Mathematics for Management -- Supplementary Electronic Materials

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Vectors & Matrices

Vectors & Matrices -- Structure

Linear Systems of Equations

Lines in the plane & systems of equations in 2 variables

The contents of this unit include: Systems of lines as systems of equations Intersection of lines as simultaneous solutions of systems of equations Elementary operations Invariance of the solution set under elementary operations Three basic types of solutions: no solution, exactly one solution, and infinitely many solutions

Lines in space & systems of equations in 3 variables

The contents of this unit include: Normal forms of planes and lines in space Solution procedure for solving systems of equations in 3 variables Augmented matrix (rows and columns) upper echelon form, pivot elements, back-substitution Elementary operations Reading the number of solutions from the upper echelon form

N-variable systems & Gaussian Elimination

The contents of this unit include: Rank of a coefficient matrix and an augmented matrix Multiplication of a matrix with a column The Gaussian Elimination algorithm

Worked-Out Exercises

worked out exercises

Determinants

Motivation: The geometry of determinants

The contents of this unit include: Position vector, direction vector, transposition Area of a parallelogram as the absolute value of the determinant function Geometric proof of the determinant formula Area of a spat as the absolute value of the determinant function Rule of Sarrus Matrices: unit matrix, matrix-addition, matrix multiplication by a scalar

General properties & computation rules

The contents of this unit include: Definition of rang revisited Linearity General definition of a determinant: multi-linear form maping the unit matrix to one and not-full rank matrices to zero Determinants and elementary row operations Computation of a determinant Determinant of a triangular matrix Transposed matrix and its determinant

Worked-Out Exercises

worked out exercises

Linear Combinations of Vectors

Vectors & their linear combinations

The contents of this unit include: Definition of a (real) vector space Linear combination, linear hull/ span, linear independence Checking for linear independence and linear dependence Basis, canonical basis, unique (coordinate) representation Dimension of a vector space

Scalar-product, vector-product, and spat-product

The contents of this unit include: (Standard) scalar product (dot product), norm Angles between two vectors, orthogonality Vector product (cross product) Spat product (mixed product)

Hessian Normal Form (HNF)

The contents of this unit include: Hessian Normal Form of a line in the plane Hessian Normal Form of a plane in space Parameter form vs. HNF

Worked-Out Exercises

worked out exercises

Matrix Computations

Matrix-matrix computations

The contents of this unit include: Definition of matrix-matrix multiplication Examples for matrix products Matrix-matrix multiplication is not commutative and not free of zero divisors

A primer on eigenvalues and eigenvectors

The contents of this unit include: Definition of eigenvectors and eigenvalues of a square matrices Characteristic polynomial Computation of eigenvalues and eigenvectors Not every matrix needs to have (real) eigenvales and eigenvectors

Inverse matrices

The contents of this unit include: Inverse of a \( 2 \times 2 \)-matrix Determinant of the inverse matrix Computation of the inverse with a modified Gaussian elimiation procedure

Worked-Out Exercises

worked out exercises

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