mab312 chapter 1 2014

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School of Mathematical Sciences

Queensland University of Technology

MAB 312

Linear Algebra

Lecture Notesc 2014 Professor Ian Turner.

CRICOS Institution Code: 00213J

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Contents

List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Introduction to Set Theory and Algebraic Systems 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Set Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Set Operations . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Foundations of Abstract Algebra . . . . . . . . . . . . . . . . 7

1.3.1 Classifications of Operations . . . . . . . . . . . . . . . 8

1.3.2 Algebraic Systems . . . . . . . . . . . . . . . . . . . . 8

1.4 Real Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 Examples of Real Vector Spaces . . . . . . . . . . . . . 13

1.5 Review of Matrix Properties, Linear Systems and Elimination

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.1 A Review of Some Basic Matrix Properties . . . . . . . 14

1.5.2 Special Matrix Types and Their Properties . . . . . . . 17

1.5.3 Review of Systems of Linear Equations . . . . . . . . . 18

1.5.4 Elimination Methods for Solving Linear Systems. . . . 201.5.5 Gaussian Elimination. . . . . . . . . . . . . . . . . . . 22

1

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List of Figures

1.1 Set inclusion A B. . . . . . . . . . . . . . . . . . . . . . . . 31.2 The union of setsA B. . . . . . . . . . . . . . . . . . . . . . 41.3 (a) The intersection of setsA B, (b) A B = . . . . . . . . 41.4 Set differenceA \ B. . . . . . . . . . . . . . . . . . . . . . . . 61.5 The complement ofA, Ac. . . . . . . . . . . . . . . . . . . . . 6

1.6 Set symmetric differenceA B. . . . . . . . . . . . . . . . . . 71.7 The algebraic systems considered in this chapter. . . . . . . . 14

2

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MAB312 Linear Algebra 3

LIST OF SYMBOLS

Greek Letters

alpha theta rho

beta kappa sigma

gamma lambda tau

delta mu phi

epsilon nu chi

zeta xi psi

eta pi omega

Set Notation

x A The objectxis in the set A(x is an element ofA)x / A The objectxis not in the set A empty setA B Set inclusion: Ais a subset ofB, each element that is a member ofA

is also a member ofB

A

B A is a proper subset ofB, there is at least one element ofB

that is not a member ofA

{x|P} The set of allxfor which propertyP holdsA B Union: A B = {x|x A or B}A B Intersection: A B = {x|x A &B}A B Cartesian product: A B = {(a, b)|a A , b B}

Logic Terms

means for all means there existsP Q means statement P implies statement QP Q means both P Q and Q Piff read if and only if and means the same asx a meansxapproaches a continuouslyx a+ means xapproaches a from the rightx a means xapproaches a from the leftx meansxincreases without bound

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Real Number System

N Natural numbers: 1, 2, . . . , n , . . .

Z Integers: . . . , n , . . . , 2, 1, 0, 1, 2, . . . , n , . . .Q Rational numbers: p

q (p, qmutually prime p Z, q N)

R Real numbers

R2 Cartesian product R RC Complex numbers

|a b| Distance between aand b

Intervals

(a, b) Open interval:{x|a < x < b}[a, b] Closed interval:{x|a x b}(a, b] Half-open interval:{x|a < x b}(a, ) Infinite interval:{x|a < x}|x a| < Neighbourhood ofa i.e. open interval a < x < a+

Functions or Mappings

f :A B Function or map from setAinto set Bf1 :B A Inverse function or map fromB into Ax f(x) denotes the image of an arbitrary element x Aunder ff g Composition of two functions f and g

Other Notation

A= [aij] Matrix

AT Transposed Matrix

In Identity Matrix

0 Zero Matrix

a Vector

[A|b] Augmented Matrixi

1z Complex conjugate ofz

det(A) = |A| Determinant ofAtr(A) =

ni=1

aiiTrace ofA

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Rn n-dimensional Euclidean space

Rmn Vector space of all m nmatrices with real coefficientsx y or xTy dot product (inner product)x y= ni=1xiyix y vector cross product inR3x norm ofx,x = (ni=1x2i ) 12C(D) space of continuous functions on domain D

Cn(D) space of functions that have n continuous derivatives on domain D

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B

U

A

Figure 1.1: Set inclusion A B.

As an example, if we let T = {x1, x2}, with x1, x2 and Sgiven above, thenT S.Immediate properties of set inclusion are

(i) A= B A B and B A.(ii) A B and B C A C.

1.2.1 Set Operations

Some structure can be added to the basic concept of a set by defining set

operations. All the sets we mention are assumed subsets of our universal set

U. In this sense, Uis our frame of reference and is selected appropriately for

the different contexts we consider throughout the unit. It is also convenient

to have available in Ua set containing no elements whatsoever, called the

empty set and written . The empty set is considered a subset of every otherset A = , for example A U.

(i) Theunion of two setsAand B is the set of all elementsxthat belong

to Aor B (refer figure 1.2), namely

A B = {x| x Aor x B} .

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U

A B

Figure 1.2: The union of sets A B.

U

A B

U

A B

(a) (b)

Figure 1.3: (a) The intersection of sets A B, (b) A B = .

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(ii) The intersection of two sets A and B is the set of all elements x

that belong to Aand B (refer figure 1.3(a)), namely

A B = {x| x Aand x B} .

(iii) Two sets are said to be disjointiffthey have no elements in common

(refer figure 1.3(b)).

(iv) The difference of two sets A and B, denoted A \ B, is the set of allelements that belong to Abut not B (refer figure 1.4), namely

A \ B = {x| x Aand x / B} .(v) Thecomplementof a setA, writtenAc, is the set of elements that do

not belong to A(refer figure 1.5), namely

Ac = {x| x / A} .

We also write Ac =U\ A.(vi) Thesymmetric differenceof two setsA and B , writtenA B, is the

set of all elements belonging to either A or B but not both A and B(refer figure 1.6), namely

A B = {x| x Aor x B, and x / A B},

also written as A B = (A \ B) (B \ A).

Remark: A set is said to be finite if it is either empty (consisting of 0

elements) or consists ofn elements for some positive integer n; otherwise it

is said to be infinite.

Classes: We refer to sets whose elements are themselves sets as classes.

We define the power set or power class of a set S, writtenP(S), to bethe class of all subsets of S. A finite set with n elements has 2n sub-

sets, including and S itself. For example, if S ={1, 2, 3} thenP(S) ={, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, S}.

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U

BA

Figure 1.4: Set difference A \ B.

U

A

A!Ac

Figure 1.5: The complement ofA, Ac.

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U

A B

Figure 1.6: Set symmetric difference A B.

EXAMPLE 1.1.

LetU= {n| n N, n 9}, A= {1, 2, 3, 4}, B = {2, 4, 6, 8}, C= {3, 4, 5, 6}.

(i) FindAc, A

C, A

C.

(ii) FindA \ B, B \ A andA B.

1.3 Foundations of Abstract Algebra

Definition 1.1. The cartesian product of two sets A and B, denoted

A B, is the set of all ordered pairs(a, b), wherea A andb B, namely

A B = {(a, b)| a A and b B} .For example, ifA1= {1, 2, 3} and A2 = {a, b} then

A1 A2= {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}.

A binary operationon a set A is a function f from a set A A into A,namely f :A A A. The familiar symbols +, , , are all examples ofbinary operations on pairs of real numbers (RR R).

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A binary operation

is said to be closed on S

A iff for a, b

S then

a b S.

1.3.1 Classifications of Operations

We list in this section only some classifications of operations, clearly others

exist. Let and be two binary operations defined on a set S and leta,b,c S.

(i) Commutativity: The binary operation is said to be commutativewhen a b= b a.

(ii) Associativity: The binary operation is said to be associative when(a b) c= a (b c).

(iii) Distributivity: The operation is said to be leftdistributive withrespect toiff a (b c) = (a b) (a c). The operationis said tobe rightdistributivewith respect to iff (b c) a= (b a) (c a).An operation is said to be distributivewith respect to if it is bothleft and rightdistributive.

(iv) Identity Element

: The set S is said to have an identity elementwith respect to the binary operation on S iff e S such thate a= a e= a,a S.

(v) Inverse Element: An element b S is called an inverseofa relativeto the binary operation iff a b = e = b a. We typically use thenotation a1 to denote the inverse ofa.

1.3.2 Algebraic Systems

The termsystemis another important concept that is difficult to define withprecision. In this section we define a system to be any well-defined collection

of mathematical objects consisting of a set together with operations on the

set, and a collection of postulates, definitions and theorems describing various

properties of the system. Perhaps the most primitive system isS = S, (setS, binary operation ), which is said to have very little structure becauseit contains only a few components. By adding additional components to a

system we are said to be adding more structure.

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We now provide a brief overview of some very important abstract mathemat-

ical systems.

1. Group: An abstract system G = G, , withG non-empty, is said to bea group iff it satisfies the following axioms

(i) operation is associative;(ii) an identity element e G;

(iii) a G, an inverse a1, called the inverse ofa.

If the operationis also commutative, the group is called an Abelian group.IfG consists of a finite number of elements then we say it is a finite group,otherwise we say it is an infinite group.

Note: the abstraction should now be apparent; we have said nothing about

the actual nature of the set G, nor the operation. Also, in axioms (ii) and(iii) we speak of the identity element and the inverse element respectively,

as if there were only one. Indeed this is the case.

Examples of Groups: We list here just a few examples of groups.

(i) Z, +, i.e., set of integers is a group under addition with e := 0 anda1 := a;

(ii)Q, +

;

(iii)R, +

;

(iv){0}, +;

(v){x | x Q and x >0}, ;

(vi){x | x R and x >0}, ;

(vii)

{1},

;

{1, 1},

.

WhilstZ, +

is a group,

Z, is not a group. As another example,{2, 1, 0, 1, 2}, + is not a group.

2. Ring: An abstract systemR = S, +, is said to be a ring iff(i)

S, +

is an Abelian group with identity element 0, which is calledzero;

(ii)

S, is closed, associative, and distributive with respect to +.

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The study of rings originated in the last half of the 19 th century. Although

Richard Dedekind introduced the concept of a ring in the 1880s, it was David

Hilbert who coined the term ring in 1892. The first axiomatic definition

of a ring was given by Adolf Fraenkel in 1914. In 1921, Emmy Noether gave

the first axiomatic foundation of the theory of commutative rings.

Examples of Rings: We list here just three examples of rings.

(i)Z, +, ;

(ii)

Q, +,

;

(iii) R, +, .3. Field: An abstract systemF= S, +, is said to be a field iff

(i) Addition:

S, +

is an Abelian group with identity element 0, which

is called zero;

(ii) Multiplication:

S\ {0}, is an Abelian group, i.e., it is closed,commutative and associative; an identity element 1 (sometimes calledthe unity), such that a 1 =a, a S; an inverse element a1, suchthat a a1 = 1, a S;

(iii) Distributivity: the binary operation is distributive with respect to+.

Although the concept of field was initially conceived in 1871 by Richard

Dedekind, it was Eliakim Hastings Moore who coined the term field. In

1881, Leopold Kronecker defined a field of polynomials and in 1893, Heinrich

M. Weber gave the first rigorous definition of an abstract field. Emil Artin

developed the relationship between groups and fields in great detail from

1928 through 1942. Roughly speaking, fields are often referred to as the

number systemsof mathematics.

Examples of Fields: We list here just two examples of fields.

(i)Q, +, ;

(ii)R, +, .

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EXAMPLE 1.2.

(i) LetA = {a, b}. Show that the systemA = A, +, with the operationsdefined by the tables

+ a b

a a b

b b a

a ba a a

b a b

is a ring.

(ii) Consider the setB of all2 2 matrices of the form

B =

a b

b a

a, b R

Determine whether the systemB= (B, +, .), where+ and . representthe usual matrix addition and multiplication, is a field.

1.4 Real Vector Spaces

We first introduce the definition of a vector space. Over the last century,

mathematicians observed that problems from different fields of mathematics

often possessed many of the same attributes and properties. This observation

led to the derivation of a unifying theory based on an abstract approach that

concentrated on only the essential facts. For such an approach, one usually

starts from a set of elements satisfying certain axioms and the nature of the

objects are purposely left unspecified. The theory then consists of logical

consequences that are argued directly from the axioms and usually stated

and proved as theorems. Such an approach leads to a rich theory that isdeveloped in an abstract manner. The beauty is that these general theorems

can be applied later to various sets satisfying those axioms. We will pursue

this thinking further in chapter 2 and study here only the vector space Rmn.

Definition 1.2. A real vector space V is a set of objects, called vectors,

together with two operations, called addition and scalar multiplication,that satisfy the following ten axioms u, v w V and c,d, R :

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Addition

A1 u v V Closure under AdditionA2 u v= v u Commutative LawA3 u (v w) = (u v) w Associative LawA4 a vector0 V such that

u 0= 0 u= u Additive IdentityA5 u a vectoru V such that

u u= u u= 0 Additive InverseScalar Multiplication

M1 c

u V

Closure under Scalar Multiplication

M2 c (u v) =c u c v Distributive Property IM3 (c+d) u = c u d u Distributive Property IIM4 (cd) u = c (d u) Associative LawM5 1 u = u Scalar identity

Depending on the application, one can generalise the scalars c and d to be

chosen from any field, for example the complex numbers C. However, in

this unit we focus solely on real vector spaces. Sometimes vector spaces are

referred to as linear spaces.

Observe that definition1.2does not specify the nature of the vectors nor the

operations - any type of object can be a vector and the addition and scalar

multiplication operators need not bear any resemblance to those associated

with the standard vector operations over R, namelyu + vand cu. Moreover,

one usually writes u vas u + vand c uas cu, being careful to keep inmind the particular operation in each case. To specify a vector space, a setV

must be provided together with the two operations satisfying the properties

of definition 1.2. A simple rule of thumb in establishing whether V may

be a vector space is to examine the closure axioms A1 and M1, because if

they do not hold we cannot have a vector space. Another important and

straightforward observation should be to check if the additive identity is in

V. Once we are satisfied that these axioms hold the remaining axioms can

then be verified, some of which will be obvious.

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1.4.1 Examples of Real Vector Spaces

The typical vector spaces we will deal with in this, and the next chapter, are:

(i) The setV = Rmn denoting the set ofm nmatrices with real entries,which is a vector space under the usual sum and scalar multiplication

operations for matrices.

(ii) The real coordinate spacesR1n = {(x1, x2, . . . , xn) | xi R, i= 1, . . . , n}and Rn1 ={(x1, x2, . . . , xn)T | xi R, i= 1, . . . , n}, which are spe-cific examples of the vector space (i). We remark for vector spaces that

it usually makes little difference whether a coordinate vector is depicted

as a row or column vector. When the row or column distinction is irrel-evant or it is clear from the context, we will adopt the commonly used

notation Rn. When it is important to specify the shape of a vector we

will explicitly write either Rn1 or R1n.

Theorem 1.1. LetV be a vector space withu V and R, then(i) 0u= 0,

(ii) 0= 0,

(iii) (1)u= u,(iv) Ifu= 0, then= 0 oru= 0.

EXAMPLE 1.3.

(i) Show thatRmn is a vector space overR.

(ii) Investigate whether the set of all ordered triplets

V ={(1, x1, x2)| x1, x2 R} with the operations for addition andscalar multiplication defined by

(1, x1, x2) (1, y1, y2) = (1, x1+y1, x2+y2),k (1, x1, x2) = (1, kx1, kx2)

is a vector space overR.

(iii) Show that the set of positive real numbersR+ with the operations of

scalar multiplication and addition defined byk x= xk, x R+, k Randx y=xy, x,y R+ respectively, is a real vector space.

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To conclude this section, we exhibit in figure 1.7 the major algebraic systems

discussed in this chapter, and how they are related to one another. For

example, all vector spaces and rings are Additive Abelian groups, but not all

vector spaces are rings.

Groups

Additive Abelian Groups

Rings

VectorSpaces

Fields

Figure 1.7: The algebraic systems considered in this chapter.

1.5 Review of Matrix Properties, Linear Sys-

tems and Elimination Methods

In this section we briefly summarise some important matrix properties, re-

view some special types of matrices, and revise linear systems and their

solution. These topics should be mainly revision from the first year math-

ematics units and the student is encouraged to revise this material as it is

assumed knowledge for the remainder of the unit.

1.5.1 A Review of Some Basic Matrix Properties

(i) A matrixA Rmn is said to have shape, or size, m n. Whenm = n(i.e., when A has the same number of rows as columns), A is called

a square matrix, otherwise it is called rectangular. The symbol Ai is

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used to denote the ith row, while Aj denotes the jth column of the

matrix A.

(ii) Two m n matrices (i.e., of the same shape) A = [aij] and B = [bij]are said to be equal ifaij =bij,i= 1, . . . , m , j = 1, . . . , n. We write

A= B. Note that even though the two arrays u=

x1x2

x3

and v =

(x1, x2, x3) describe the same point in R3, they cannot be considered as

equal because they have different shapes.

(iii) IfA and B are matrices of the same size, then A + BandAare alsomatrices of the same size having their ijth component as aij+bij and

aij respectively. These two properties are known as closure laws for

matrix addition and scalar multiplication respectively.

(iv) IfA Rnr and B Rrm, then the product ABis an n m matrixwhoseij th component is given by

rk=1

aikbkj . MatricesAand B are said

to be conformable for multiplication in the order ABwhenever Ahas

exactly as many columns as B has rows, i.e, when it is possible to form

the product.

(v) A matrix that has all components equal to zero is called thezero matrix

and is denoted by the symbol 0. When it is necessary to specify the

precise shape of the zero matrix we often write 0mn to indicate that

0 Rmn. Note that 0A= A0= 0, A A= 0, 0 A =A andA + 0= 0 + A= A. 0 is called the additive identity ofA andA isthe additive inverse ofA.

(vi) Matrix products generally do not commute, i.e., AB =BA in general.(vii) If A Rnm and B Rmp and C Rpq, then the Associative Law

for matrix multiplication is given byA(BC) = (AB)Cwhere both the

LHS and RHS are n q matrices.(viii) Provided that all of the products are defined, then the Distributive

Lawsfor Multiplication are A(B + C) = AB + AC(left hand) and

(A + B)C= AC + BC(right hand).

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Let A

Rmn and B

Rnr. We note the following three important cases

of block matrix multiplication for computing the product AB Rmr:1. Let B= (B1|B2), B1 Rnk, B2 Rn(rk) then AB= (AB1|AB2).

2. Let A =

A1

A2

where A1 Rkn, A2 R(mk)n then AB =

A1B

A2B

.

3. Let A= (A1, A2) and B= B

1B2

where A1 Rmk, A2 Rm(nk)

and B1 Rkr, B2 R(nk)r thenAB= A1B1+ A2B2.

1.5.2 Special Matrix Types and Their Properties

(i) A square matrix A Rnn is called symmetric if AT = A and skew-symmetric ifAT = A.

(ii) A square matrix in which all off diagonal entries are zero (aij = 0 ifi =j) is known as adiagonal matrix, denoted by D= diag (d11, d22, . . . , dnn)

or D = [d11e1|d22e2| . . . |dnnen], where ei is the ith vector in the stan-dard basis (also known as the canonical basis or the natural basis) ofRn.Clearly, with this notation, we see that AD= [d11A1|d22A2| . . . |dnnAn]and conclude that multiplication by a diagonal matrix on the rightscales the columns of A. Furthermore,

DA =

d11eT1d22e

T2

...

dnneTn

A =

d11A1

d22A2...

dnnA1

so that multiplication by Don the left scales the rows ofA.

(iii) An n n diagonal matrix with 1s on the main diagonal is known asthe identity matrixand is denoted by In. Note that InA= AIn = A.

The subscript on In can be omitted whenever the size is obvious from

the context.

(iv) A square matrix A Rnn is said to beidempotentif it has the propertythat A2 =A.

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(v) A square matrixA

Rnn is said to beinvolutoryif it has the property

that A2 =I.

(vi) A square matrix A Rnn is said to be nilpotent of index k if it hasthe property that Ak = 0, but Ak1 =0.

(vii) A square matrix in which all entries above the diagonal are zero is called

a lower triangularmatrix denoted as L, that is the componentsij = 0

for i < j.

(viii) A square matrix in which all entries below the diagonal are zero is

called anupper triangularmatrix denoted asU, that is the components

uij = 0 for i > j.(ix) The product of two lower triangular matrices is lower triangular and

the product of two upper triangular matrices is upper triangular.

(x) A system of linear equations is called sparse if only a relatively small

number of its matrix elements are nonzero.

(xi) Ifu, v Rn then the outer productof the two vectors uvT produces afull n n matrix.

(xii) Let U = [u1, . . . , un] Rmn and V = [v1, . . . , vn] Rkn, then them k matrix UV

T

= u1vT1 + +unv

Tn is called an outer product

expansion, because it produces a sum of vector outer products.

1.5.3 Review of Systems of Linear Equations

In general, a finite set of linear equations in the variables x1, x2, x3, . . . , xn is

known as alinear system of equationsor simply, alinear system, an example

of which is the system ofmequations in nunknowns:

a11x1+a12x2+a13x3+ +a1nxn = b1a21x1+a22x2+a23x3+ +a2nxn = b2a31x1+a32x2+a33x3+ +a3nxn = b3

... ...

am1x1+am2x2+am3x3+ +amnxn = bm

(1.1)

Interestingly, if we represent each equation in the linear system (1.1) as a

hyperplane of the formHi ={x |Aix= bi} , i = 1, 2, . . . , n, where Ai =

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(ai1, ai2, . . . , ain) denotes the ith row of the coefficient matrix, then the solu-

tion lies in the intersection of the mhyperplanes, namely H1H2 Hm.The set of all solutions to the linear system (1.1) is known as the solution set.

Two linear systems involving the same variables are said to be equivalent if

they have the same solution set.

In the system (1.1) we define the coefficient matrix A Rmn as follows:

A=

a11 a12 a13 a1na21 a22 a23 a2na31 a32 a33

a3n

......

... . . . ...

am1 am2 am3 amn

. (1.2)

Define further then 1 column vectorx(writtenx Rn1 or simply x Rn)and the m 1 column vector b(written b Rm1 or b Rm) as

x=

x1

x2

x3...

xn

and b=

b1

b2

b3...

bm

(1.3)

so that the matrix form of the linear system is given by the equation

Ax= b. (1.4)

Note that the matrix A consists ofmrows andncolumns or more precisely,m

(row) vectors in Rn andn(column) vectors in Rm. The vectorbis sometimes

referred to as the right-hand side (RHS) for the system.

Every linear system of the form (1.4) has either no solution, in which case

the system is said to be inconsistent, orat least one solution(exactly oneor infinitely many), in which case the system is said to be consistent. A ma-

jor job in dealing with linear systems is deciding which of these possibilities

is true for a given system.

Another important observation we can make at this point is that the product

Axmay be viewed as a linear combination of the columns of A

Ax= x1A1+ +xnAn.

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Furthermore, we see that the linear system (1.4) can be expressed as

x1A1+ +xnAn=b,from which it can be concluded that when a solution x of (1.4) is sought, we

are looking for the vector x that resolves b as a linear combination of the

columns of A. If there exists no such x then the system is inconsistent.

Definition 1.3.

If the vector b = 0 (with 0 Rm1 the zero vector), then the associatedhomogeneous system is given as

Ax= 0. (1.5)

Definition 1.4.

The matrix formed by augmenting the RHS vectorb to theA matrix is called

the augmented matrix for system (1.4) and is written

[A|b] =

a11 a12 a13 a1n b1a21 a22 a23 a2n b2a31 a32 a33 a3n b3

......

... . . .

......

am1 am2 am3 amn bm

. (1.6)

Usually the matrix and the RHSvector are separated by a vertical dashed,or solid, line to signify an augmented matrix.

Notes:1. A linear system is said to be overdetermined if there are more equations

than unknowns (m > n). Overdetermined systems are usually (but not

always) inconsistent.

2. A linear system is said to be underdetermined if there are fewer equa-

tions than unknowns (m < n). Although it is possible for such systems

to be inconsistent, they are usually consistent with an infinite number

of solutions.

1.5.4 Elimination Methods for Solving Linear Systems

Elimination methods transform, or reduce, the original augmented system

(1.6) into a structure that can easily be solved. In order to row reduce the

matrix to what is known as(Reduced) Row Echelon Form, three elementary

row operations are necessary.

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Elementary Row Operations

(i) Type I: interchange the ith and jth rows, i.e., Ri Rj.(ii) Type II: the ith row of a matrix can be scaled by a nonzero number ,

i.e., Ri Ri.(iii) Type III: a multiple of the ith row can be added to the jth row, i.e.,

Rj Rj+Ri.

Annnmatrix, usually denoted by E, is defined to be anelementary matrixif it can be obtained from the n nidentity matrix Inby performing a singleelementary row operation.

Theorem 1.2. Let A Rmn. If an elementary matrix E results fromperforming a certain row operation on Im, then the product EA (i.e., when

used as a left-hand multiplier) is the matrix that results when the same row

operation is performed onA.

Definition 1.5. (Equivalence)

Whenever B is obtained from A by performing a finite sequence of elemen-

tary row operations we writeArow

B and say that the two matrices are row

equivalent, i.e. A row B PA = B, where P is the product of these finitenumber of elementary matrices.

Note that in some texts the notation A B is used often to indicate rowequivalent matrices. We use these notations synonymously throughout the

unit.

Definition 1.6. (Reduced Row Echelon Form)

A matrix is deemed to be in Reduced Row Echelon Form (RREF) if the

following four conditions hold:

(i) All zero rows (if, in fact, they arise during the elimination process)

appear at the bottom of the matrix.

(ii) The first nonzero number (moving from the left) in any nonzero row is

1.

(iii) For any two successive nonzero rows, the leading 1 in the lower row

occurs farther to the right than the leading 1 in the higher row.

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(iv) Each column that contains a leading 1 has zeros everywhere else.

Definition 1.7. (Row Echelon Form)

A matrix is said to be in Row Echelon Form (REF) if conditions (i)-(iii) of

definition1.6hold.

Note that the Row Echelon Form of a matrix A is not unique, however,

the Reduced Row Echelon Form is unique and it is often denoted by EA

(the notation we use throughout this unit) when it is derived from Asolely

by means of row operations. Sometimes we relax the requirement that 1s

appear in the pivotal positions for REF.

Definition 1.8. (Basic Variables and Basic Columns)

The variables corresponding to the first leading 1s (or the locations of the

pivotal positions) occurring in each nonzero row coincide with the basic vari-

ables and the associated columns in the coefficient matrix are called the basic

columns. The remaining variables are the free variables and the correspond-

ing columns in the coefficient matrix are called the nonbasic columns. In fact,

if there arer nonzero rows in the reduced augmented matrix andn unknowns

then there will ber basic variables and(n r) free variables.

Definition 1.9. (Rank of a Matrix)

If r is the number of nonzero rows (r basic variables) in any matrix in a

row echelon form to whichA is equivalent, thenrank(A) =r. Furthermore,

rank(A) can be interpreted as the number of basic columns inA.

Note that elementary row operations performed onAdo not changerank(A)

i.e., A B iff rank(A) = rank(B). Note further that transposition doesnot change the rank, i.e., m n matrices A,rank(A) =rank(A

T

).

1.5.5 Gaussian Elimination

Gaussian Elimination effectively transforms the augmented system for the m

equations inn unknowns into a particular REF, which is sometimes called a

jagged or stair-case type of triangular form, ifm > n, or anupper triangular

form, ifm= n so that the unknown variables can be obtained by a process

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known as back substitution. Thus, the Generalised Gaussian Elimination

procedure is carried out in two distinct stages:

Stage 1: (Elimination) Ax = b reduce Rx = y, where R = PA,

y= Pb and P Rmm is a product of elementary matrices.The new rows of thekth sub-augmented matrix [R|y] are computed using theappropriate Type I and Type IIIoperations. Typically, the nonzero entriesin an echelon form must lie on or above a stair-case line that emanates fromthe upper-left-hand corner and slopes downward to the right, an example ofwhich is shown in the following figure:

.

Stage 2: (Back Substitution) Solve Rx= y for x.

Before the back substitution process is commenced it is necessary to check

the consistency of the system. If there are any zero rows in the eliminated

system that have nonzero entries in the corresponding y vector element,

then there is no solution to the system because the system is inconsistent,

i.e., rank(A|b)= rank(A). If the system is consistent, i.e., rank(A|b) =rank(A), then one proceeds with the standard back substitution process

discussed in the elementary linear algebra unit.

EXAMPLE 1.4.

(i) Make use of the idempotent sub-matrices of

A=

1 0 1/2 1/2

0 1 1/2 1/2

0 0 1/2 1/2

0 0 1/2 1/2

to findA100.

(ii) Find a parametric description of the intersection of the following three

hyperplanes inR4

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x1

x2+ 2x3+ 3x4 = 2

2x1+x2+x3 = 1

x1+ 2x2 x3 3x4 = 1

(iii) Which of the following linear system solution sets, with usual addition

inR3, are vector spaces?

(a)

1 0 1

2 3 1

3 3 0

x1

x2

x3

=

0

0

0

, (b)

1 2 2

3 6 5

1 2 0

x1

x2

x3

=

1

2

1

.

mab312 chapter 1 2014

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