Department of Economics
College of Liberal Arts

Linear Algebra 

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                                The notes are prepared by Christos A Ioannou. 

                                If you find an error, please email me at christos@econ.umn.edu

What is a Linear Equation

Consistent versus Inconsistent Systems

Solving a Linear System

Echelon Form and Reduced Echelon Form

Basic and Free Variables

Vectors

Linear Combinations of Vectors

Vector Equations

Linear Combinations in a Crusoe Economy

Matrix Equations

Equivalence Between Vector Equations and Matrix Equations

Homogeneous Linear Systems

Linear Independence

Determinants

Invertible Matrices

Characterization of Invertible Matrices (The Invertible Matrix Theorem)

An Algorithm for Finding the Inverse

Definiteness of Matrices

Subspaces

The Dimension of a Subspace

Column Space and Null Space of a Matrix

Basis for a Subspace

The Rank of a Matrix

The Rank Theorem

Diagonalization

The Diagonalization Theorem

The LU Decomposition

Significance of the LU Decomposition

Eigenvectors, Eigenvalues and Eigenspace

The Characteristic Equation

The Inner Product

Properties of Inner Products

The Length of a Vector

Distances

Orthogonal Vectors

Least-Squares Solution

Exercises

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