Maths course notes week 7

Week 7 Course Notes: Vectors and Matrices in Economics and Finance

Week Introduction

  • Algebraically, vectors and matrices are lists or tables of numbers, serving as storage for many interacting variables in economics and finance.

  • Geometrically, a vector is a point in high-dimensional space; a matrix provides instructions for mapping points from one space to another.

  • Both perspectives (algebraic and geometric) are essential for intuitive understanding of linear algebra concepts.

Part I: Vector Fundamentals

Vectors

  • Example: Prices of apples (P1) and bananas (P2) represented as a column vector:

    • P = [P1, P2]

  • Underlined variables indicate vectors; without underlining, it refers to the magnitude of the vector.

Zero Vectors or Null Vectors

  • Denoted as 0_n, where all entries are zero. Example in R^2: 0_2 = [0, 0].

  • Represents the origin in R^n.

Vector Addition

  • For two n × 1 vectors a = [a1, ..., an], b = [b1, ..., bn], the sum is:

    • a + b = [a1 + b1, ..., an + bn].

  • Geometric interpretation involves adding arrows originating from the origin.

Scalar Multiplication

  • A scalar is a single number. Scalar multiplication modifies vector length and direction based on scalar value.

Linear Combinations

  • A linear combination of vectors involves scaling and adding vectors. If x, y ∈ R^n, then z = ax + by is a linear combination of x and y (a, b are scalars).

A Coordinate System for ℝⁿ: the Canonical Basis

  • Vectors can be expressed as linear combinations of basis vectors.

  • E.g., in R^2, canonical basis is {e1 = [1, 0], e2 = [0, 1]}.

Dot Product (Inner Product, Scalar Product) of Two Vectors

  • Defined as a ⋅ b = a1b1 + a2b2 + ... + anbn.

  • Example: For a = [1, 3, -2] and b = [2, 7, 5], a ⋅ b = 2 + 21 - 10 = 13.

Magnitude of a Vector

  • Distance from the origin given by ||x|| = √(x ⋅ x).

  • Example: For x = [4, -7, 3], ||x|| = √(4^2 + (-7)^2 + 3^2) = √74 ≈ 8.6.

Distance Between Two Vectors

  • Distance between x and y is given by ||x - y||.

Angle Between Two Vectors

  • The cosine of the angle θ between two vectors is defined by the equation a ⋅ b = ||a|| ||b|| cos(θ).

Orthogonal Vectors

  • Two vectors are orthogonal if a ⋅ b = 0, indicating a right angle.

Part II: Matrix Fundamentals

Matrices

  • An m × n matrix A is a rectangular array of numbers with m rows and n columns.

Multiplying a Matrix by a Scalar

  • If A is a matrix and λ is a scalar, the new matrix λA is defined by multiplying each element of A by λ.

Matrix Addition

  • For two m × n matrices A and B, their sum A + B is obtained by adding the corresponding elements.

Matrix Times Vector

  • Given matrix A and vector x, the product Ax results in a new vector.

Matrix Multiplication: Matrix Times Matrix

  • If A is m × n and B is n × p, the product AB is m × p. The operation is associative but not commutative (AB ≠ BA) and must respect dimensionality.

Identity Matrices

  • The identity matrix I has ones on the diagonal and zeros elsewhere; it behaves as a multiplicative identity.

Transposes

  • The transpose of an m × n matrix A, denoted A^T, is an n × m matrix whose (i,j) entry is the (j,i) entry of A.

Part III: The Total Probability Theorem

Probability Fundamentals

  • Introduces simple outcomes and events in the context of probability theory.

  • Conditional probability P(A|B) is the probability of event A given that event B has occurred.

Total Probability Theorem

  • The formula states that if A and B are events, then P(A) = P(A|B) × P(B) + P(A|Bᶻ) × P(Bᶻ).

Part IV: Application: Markov Processes

Markov Processes Overview

  • A Markov process consists of state transitions based solely on the present state, not on the sequence of events that preceded it.

Transition Matrix

  • Utilizes a transition matrix to store probabilities of switching between states.

Examples and Applications of Markov Processes

  • The separation of calculations involving individual states and the transition matrix facilitates effective computations in economic modeling, such as for labor forces.

Week 7 Essentials

  • Understanding algebraic and geometric interpretations of vectors and matrices.

  • Key operations: scalar multiplication, addition, matrix multiplication (non-commutative, associative).

  • Introduction to Markov processes and their application in modeling probabilities.