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 x1……… xn is an equation that can be

written in the form

• b and the coefficients a1……..an 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 AT j i

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

More on Invertible Matrices

Using Row Operations to find A-1

Begin with:

Use successive row operations to produce:

Linear Systems and Invertible Matrices

A . x = B

A-1 . A . x = A-1 . B

A . x = B

A-1 . A . x = A-1 . B

x = A-1 . B

X3 free variable

If X3 = t

X1 = -4/3 t

X2 = 2

The concept of a network appears in a variety of

applications.

A network is a set of branches through which

something “flows.”

The branches might be:

• electrical wires through which electricity flows,

• pipes through which water or oil flows,

• traffic lanes through which vehicular traffic flows,

• economic linkages through which money flows,

Applications of Linear Systems

In most networks, the branches meet at points,

called nodes or junctions, where the flow

divides

We will restrict our attention to networks in which

there is flow conservation at each node,

by which we mean that the rate of flow into any

node is equal to the rate of flow out of that node.

This ensures that the flow medium does not

build up at the nodes and block the free

movement of the medium through the network.

Applications of Linear Systems

Leontief Input-Output Models

• In 1973 the economist Wassily Leontief was awarded the Nobel prize

• for his work on economic modeling

• in which he used matrix methods to study

• the relationships between different sectors in an economy

A Homogeneous System in Economics

Leontief “input–output” (or “production”) model

The equilibrium price vector for the economy has the form

0.2 Pc + 0.8 Pf + 0.4 Pm = Pc

0.3 Pc + 0.1 Pf + 0.4 Pm = Pf

0.5 Pc + 0.1 Pf + 0.2 Pm = Pm

0.8 Pc - 0.8 Pf - 0.4 Pm = 0

-0.3 Pc + 0.9 Pf - 0.4 Pm = 0

-0.5 Pc - 0.1 Pf + 0.8 Pm = Pm