Elementary Linear Algebra

Linear algebra is the study of; linear sets of equations and their transformation properties. Linear algebra allows the analysis of; rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering.

Chapter 1 Systems of Linear Equations and Matrices 1.1 Introduction to Systems of Linear Equations 1.2 Gaussian Elimination 1.3 Matrices and Matrix Operations 1.4 Inverses: Algebraic Properties of Matrices 1.5 Elementary Matrices and a Method for finding A -1 1.6 More on Linear Systems and Invertible Matrices 1.7 Diagonal, Triangular, and Symmetric Matrices 1.8 Applications of Linear Systems 1.9 Leontief Input-Output Models

A linear equation in the variables x 1 x n is an equation that can be written in the form b and the coefficients a 1..a n are real or complex numbers, Usually known in advance. The subscript n may be any positive integer. In textbook examples and exercises, n is normally between 2 and 5. In real-life problems, n might be 50 or 5000, or even larger.

Linear Systems in Two Unknowns

A system of linear equations has 1. no solution, or 2. exactly one solution, or 3. infinitely many solutions.

The Equation of a Plane What is x=4 in 2D and 3D? What is x=2 and y=2 in 2D and 3D? Find the equation of the plane passing through A(2,0,0) B(3,0,0) C(4,0,0)

Linear Systems in Three Unknowns

Matrix Notation The essential information of a linear system can be recorded compactly in a rectangular array called a matrix

Solving a Linear System Describe an algorithm, or a systematic procedure, for solving linear systems. The basic strategy is to replace one system with an equivalent system (i.e., one with the same solution set) that is easier to solve

Elementary Row Operations 1. Multiply a row through by a nonzero constant. 2. Interchange two rows. 3. Add a constant times one row to another

Section 1.2 Gaussian Elimination Row Echelon Form Reduced Row Echelon Form: Achieved by Gauss Jordan Elimination

We can find all the variables. So a solution exists; the system is consistent. So the solution is unique.)

Homogeneous Systems All equations are set = 0 Theorem 1.2.1 If a homogeneous linear system has n unknowns, and if the reduced row echelon form of its augmented matrix has r nonzero rows, then the system has n r free variables Theorem 1.2.2 A homogeneous linear system with more unknowns than equations has infinitely many solutions

Matrices and Matrix Operations Definition 1 A matrix is a rectangular array of numbers. The numbers in the array are called the entries of the matrix. The size of a matrix M is written in terms of the number of its rows x the number of its columns. A 2x3 matrix has 2 rows and 3 columns

Arithmetic of Matrices A + B: add the corresponding entries of A and B A B: subtract the corresponding entries of B from those of A Matrices A and B must be of the same size to be added or subtracted ca (scalar multiplication): multiply each entry of A by the constant c

Multiplication of Matrices

Diagonal, Triangular and Symmetric Matrices

Transpose of a Matrix AT

Ai j T A ji

Transpose Matrix Properties

Trace of a matrix

Algebraic Properties of Matrices

Find if AB = BA

The identity matrix and Inverse Matrices

Inverse of a 2x2 matrix