Unit 0 Mathematics for Business I-merged

Presentation Overview

• Unit: Mathematics for Business I

• Professor: Sarai Vera Rodríguez

• Institution: UCAM Universidad Católica de Murcia

Syllabus by Units

• Unit 1: Vector Spaces

• Unit 2: Matrix and Matrix Calculus

• Unit 3: Linear Equations System

• Unit 4: Linear Applications

• Unit 5: Real Matrix Diagonalization

• Unit 6: Quadratic Real Forms

Assessment Structure

Theoretical Part (80% Total Mark)

• Two Written Exams:

  • Midterm Exam: 30%

  • Final Exam: 50% (Both Parts 1 and 2)

  • Dates:

    • Midterm: November 12, 2024

    • Ordinary Call Final: January 14, 2025

    • Recovery Call: July 2, 2025

Practical Part (20% Total Mark)

• Evaluation based on Individual Assignments

• Each Unit requires one submission in .pdf format, which must be legible and well-written.

• The average grade must be 5 or greater for passing.

Passing Criteria

• Ordinary Call:

  • Must achieve a weighted average of 5 or higher across all evaluation components.

  • If any component with a weighted average of 20% is below 5, the course fails.

• Recovery Call:

  • Passed components do not carry over to subsequent academic years if the course is failed.

Contact Information

• Email: svera@ucam.edu

• Encouragement to reach out for questions/suggestions and wish for good luck in the course.

Key Concepts in Vector Spaces

Definition of Vector Space

• An algebraic structure consisting of a non-empty set where vector addition and scalar multiplication are defined.

Properties of Vector Addition and Scalar Multiplication

1. Addition Properties:

   • Associativity:

     • For any vectors u, v, w: (u + v) + w = u + (v + w)

   • Identity Element:

     • There exists a zero vector 0 such that u + 0 = u

   • Inverse Element:

     • For every u, there exists -u such that u + (-u) = 0

   • Commutativity:

     • u + v = v + u

2. Scalar Product Properties:

   • Associativity:

     • α(βu) = (αβ)u

   • Identity Element:

     • 1 · u = u

   • Distributive:

     • α(u + v) = αu + αv

Example of Vectors

• A vector in R^n is represented as u = (x1, x2, ..., xn).

• Addition of vectors: u + v = (x1 + y1, x2 + y2, ..., xn + yn).

• Scalar multiplication of u: αu = (αx1, αx2, ..., αxn).

Matrix Concepts

Definition of Matrix

• A matrix consists of elements organized in rows and columns, with real numbers being the common elements used.

Types of Matrices

• Rectangular, Row (one row), Column (one column), Square (equal rows and columns), Triangular (superior/inferior), Diagonal, Scalar, Unit (identity).

Matrix Operations

1. Matrix Addition:

   • Two matrices can be added if they have the same order.

   • Resulting matrix A + B consists of sums of corresponding elements: (A + B)ij = Aij + Bij.

2. Scalar Multiplication:

   • Defined as k·A where each element of A is multiplied by k.

3. Matrix Multiplication:

   • Matrices A and B can be multiplied if the number of columns of A equals the number of rows of B, resulting in matrix C.

Determinant of a Matrix

• The determinant provides essential properties about a matrix, including its invertibility and properties when assessing its eigenvalues and spans.

Systems of Linear Equations

Definition

• A system of linear equations consists of equations relating multiple variables and can be expressed in matrix form AX = B, where A is the coefficient matrix, X is the vector of variables, and B is the constant vector.

Types of Solutions

1. Consistent: Has at least one solution

2. Inconsistent: No solutions

3. Determined: Unique solution

4. Undetermined: Infinite solutions

Gauss-Jordan Elimination Method

• A systematic procedure used to simplify a matrix into a diagonal form to solve systems of linear equations.

Cramer’s Rule

• Provides a formula for finding the solution of a system of equations with a unique solution using determinants.