Bsc 6th sem m-1 full notes

Curriculum for B.Sc. Mathematics Program of RCUB as per NEP 2020 w.e.f. 2021-22

SEMESTER – VI

Course Title: 6.1Linear Algebra
Sem. VI

             Formative                                        Summative                                          Duration of 

             Assessment                                      Assessment                                             ESA:

             Marks: 40                                         Marks: 60                                             02 hrs.

 Course           Course Learning Outcomes: The overall expectation from this course is that the  Outcomes          student  will build a basic understanding in few areas of linear algebra such as                                          vectors spaces,  linear transformations. Some broader course outcomes are listed                                     as follows. At the end of this course, the student will be able to

                     

•              Understand the concepts of Vector spaces, subspaces, bases dimension and their

                       properties.

•              Become familiar with the concepts of Eigen values and Eigen vectors, linear

                        transformations etc.

•               Prove various statements in the context of vectors spaces.

Unit I     

              Rings and integral domains: Rings, Properties of rings, sub rings,

              ideals, principal and maximal ideals in commutative ring, quotient

              ring, homomorphism and isomorphism, and integral domains.

Unit II

              Vector spaces: Definition, examples and properties; Subspaces -

              Examples, criterion for a sub-set to be a subspace and some properties;

              Linear Combination-Linear span, Linear dependence and Linear

              independence, basic properties of linear dependence and

              independence, techniques of determining linear dependence and

              independence in various vector spaces and related problems; Basis and

              dimension - Co-ordinates, ordered basis, some basic properties of

              basis and dimension and subspace Spanned by given set of vectors;

              Quotient space. Dimension of quotient space (derivation in finite

              case); Sum and Direct sum of subspaces - Dimensions of sum and

              direct sum spaces (Derivation in finite case).

Unit III

             Linear transformations: Definition, examples, equivalent criteria,

             some basic properties and matrix representation and change of basis

             and effect on associated matrix, similar matrices; Rank - Nullity

             theorem - Null space, Range space, proof of rank nullity theorem and

             related problems.

Unit IV

             Isomorphism, Eigen values and Diagonalization:

             Homomorphism, Isomorphism and automorphism-Examples, order of

            automorphism and Fundamental theorem of homomorphism; Eigen

            values and Eigen vectors-Computation of Eigen values, algebraic

            multiplicity, some basic properties of Eigen values, determination of

            eigenvectors and Eigen space and geometric multiplicity.

            Diagonalizability of linear transformation - Meaning, condition based

            on algebraic and geometric multiplicity (mentioning) and related

            problems(Only verification of diagonalizability).

Recommended Leaning Resources

References:

                    1.   I. N. Harstein, Topics in Algebra, 2nd Edition,Wiley.

                    2.   Stephen H. Friedberg, Arnold J. Insel & Lawrence E. Spence (2003), Linear

                          Algebra (4th Edition), Printice- Hall of India Pvt. Ltd.

                    3.   F. M. Stewart, Introduction to Linear Algebra, Dover Publications.

                    4.   S .Kumaresan, LinearAlgebra, Prentice Hall India Learning Private Limited.

                    5.   Kenneth Hoffman & Ray Kunze (2015), Linear Algebra, 2ndEdition), Prentice                                Hall India Leaning Private Limited.

                    6.   Gilbert. Strang (2015), Linear Algebra and its applications, (2ndEdition), Elsevier.

                    7.   Vivek Sahai & Vikas Bist(2013), Linear Algebra (2nd Edition) Narosa Publishing.

                    8.   Serge Lang (2005), Introduction to Linear Algebra (2ndEdition), Springer India.

                    9.   T. K. Manicavasagam Pillaiand K S Narayanan, Modern Algebra Volume2.

Practical/Lab Work to be performed in Computer Lab (FOSS)

Suggested Software’s: Maxima/ Scilab /Python/R.

Suggested Programs:

1. Program on multiple product of vectors–Scalar and Cross product.

2. Program on vector differentiation and finding unit tangent.

3. Program to find curvature and torsion of a space curve.

4. Program to find the gradient and Laplacian of a scalar function,

divergence and curl of a vector function.

5. Program to demonstrate the physical interpretation of gradient, divergence

and curl.

6. Program to evaluate vector line integral.

7. Program to evaluate a surface integral.

8. Program to evaluate a volume integral.

9. Program to verify Green’s theorem.

10. Program to find equation and plot sphere, cone and cylinder

11. Program to find distance between a straight line and a plane.

12. Program to construct and plot some standard surfaces.

We can provide BSc 6th sem Mathematics-1 full documents 

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Unit I     

              Rings and integral domains: Rings, Properties of rings, sub rings,

              ideals, principal and maximal ideals in commutative ring, quotient

              ring, homomorphism and isomorphism, and integral domains.

5.1 Introduction

In abstract algebra, ring theory is the study of rings -

algebraic structures in which addition and multiplication are

defined and similar properties to those operations defined for the

integers. Ring theory studies the structure of rings, their

representations, modules, special classes of rings (group rings,

division rings) as well as properties.

In this topic we will study about Rings, different types of

rings integral domain and fields. Before that the

properties of rings are important for studying the ring theory."

Commutative ring (abelian ring), division ring, unity of

ings (invertible of ring) and ring with zero divisors and without

mro divisors are the main concept to study in this chapter.

5.2 Definition of Ring

A ring is a non-empty set R together with two binary

operations usually called addition and multiplication such that, it

satisfy the following axioms,

i) (R, +) is an abelian

il) a(bc) = (ab)c, (Multiplication associative law)

iii)a(b+c) = ab+ac, Va, b,ce R (Left distributive law)

and (b+c)a= ba+ca, Va, b, ce R (Right distributive law)

5.3 Abelian Ring or Commutative Ring

A ring (R, +, . ) is said to be abelian ring, if it satisfies

the following axioms.

1) Closure law

2) Associative law

3) Inverse law

4) Identity law

5) Commutative law

5.4 Ring with Unity and without Unity

A ring R is said to be ring with unity (identity) if there is

an element eER such that e.a=a.e, Va E R

Then elearly e is unique this e is called the multiple

identity or just identity element in R.

Sometimes we write e=1, the additive identity of a ring

is called the zero element and it is denoted as zero.

In a ring of integers the unit element is the integral

whereas in the ring of matrices the unit element is the unit ring

of suitable order. In the ring of all even integers there is no

element and as such it is a ring without unity,

Zero Ring

Let R=(0) with 0+0=0 and 0.0=0 clearly R is aring

with multiplicative identity equal to zero

L.e .,

e=1=0

.. R is called the zero ring.

5.5 Finite and Infinite Rings

Finite Ring : A ring containing finite number of elements

then it is called finite ring.

Example : Z, is a finite ring.

Infinite Ring : A ring containing Infinite number

elements than it is called infinite ring.

In other words, A ring which is not finite is called infinite

ring.

5.6 Units in a Ring or Invertible

Let R be a ring with identity 'e' is an element then 

is said to be left invertible if there exist be R,

where "b' is called right invertible.

If a is said to be right invertible, if there exist 

that c.a -c.

where c'is also called left invertible.

If a E R is both left invertible and right invertible then we

say that "a' Is invertible or a unit or regular.

 

Theorem:If 'a' is a unit then its left and right inverse(Invertible) are same.

Proof : Let 'a' be a unit. Then,

ab= e (left invertible)

ca = e (right invertible)

 b = eb

    = (ca)b

    = c(ab)

    = ce

 b =c

Thus if n e Ris a unit then its left and right invertible of

he same element which is denoted by a-l called inverse of 'a'.

Group of units in R

Let R be a ring with identity. Let U(R) denote the set of

il units in R, then clearly U(R) is a group under multiplication

called group of units in R. Also U(R)= 0

If a € U(R), then 

1) if   a€ U(R) = aa"! = e'= a-la

                     a = (a)-1 = a"1 € U(R)

2) if   a,bE UR = (a b) (a b) -! = (a b)(b"ta -! )  

             => (ab) (ab)"! = c

              => b-l * a-1

              => ab E U(R)

5.7 Ring with Zero Divisors and Without Zero Divisors
A ring with zero divisors :

Let R be a ring, an element a = 0 in R is said to be a ring

with zero divisor, if it has either left zero divisor or right zero

divisor.

In other words product of two non-zero elements is zero

then it is said that the ring with zero divisors I.e ., c.d=0 where

'c' and 'd' being the divisors of zero. where c # 0 and d = 0

Left zero divisor : Let R be a ring, an element a = 0 in

R is said to be a left zero divisor, if there exist an element > = 0

in R such that a.b=0

5.8 Properties of a Ring

Let R be a ring, then for all a, b, c E R, then

i) a.0=0.a=0

il) a(-b) = - (ab) = (-a)b

iii)(-a) (-b) = ab

iv)a(b-c) = ab-ac

v) (b-c)a = ba-ca

Proof:-

i) We have,

a.0 = a(0+0) [- 0+0=0]

= a.0 + a.0 [By left distributive law]

.. 0+a.0 = a.0 + a.0 [. a.0ER and 0+2.0=a.0]

Now R is a group with respect to addition, therefore by

applying right cancellation law for addition in R we get 0=a.0.

Similarly we have 0.a = (0+0)a

= 0.a + 0.a [By right distribution]

.. 0+0.a= 0 a + 0 a

Applying right cancellation law for addition is 

We get, 0 = 0.a

Hence a.0=0.a=0

ii) We have,

al(-b)+b] = a.0

= a(-b) +ab = 0

[by using left di

and

= a(-b) = - (ab) 

Similarly we have (-a+a)b = 0.b

(-a)b+ab = 0

= (-a)b = - (ab)

iii) We have,

(-a)(-b) = - [(-a)b],

             = - [-(ab)]

 .'.  (-a)(-b) = ab

iv) We have,

a(b-c) = a[b+(-c)]

=. ab+a(-c)

[left dist

= ab+[-(ac)]

a(b-c) = ab-ac

v) We have,

(b-c)a = [b+(-c)]a

           = ba +(-c)al- right distributive law]

           = ba+[-(ca)]

     .'. (b-c)a = ba-ca

 

5.9 Sub-Rings

Definition : Let R be a ring. A non-empty sub-set S of the

set R is said to be a sub-ring of R if it itself is a ring under the

two induced operations.

For Example :

1.    Consider <R. +. . >.

      We know that R forms a ring under '+' and '."

         Consider    Hn = { ...- 2n, - n. O. n. 2n .-. . }

Clearly     Hn C R    

Also < H ,. +, - > is a ring.

Hence H ,, is a subring of R.

2.     The set or integers is a sub-ring of the ring of rational

        numbers.

  Remember :

i) If a ring is without zero divisors, then the sub-ring must

also be without zero divisors.

il) If a ring is commutative, then the sub-ring must also be

commutative.

(iii) If a ring is with unity, then the sub-ring may be without

unity.

5.10 Homomorphism and Isomorphism of Rings

a) Homomorphism of Two Rings

Definition : A mapping $ from the ring R into the ring

R'(@ : R-> R') is said to be homomorphism if

i) "(a+b) = q(a) +((b) ii) (ab) = ((a)(b), Va,bE R

b) Isomorphism of Two Rings

Definition : Two rings R and R' are said to be isomorphic to

each other if there exists one-one and onto mapping f : R -+ R'

such that

f(a+b) = f(a)+ f(b) and f(a.b) = f(a)- f(b),V a, bE R& we

write it as R = R' and R' is called the isomorphic image of R.

This mapping f is then said to be isomorphism of R once

R' , i.e ., a mapping which is one-one and onto and preserves the

two compositions of the ring.

 

5.11 Certain Theorems on Isomorphism of Rings

If f is an isomorphism of a ring R onto a ring .R' such the

f(a+b) = f(a) +f(b)                        .........(1)

f(a .b) = f(a). f(b)       \/  a, bE R  .........(2)

then prove the following.

5.12 Properties of Homomorphism

If @ is a homomorphism of R into R', then

i) @(0) = o'       ii) @(-a) = -@(a),   \/ a E R

5.13 Ideals and Quotient Rings

a)Two-Sided Ideal : Definition : A non-empty sub-set U of

a ring R is said to be (two-sided) ideal of R if

i) U is a sub-group of R under addition

il) V ue U and re R, both ur and ru are in U.

b) Left Ideal : A non-empty sub-set S of a ring R is said to

be left ideal of R if

i) S is a sub-group of R under addition

il) V SE S and re R. srE S

5.16 Maximal Ideal

Definition : An ideal S # R in a ring R will be said to be a maximal ideal of R if whenever T is an ideal or R such that

SCT CR then either S = T or R =T.

In simple language above amounts to saying that ideal S

will be maximal ideal if it is impossible to find another ideal

which lies between S and the whole ring R.

Hence if S is a maximal ideal then there shall not exist any

other ideal T properly contained in R and which properly contains

5.17 Integral Domain

A ring R is said to be an integral domain if it has 

(1) commutative, (ii) unit element, (iii) without zero divisors

Examples :

1) The set of integers Z is an integral domain.

2) The algebraic structure (C. +, . ). (Q. +, ),

(R. +, . ) are all integral dormins.

3) A ring ((0. 1. 2, 3, 4), ( ,. @,) is an infinite integral

domain.

👇Bsc 6th sem m-1 Unit-1 full notes👇

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Unit II

              Vector spaces: Definition, examples and properties; Subspaces -

              Examples, criterion for a sub-set to be a subspace and some properties;

              Linear Combination-Linear span, Linear dependence and Linear

              independence, basic properties of linear dependence and

              independence, techniques of determining linear dependence and

              independence in various vector spaces and related problems; Basis and

              dimension - Co-ordinates, ordered basis, some basic properties of

              basis and dimension and subspace Spanned by given set of vectors;

              Quotient space. Dimension of quotient space (derivation in finite

              case); Sum and Direct sum of subspaces - Dimensions of sum and

              direct sum spaces (Derivation in finite case).

vector spaces :-  In physics divector is define as a quantity having both magnitude & direction A vector is a collection of objects for which Such addition & multiplication is defined.

Internal composition:

Ict v be a non Empty set A mapping form VXV into V is an internal composition

External composition: 

· Iet F&V be two non Empty sets a mapping from Fxv into V is called an External

compositionin vover F

* Vector Sub space :

let V be the vector space over field F and w be a non empty subset of v. if wis the is vector space over F under the

operation of v. then w is called subspace

Ex :- V = R3 (R) is a vector Space over R

12t W= 2 (x1 , x2 0), x= 12 ERS

then

w is a subspace of v)

EX :- V=R(X) is a vector spaces of all polynomials over R.

if the w is set of all polynomials over

of degree at most N then w is a subspace

of R(2)

smallest subspace:-

is let, V(F) be a vector space tot &s be a

any Subset of V(F) if wis a subspace of

V containg is and s itself is contained

in Every Subspace of V. containg 8. then

w is called the smallut subspace of V

containg S. w is smallest Subspace of von

genrated by s

Linear span of set :-

let V(F) be a vector space and & be a non Empty subset of V1, then the set of all linear combinations, 2 of finite number of Element is called linear span of (s) set

👇Bsc 6th sem m-1 Unit-2 full notes👇

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