MATB 314 & MATB 253 - LINEAR ALGEBRA

2009 Bulletin Data: Systems of Linear Equations and Matrices,

Determinants, Euclidean Vector Spaces,

General Vector Spaces, Inner Product

Spaces, Eigenvalues and Eigenvectors,

Applications.

Textbook: Anton H. and Rorres C.: Elementary Linear

Algebra (Applications Version), 9th Edition,

John Wiley & Sons, Inc, 2005.

Objectives:

At the end of the course, students should be able to solve systems of linear equations using the Gaussian/ Gauss-Jordan elimination, Cramer’s rule and the inverse of a matrix, calculate the determinants, find the standard matrix of linear transformations from Rn to Rm , determine whether a set of objects together with operations defined on it form a vector space, test for a subspace, show whether a set of vectors is a basis, determine the dimension of a vector space, find a basis for the row space, column space and nullspace of a matrix, calculate the rank and nullity of a matrix, give examples of inner product spaces, use the Gram- Schmidt process to find an orthonormal basis, find the eigenvalues and the corresponding eigenvectors of a square matrix, how to diagonalize a matrix. Some applications of linear algebra to engineering are discussed.

2

Chapter 1

SYSTEMS OF LINEAR EQUATIONS AND MATRICES

(2 ½ weeks)

Contents

1.1 Introduction to Systems of Linear Equations...................................................3

1.2 Gaussian Elimination.......................................................................................9

1.3 Matrices and Matrix Operations.....................................................................20

1.4 Properties of Matrix Operations.....................................................................27

1.5 Elementary Matrices and a Method for Finding 1A ......................................33

1.6 Further Results on Systems of Equations and Invertibility............................42

1.7 Diagonal, Triangular, and Symmetric Matrices.............................................48

3

Introduction to Linear Algebra

Sample of Linear Algebra Application.

Find the currents in the circuits

SYSTEMS OF LINEAR EQUATIONS AND MATRICES

1.1 Introduction to Systems of Linear EquationsLinear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations in finite dimensions. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract

4

algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by a linear model.

In mathematics and linear algebra, a system of linear equations is a set of linear equations such as

A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. The existence of a solution depends on the equations, and also on the available values (whether integers, real numbers, and so on).

There are many different ways to solve systems of linear equations, such as substitution, elimination, matrix and determinants. However, one of the most efficient ways is given by Gaussian elimination (matrix)

5

In general, a system with m linear equations and n unknowns can be written as

where are the unknowns and the numbers are the coefficients of the system.

We can collect the coefficients in a matrix as follows:

If we represent each matrix by a single letter, this becomes

where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries.

6

Gauss-Jordan elimination applies to all these systems, even if the coefficients come from an arbitrary field.

If the field is infinite (as in the case of the real or complex numbers), then only the following three cases are possible (exactly one will be true)

For any given system of linear equations:

the system has no solution (the system is over determined)

the system has a single solution (the system is exactly determined)

the system has infinitely many solutions (the system is underdetermined).

7

A system of equations that has at least one solution is called consistent; if there is no solutions it is said to be inconsistent.

A system of the form is called a homogeneous system of linear equations. The set of all solutions of such a homogeneous system is called the nullspace of the matrix A.

Example 1 :

If the system is homogeneous and then we have a trivial solution.

If the system is homogeneous and at least one x i ≠ 0 then we have a nontrivial solution.

Because of a homogeneous linear system always has the trivial solution; there are only 2 possibilities for its solution.

(a) The system has only the trivial solution

(b) The system has infinitely many solutions in

addition to the trivial solution.(Howard,2005)

8

____________________________________________________

Augmented Matrices

Linear Equations matrix form

Augmented Matrix

______________________________________________________________________________

Theorem A homogeneous system of linear equations with more unknowns than equations has infinitely many solution.

9

Elementary Row Operation

An elementary row operations (ERO) on a matrix A is one of the following :

1. Multiply a rows by a nonzero constant, c cRi Ri

2. Switching two rows Ri Rj

3. Add c times of one row to another row cRi + Rj Rj

Elementary row operations are used to reduce an augmented matrix or matrix to row echelon form or reduced row-echelon form. Reducing the matrix to row echelon form is called Gaussian elimination and to reduced row-echelon form is called Gauss–Jordan elimination . Gaussian elimination is an efficient algorithm for solving systems of linear equations. An extension of this algorithm, Gauss–Jordan elimination, reduces the matrix further to reduced row echelon form.

In mathematics, Gauss–Jordan elimination is a version of Gaussian elimination that puts zeros both above and below each pivot element as it goes from the top row of the given matrix to the bottom. (http://en.wikipedia.org)

______________________________________________________________________________

10

1.2 Gaussian Elimination

Echelon Forms Reducing the augmented matrix of a system to “row-echelon form” To be in this form, a matrix must have the following properties:

1. If a row does not consist entirely of zeroes, then the first nonzero number in the row is 1, We call this a leading 1.

2. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix

3. If any two successive rows that do not consist entirely zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.

4. In Reduced Row-Echelon, each column that contains a leading 1 has zeros everywhere else in the higher row.

____________________________________________________________________________

Row-Echelon

Example 2 :

11

Reduced Row-Echelon

Example 3 :

, ,

____________________________________________________

The augmented matrix for a system of linear equations is put in reduced row-echelon form, and then the solution set of the system will be evident by inspection or after a few simple steps.

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form.

Example 4:

(a)

means by

12

(b) means by

(c)

use first 3 rows to solve the system of linear equation

(d) means no solution

____________________________________________________

13

Solutions of Linear Systems

Elimination Methods : Gaussian Elimination and Back – Substitution

Example 5 :

Solve the system by Gaussian Elimination

1.

Solution Gaussian Elimination

Back – Substitution

14

______________________________________________________________________________

2.

Solution

15

Gauss - Jordan Elimination

Example 6 :

(a) Consider the linear system .

Solve the system using Gauss-Jordan elimination method.

Solution (a)Gauss - Jordan Elimination

17

(b) Consider .

Solve the system using Gauss-Jordan elimination method.

Solution (b)Gauss-Jordan elimination

18

__________________________________________________________

Homogeneous Linear Systems

Example 7 :

(a) Solve

Solution (a)

19

Since last row is entirely zero entries, the homogeneous system has nontrivial solution (many solutions)

20

(b) Solve

Solution (b)

21

Therefore the system has exactly one solution. (Trivial solution)

____________________________________________________

1.3 Matrices and Matrix Operations

Matrix Matrix is a rectangular array of numbers or, more

generally, a table consisting of abstract quantities that can be added and multiplied.

The numbers in the array are called the entries in the matrix

Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on two parameters.

Matrices can be added, multiplied, and decomposed in various ways, making them a key concept in linear algebra and matrix theory.

Definitions and notations The horizontal lines in a matrix are called rows and the

vertical lines are called columns.

22

A matrix with m rows and n columns is called an m-by-n matrix (written m×n) and m and n are called its dimensions.

The dimensions of a matrix are always given with the number of rows first, then the number of columns.

The entry of a matrix A that lies in the i -th row and the j-th column is called the i,j entry or (i,j)-th entry of A. This is written as aij or (A)ij. The row is always noted first, then the column.

We often write to define an m×n matrix A with each entry in the matrix [aij]mxn called aij for all 1 ≤ i ≤ m× and 1 ≤ j ≤ n

(Adapted from http://en.wikipedia.org)

A matrix where one of the dimensions equals one is often called a vector, and interpreted as an element of real coordinate space.

23

A 1 × n matrix (one row and n columns) is called a row vector, and an m × 1 matrix (one column and m rows) is called a column vector.

____________________________________________________

Example 8 :

Row vector Column vector

A square matrix is a matrix which has the same number of rows and columns.

A square matrix of order n and the entries a11, a22, . . . , ann are the main diagonal of A.

The unit matrix or identity matrix In, with elements on the main diagonal set to 1 and all other elements set to 0, satisfies M In=M and In N=N for any m-by-n matrix M and n-by-k matrix N.

Example 9 :

24

if n = 3: I3 =

__________________________________________________________

Operations on Matrices

In matrix notation, if A= [aij] and B = [bij] have the same size, then A = B if and only if (A)ij= (B)ij

Example 10

Addition and Subtraction ( Only on same size matrices)

DefinitionTwo matrices are defined to be equal if they have the same size and their corresponding entries are equal.

25

Given m-by-n matrices A and B , their sum A + B is the m-by-n matrix computed by adding

corresponding entries (A + B)ij = (A)ij + (B) ij = a ij + b ij

their difference A B is the m-by-n matrix computed by subtracting corresponding entries (A - B)ij = (A)ij - (B)ij

= a ij - b ij

Example 11 :

1. Addition

2. Subtraction

______________________________________________________________________________

Scalar Multiples

26

If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of the matrix A by c. The matrix cA is said to be a scalar multiple of A. In matrix notation, if A = aij , then (cA)ij = c(A)ij = caij

Example 12 :

Linear combination c1A1 + c2A2 + . . . + cnAn

______________________________________________________________________________

Multiplying Matrices

Multiplication of two matrices is well-defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix.

If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix (m rows, p columns) given by

27

Example 13

______________________________________________________________________________

Matrix Products as Linear Combination

28

Then

______________________________________________________________________________

Matrix Forms of a Linear System

Let Ax = b as an augmented matrix

Matrix Defining Functions and the product y = Ax is

29

______________________________________________________________________________

1.4 Properties of Matrix Operations

Properties of Matrix ArithmeticAssuming that the sizes of the matrices are such that the indicated operations can be performed, the following rules of matrix arithmetic are valid. A, B ,C are matrices and a ,b, c are any constant

(a) A + B = B + A (Commutative law for addition)

(b) A + (B + C) = (A + B) +C (Associative law for addition)

(c) A(BC) = (AB)C (Associative law for multiplication)

(d) A (B + C) = AB + AC (Left distributive law)

(e) (B + C) A = BA + CA (Right distributive law)

(f) A (B – C) = AB – AC(g) (B – C) A = BA – CA(h) a (B + C) = aB + aC (i) a (B C) = aB – aC (j) (a + b)C = aC + bC

30

(k) (a b)C = aC bC (l) a(bC) = (ab) C(m) a(BC) = (aB)C = B(aC)

______________________________________________________________________________

Zero Matrices

Properties of Zero Matrices

(a) A + 0 = 0 + A = A(b) A-A = 0(c) 0 – A = –A(d) A0 = 0 ; 0A =0

______________________________________________________________________________

Identity Matrices

TheoremIf R is the reduced row-echelon form of an nxn matrix A, then either R has a row of zeros or R is the identity matrix In.

31

Inverse of A

Example 14

Solution

DefinitionIf A is a square matrix, and if a matrix B of the same size can be found such that AB = BA= I, then A is to be invertible and B is called an inverse of A. If no such matrix B can be found, then A is said to be singular.

32

Note: A

– 1

= B and B

– 1

= A_________________________________________________________

Method of finding inverse of 2 x 2 invertible matrix

Properties of Inverses

1) If B and C are both inverses of the matrix A, then B = C

2) A A−1 = A−1A = I

3) If A and B are invertible matrices of the same size , then AB is invertible and (A B)−1 = B−1A−1

_________________________________________________________________________________________

Power of a Matrix

Definition If A is a square matrix, then we define the nonnegative integer powers of A to be

Theorem

The matrix is invertible if , in

which case the inverse is given by the formula

33

a. A0 = I

b. An = A . A . A . . . A (n> 0) n factors

c. A−n = (A−1)n = A−1 . A−1 . . . A−1

n factors

Laws of Exponents

a . If A is a square matrix and r and s are integers, then Ar As = Ar+s

b. ( Ar ) s = A r s

c. If A is an invertible matrix, then : A−1 is invertible and

( A−1 ) −1 =A

d. If A is an invertible matrix, then : An is invertible and

( An ) −1 = (A−1 )n for n = 1, 2, …

e. For any nonzero scalar k, the matrix kA is invertible and

(kA)−1 = (1/k) A−1

__________________________________________________________________________________________

34

Transpose of a Matrix (AT)i j = (A)j i

Example 15 :

Properties of the Transpose

If the sizes of the matrices are such that the stated operations can be performed, then

(a) ((A)T)T = A

(b) (A+B)T=AT + BT and (A − B)T=AT − BT

(c) (k A)T= kAT , where k is any scalar

(d) (AB)T= BT AT

DefinitionIf A is any m x n matrix, then the transpose of A, denoted by AT, is defined to be the n x m matrix that the results from interchanging the rows and columns of A; that is, the first column of AT is the first row of A, the second column of AT

is the second row of A, and so forth.

35

Invertibility of a Transpose

Example 16 :

Let

_____________________________________________________________________________________

1.5 Elementary Matrices and a Method for Finding 1A

Elementary Matrices

.

TheoremIf A is an invertible matrix, then AT is also invertible and (AT)−1 =( A−1)T

DefinitionAn n x n matrix is called an elementary matrix if it can be obtained from the n x n identity matrix In by performing a single elementary row operation

36

Example 16 : I E1

Row Operations and Inverse Row Operations

Row operation on I that produces E

Example Row operation on E that reproduces I

Example

Multiply row i by c 0

[I] R1 = 3R1 [E] Multiply row i by 1/c

[I] R1 = 1/3R1[E]

Interchange rows i and j

[I] [E] Interchange rows i and j

[I] [E]

Add c times row i to row j

[I] R1 = R1 +2R3 [E] Add –c times row i to row j

[I] R1= R12R3[E]

To find the inverse of an invertible matrix A, find a sequence of elementary row operations that reduces A to the identity and then perform this same sequence of operations on In to obtain A1

37

__________________________________________________________

Example 17 :

Given matrix . Find the product of EA.

Let E is obtained from .

Solution

Theorem ROW OPERATIONS BY MATRIX MULTIPLICATION

If the elementary matrix E results from performing a certain row operation on Im and if A is an m x n matrix, then the product EA is the matrix that results when this same row operation is performed on A.

38

By Theorem

______________________________________________________________________________

A method for Inverting Matrices

Ek . . . E2 E1 A = In A−1 = Ek . . . E2 E1 In = Ek . . . E2 E1

A−1

A = E1 −1 E2 −1 . . . Ek −1 In = E1 −1 E2 −1 . . . Ek −1

__________________________________________________________

Example 18 :

Theorem Every elementary matrix is invertible, and the inverse is also an elementary matrixTheorem EQUIVALENT STATEMENTS

If A is an n x n matrix, then the following statements are equivalent, that is, all true or all false.

(a) A is invertible(b) Ax = 0 has only the trivial solution(c) The reduced row-echelon form of A is In (d) A is expressible as a EA

39

Let A be a matrix such that where is obtained from by performing the operation

is obtained from by performing the operation

is obtained from by performing the operation

Find A and A 1 using elementary matrices.

Solution

40

Try this

=

__________________________________________________________

Row Operation to find A 1 [ A | I ] ® [ I |A 1]

41

Example 19 :

Find the inverse of

Solution

42

Try this

______________________________________________________________________________

Showing That a Matrix Is Not Invertible

Example 20 :

Show that A is not invertible

Solution

43

There is a row of zeros. Therefore A is not invertible.

A Consequence of Invertibility

Let A = . If A is invertible , then

1. the homogeneous system has only the trivial solution.

2. the nonhomogeneous system has exactly one solution.

3. the linear system is consistent (the linear system has a solution)

44

__________________________________________________________

1.6 Further Results on Systems of Equations and Invertibility

Basic Theorem

TheoremEvery system of linear equations has no solutions, or has exactly one solution, or has infinitely many solutions.

Linear Systems by Matrix Inversion

Theorem If A is an invertible n x n matrix, then for each n x 1 matrix b , the system of equations Ax =b has exactly one solution, namely , x = A−1b

Solution of a Linear System Using A − 1

45

Write as Ax = b. Find A−1 then A−1(Ax) = A−1b x = A−1b

Example 21 :

Consider the system of linear equations .

Solve using matrix inversion.Solution

A = and A1 = (from Example 18)

x = 3 , y = 2 , z = 4

Solving Two Linear System at Once

Example 22 :

Solve the systems and

Solution

Linear system with a common coefficient matrix

46

First system : x = 3 , y = 0 , z = 5

Second system : x = 2 , y = 0 , z = 3__________________________________________________________

Properties of Invertible Matrices

AB = I and BA = I for A & B are square matrices

______________________________________________________________________________

47

__________________________________________________________

Determining Consistency by Elimination

Example 23:

Theorem Let A be a square matrix, (a) If B is a square matrix satisfying BA = I, then B = A−1

(b) If B is a square matrix satisfying AB = I, then B = A−1

Theorem EQUIVALENT STATEMENTS

If A is an n x n matrix, then the following statements are equivalent, that is, all true or all false.a) A is invertibleb) Ax = 0 has only the trivial solutionc) The reduced row-echelon form of A is In d) A is expressible as a product of elementary matrices. EA e) Ax = b is consistent for every nx1 matrix bf) Ax = b has exactly one solution for every n x 1 matrix b

TheoremLet A and B be a square matrices of the same size. If AB is invertible, then A and B must also be invertible.

48

Find the conditions that the b’s must satisfy the system to be consistent

(a)

(b)

Solution

(a)

49

______________________________________________________________________________

(b)

50

Therefore b3 has no condition for the linear system to be consistent.

1.7 Diagonal, Triangular, and Symmetric Matrices

Diagonal Matrices Example 24

_____________________________________________________________________________________________________________________

Triangular Matrices

Example 25

Upper Triangular Lower Triangular

51

___________________________________________________________________________

Symmetric Matrices A = AT (row i = column j )

Example 26 :

Products AA T and A T A

Example 27 :

and are equivalent matrices

52

*************************** END OF CHAPTER 1 *****************************