorem (IVT): If $f$ is continuous on $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, there exists $c \in (a, b)$ such that $f(c) = k$.

1.3 Differentiation

• Definition: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.
• Geometric Meaning: The slope of the tangent line to the curve at $x=a$.
• Rules:
  • $\frac{d}{dx}[c] = 0$.
  • $\frac{d}{dx}[x^n] = nx^{n-1}$.
  • $\frac{d}{dx}[f \cdot g] = f'g + fg'$ (Product Rule).
  • $\frac{d}{dx}[f/g] = \frac{f'g - fg'}{g^2}$ (Quotient Rule).
  • $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ (Chain Rule).
• Implicit Differentiation: Differentiating both sides of an equation with respect to $x$, treating $y$ as a function of $x$, and solving for $\frac{dy}{dx}$.
• Transcendental Functions:
  • $\frac{d}{dx}[\sin x] = \cos x$, $\frac{d}{dx}[\cos x] = -\sin x$, $\frac{d}{dx}[\tan x] = \sec^2 x$.
  • $\frac{d}{dx}[e^x] = e^x$.
  • $\frac{d}{dx}[\ln x] = \frac{1}{x}$.

1.4 L’Hôpital’s Rule

• If $\lim_{x \to a} \frac{f(x)}{g(x)}$ yields $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (provided the latter exists).
• Other Indeterminate Forms: $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, $\infty^0$. These must be algebraically converted to $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

Chapter 2: Applications of Derivative

• Extrema:
  • Absolute Extrema: The highest/lowest values on an interval. On a closed interval, checking critical numbers and endpoints.
  • Critical Number: $c$ where $f'(c) = 0$ or $f'(c)$ is undefined.
• Rolle’s Theorem: If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, there is a $c \in (a, b)$ such that $f'(c) = 0$.
• Mean Value Theorem (MVT): There exists $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.
• Monotonicity: $f' > 0 \implies$ increasing; $f' < 0 \implies$ decreasing.
• Concavity: $f'' > 0 \implies$ concave up; $f'' < 0 \implies$ concave down.
• Inflection Point: Point where concavity changes sign.
• Optimization: Converting word problems to functions, finding critical points to maximize/minimize quantities (e.g., volume, area, time).
• Related Rates: Finding the rate of change of one quantity based on the rate of change of others using the Chain Rule (e.g., area of a growing disc, ladder sliding down a wall).

Chapter 3: Integration

Antiderivatives and Indefinite Integrals

• Definition: $F$ is an antiderivative of $f$ if $F'(x) = f(x)$.
• Notation: $\int f(x) dx = F(x) + C$.
• General Formulas:
  • $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$).
  • $\int \frac{1}{x} dx = \ln|x| + C$.
  • $\int e^x dx = e^x + C$.
  • $\int \sin x dx = -\cos x + C$.

Techniques of Integration

• Substitution (u-substitution): $\int f(g(x))g'(x)dx = \int f(u)du$.
• Integration by Parts: $\int u dv = uv - \int v du$.
• Partial Fractions: Decomposing rational functions $\frac{P(x)}{Q(x)}$ into sums of simpler fractions based on the factors of $Q(x)$.

Definite Integrals

• Riemann Sum: The limit of sums of rectangular areas under a curve.
• Fundamental Theorem of Calculus (FTC):
  • Part I: $\frac{d}{dx} \int_a^x f(t) dt = f(x)$.
  • Part II: $\int_a^b f(x) dx = F(b) - F(a)$.

Chapter 4: Applications of Integration

• Area Between Curves: $A = \int_a^b [f(x) - g(x)] dx$.
• Work Done by Force: $W = \int_a^b F(x) dx$. Use Hooke's Law for springs: $F(x) = kx$.
• Volume of Solids of Revolution:
  • Disk Method: $V = \int_a^b \pi [f(x)]^2 dx$.
  • Washer Method: $V = \int_a^b \pi ([f(x)]^2 - [g(x)]^2) dx$.
• Arc Length of Plane Curves: $\ell(C) = \int_a^b \sqrt{1 + [f'(x)]^2} dx$.