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Elementary Linear Algebra Howard Anton Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved. Chapter 1

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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

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Linear Equation a 1 x 1 +a 2 x 2 +a 3 x 3 +…+a n x n =b 1 System of Linear Equations a 11 x 1 +a 12 x 2 +a 13 x 3 +…+a 1n x n =b 1 a 21 x 1 +a 22 x 2 +a 23 x 3 +…+a 2n x n =b 2 a 31 x 1 +a 32 x 2 +a 33 x 3 +…+a 3n x n =b 3 a m1 x 1 +a m2 x 2 +a m3 x 3 +…+a mn x n =b n …………

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Solution of Linear system A linear system may have One solution No solution Infinitely many solutions

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Linear Systems in Two Unknowns Parallel LinesIntersectionOverlapping

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Linear System Involving Three Unknowns a 11 x 1 +a 12 x 2 +a 13 x 3 =b 1 a 21 x 1 +a 22 x 2 +a 23 x 3 =b 2 a 31 x 1 +a 32 x 2 +a 33 x 3 =b 3 Plane in 3D

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Linear Systems in Three Unknowns

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Homogeneous Linear System m Equation involving n Unknowns a 11 x 1 +a 12 x 2 +a 13 x 3 +…+a 1n x n =0 a 21 x 1 +a 22 x 2 +a 23 x 3 +…+a 2n x n =0 a 31 x 1 +a 32 x 2 +a 33 x 3 +…+a 3n x n =0 a m1 x 1 +a m2 x 2 +a m3 x 3 +…+a mn x n =0

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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 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

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

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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

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Section 1.3 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

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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

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Addition and Subtraction

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Multiplication of Matrices

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Matrix Partition

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Matrix Multiplication by Columns and by Rows --------8 --------9

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Entry Method, Row Method, Column Method

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Linear Combinations

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Linear Combination Theorem

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Linear Combination Example

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Column Row Expansion

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Matrix form of a Linear System

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Transpose of a Matrix A T

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Transpose Matrix Properties

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Trace of a matrix

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Section 1.4 Algebraic Properties of Matrices

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The identity matrix and Inverse Matrices

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Inverse of a 2x2 matrix

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More on Invertible Matrices

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Section 1.5 Using Row Operations to find A -1 Begin with: Use successive row operations to produce:

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Section 1.6 Linear Systems and Invertible Matrices

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Section 1.7 Diagonal, Triangular and Symmetric Matrices

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