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Math Ed 526�LINEAR ALGEBRA�Presented �By�Abatar Subedi� July-08, 2015 �

Prerequisites

• Abelian group
• Ring and field
• Linear transformation
• Vector Spaces with examples
• Basis and dimension of the vector space
• Scalar product and hermitian product of vectors
• System of linear equations
• Matrix and determinant

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Linear Algebra

• The branch of mathematics that deals with the theory of systems of linear equations, matrices, vector spaces, determinants, and linear transformation. Or the part of algebra that deals with the theory of linear equations and the linear transformations. Branch of algebra concerned with method of solving system of linear equations, more generally, the mathematics of linear transformation and vector spaces.

Some remarks on the history of linear algebra�

Determinants

• The origins of the concepts of a determinant and a matrix, as well as an understanding of their basic properties, are historically closely connected. Both concepts came from the study of systems of linear equations.
• Leibniz (1646-1716) had considered patterns of coefficients in such systems and represented them with pairs of numbers.

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• Around 1729 Maclaurin presented the solution of simultaneous linear equations in two, three, and four unknowns. The rule he gave was the one given by Cramer (1750), who studied the Coefficients of the general conic A + By + Cx + Dy² + Exy + x² = 0 pasisng through five given points. Cramer gave the solution in terms of the ratios of determinants, precisely what is known today as Cramer's rule.

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• Vandermonde (1772) was the first to give an exposition of the theory of determinants (i.e. apart from the solution of linear equations although such applications were also made by him). He gave a rule for expanding a determinant by using second order minors and their complementary minors. Also in 1772, Laplace expanded Vandermonde's rule by using a set of minors of r rows and the complementary minors.

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• Later, the idea of the determinant found applications not only in systems of linear equations, but also in the simultaneous solution of equations of higher degree (known as elimination theory), in the transformation of coordinates, in the change of variables in multiple integrals, in the solution of systems of differential equations arising in planetary motion, and in the reduction of quadratic forms to standard forms in 19^th century.

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• The word determinant was already used by Gauss (1777-1855) for the discriminant of the quadratic form ax² + 2bxy + cy² in his number theoretic investigations. The word was later applied by Cauchy (1789-1857) to the determinants that had already appeared in the 18th century work. In his 1815 paper he introduced the idea of arranging the elements in a square array, and used the double subscript notation. A third order determinant would then appear as

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a_ll a₁₂ a₁₃

a₂₁ a₂₂ a₂₃

a₃₁ a₃₂ a₃₃

• The two vertical lines as we know it today were introduced by Cayley in 1841. In the 1815 paper Cauchy gave the first systematic and almost modern treatment of determinants. One of the major results was the multiplication theorem for determinants. This had been obtained by Lagrange in 1773 for third order determinants.

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• The mathematician James Joseph Sylvester (1814-1897) worked in the theory of determinants over a period of 50 years. Jacobi applied the method of determinants to the study of the change of variables in multiple integrals. He also studied determinants whose entries are functions (1841).

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• There is intimate connection between quadratic form and determinants, was justified by Sylvester (1852) for his law of inertia of quadratic form. The law was later rediscovered and proved by Jacobi in 1857. The further study of the reduction of quadratic forms involves the notion of the characteristic equation of a quadratic form or of a matrix. A quadratic form in three variables is written as Ax² + By² + Cz² + 2Dxy + 2Exz + 2Fyz. The coefficient of this equation gives the matrix H associated with it which is symmetric. The characteristic equation of the form or of the matrix is then det(H - ti) = 0 which gives the characteristics roots. The notion of the characteristics equation appears in the work of Euler (1748) when he dealt with the above problem. The term characteristic equation is due to Cauchy (1840).

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• The study of the reduction of quadratic forms and the theory of bilinear forms was later done by Weierstrass (1858, 1868). He gave a general method of reducing two quadratic forms simultaneously to sums of squares.

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Matrices�

• The word was first used by Sylvester (1850) when he wished to refer to a rectangular array of numbers. Cayley (1855) insisted that logically the idea of a matrix preceded that of the determinant but it was reversed order historically. Together with Sylvester, cayley was the creator of the theory of matrices and founder of the theory of invariants.

Vectors�

• By the year 1800 mathematicians were using freely the various types of real numbers and even complex numbers, but the precise definitions of these various types of real numbers and even complex numbers were not available nor was there any logical justification for the mathematical operations (such as addition and multiplication) used on them. By the middle of the 19th century the mathematical community generally accepted the following axioms:

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• 1. Equal quantities added to the third yield equal quantities.
• 2. a+ (b + c) = (a + b) + c
• 3. a + b = b + a
• 4. Equals added to equals give equals
• 5. a(bc) = (ab)c
• 6. ab = ba
• 7. a(b +c) = ab + ac .

These axioms constituted the Principle of the Permanence of Form.

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• The notion of a vector was already used by Aristotle to represent forces. He was already aware of the parallelogram law. Gauss and others had also introduced the geometric representation of complex numbers.
• In 1837 Hamilton suggested that complex numbers be expressed as ordered pairs of real numbers satisfying the conditions that if a + bi and c + di are represented as (a, b) and (c, d), then
• (a, b) + (c, d) = (a + c, b + d) (similarly for -)
• (a, b), . (c, d) (ac - bd, ad+ bc)
• (a, b) / (c, d) = ((ac + bd)/(c² + d² ),(bc - ad)/(c² +d²)).

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• The use of complex numbers to represent vectors in the plane became well known around the 1830's. From Pythagoras to the mid19^th century, the fundamental problem of geometry was to relate numbers to geometry. It played a key role in the creation of field theory (via the classical
• Construction problems), and quite differently, in the creation of linear algebra.

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• Course Title: Linear Algebra
• Nature of course: Theoretical
• Course No : Math Ed 527
• Credit hours: 3
• Level: M. Ed
• Teaching hours: 48
• Semester: Second

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Course Description�

This is a specialization course designed for the students majoring Mathematics Education at master level. This is one semester course deals with Linear algebra. It covers Vector spaces, Inner product Spaces, linear mapping & their algebraic properties, bilinear form & Standard operators, Spectral Theorem & primary decomposition theorem with Jordan Canonical Form and Module Theory.

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General Objectives�

• To provide students the deeper understanding of theoretical concepts of linear algebra including module theory.
• To facilitate students to develop computing power in linear algebra.
• To help students develop positive attitude towards linear algebra.
• To make students understands and explain the concepts of modules in any ring and distinguish it with vector space.

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Unit I : Linear Maps and Matrices 4 hrs

Specific Objectives

• Review the concepts of vector space, subspace, bases and dimensions of vector space and illustrate them with examples.
• Find the linear map associated with the matrix.
• Find the matrix associated with linear maps.
• Explain the relation of bases matrices and linear map in the vector space.

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Contents

• The linear map associated with a matrix.
• The matrix associated with linear maps.
• Bases, matrices, and linear maps.

Unit II: Bilinear Forms and Standard Operators 10 hrs

Specific Objectives

• Review the concepts of Scalar product, Hermitian Product ,Bilinear maps, Linear functional and Dual space
• Define bilinear forms and standards operators.
• Prove the properties of bilinear forms and standard operators.
• State and prove Sylvester’s Theorem and find the index of positivity and nullity.

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Contents

• Bilinear forms.
• Quadratic forms.
• Symmetric Operators
• Hermitian Operators
• Unitary Operators
• Sylvesters’ Theorem

Unit III: Algebraic Properties of Linear Transformation 10 hrs�

Specific Objectives

• Define Eigenvalues and eigenvectors with examples
• Prove the properties of Eigenvectors and Eigenvalues.
• Define characteristics polynomials of matrices and find characteristics polynomials of the matrices.
• Determine Eigenvalues and Eigenvectors of the matrices and linear maps.
• Determine the triangulizable and diagonalizable matrices.
• State and prove Hamilton Cayley Theorem.

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Contents

• Eigen values and eigenvectors
• Characteristics polynomial.
• Triangulation of matrices and linear maps.
• Diagonalization of unitary

matrices.

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Unit IV: Spectral Theorem and Primary Decomposition Theorem 10 hrs�

Specific Objectives

• Prove the properties of symmetric linear maps.
• State and prove the Spectral Theorem.
• Apply the standard properties of polynomials to decompose the vector spaces.
• Define S-invariant subspace and simple S-space.
• State and prove Schur’s lemma.
• Define Jordan canonical form and reduce the matrices in Jordan canonical forms.

Contents

• Eigenvectors of symmetric

linear maps.

• The Spectral Theorem.
• The unitary operator.
• Application of polynomial to decomposition of vector spaces.
• Schur’s Lemma and Jordan normal form.

Unit V: Module Theory 14 hrs

Specific Objectives

• Define modules, sub modules, Quotient modules and module homomorphism and illustrate them with examples.
• Prove the elementary properties of modules and sub modules.
• State and prove fundamental theorem of module homomorphism.
• Define direct sum of modules and prove its basic properties.
• Define torsion and torsion free modules and illustrate them with examples.
• Define exact sequence and establish the fundamental properties of module homomorphism.
• Explain free modules and prove the elementary properties of free module.
• Define projective and injective modules and prove elementary properties of them

Contents

• Modules and sub modules
• Module homomorphism
• Quotient module
• Direct sum of modules
• Torsion modules
• Exact sequences
• Free modules
• Projective and Injective modules
• Homomorphism and Duality.

Instructional Techniques

This course is theoretical in nature and thus the teacher-centered instructional techniques will be dominant in teaching learning process. However, the instructional technique for this course is divided into two groups. The first group consists of general instructional techniques applicable to most of the contents. The second group consists of the specific instructional techniques applicable to specific contents of each chapter.

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General Techniques

Following instructional techniques will be adopted according to the need and nature of the lesson.

• Lecture with illustration
• Discussion
• Question-answer
• Group work presentation and participation

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Specific Instructional Techniques

For Unit I

• Group discussion for the matrix and linear maps.
• Individual work and group work Presentation.
• Individual assignment on solving problem of exercise.

For Unit II

• Individual work and group work Presentation to explore bilinear form and its associated matrix.
• Group work assignment on solving some problem of exercise and then group presentation.

For unit III

• Individual work and group work to explore polynomial of matrix and linear maps.
• Individual assignment to find the solution of numerical problem related to theorems of this unit and presentation.
• Group tasks to solve the problem of exercise and discussion in small groups.

For Unit IV

• Individual work and group work Presentation
• Problem solving exercise and group presentation.

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For Unit V

• Individual work and group work Presentation
• Solving problem of exercise
• Connecting examples with theorems and facilitate to find related examples.

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Evaluation�

Internal Evaluation 40 %

Internal evaluation will be conducted by course teacher based on following activities

Attendance 5 Point

Participation in learning activities 5 Points

First assignment/ midterm exam 10 Points

Second assignment/ assessment 10 points

Third assignment/assessment 10 Points

Total 40 Points

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External Examination (Final examination) 60%�

Examination Division of the Dean office, Faculty of Education will conduct final examination at the end of the semester

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• Objective questions (multiple choice 10 × 1) 10 points
• Short answer question (6Questions × 5 ) 30 points
• Long answer questions (2Questions × 10 ) 20 points

Total 60 points

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Recommended Books�

• Bhattacharya, P.B, Jain, S.K and Nagpaul, S.R (2008). First Course in Linear Algebra. india: Lew Age International House. (For Chapter III and IV).
• Lang, s. (1973). Linear Algebra.New York: Addision Wesley. (For Chapter I to IV )
• Bhattacharya, P.B, Jain, S.K and Nagpaul, S.R (2007). Basic Abstract Algebra,(Printed in india): Cambridge University Press.( For Chapter V).
• Hungerford, T.W (1974). Algebra. New YorK: New York Inc.Springer Verlag. (For Chapter V).

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Reference Books�

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• Maharjan, H.B. (2008). Rings and Modules. Kathmandu: Bhunipuran Prakasan.
• Bhattarai,B.N. (2011).Introduction of Rings and Modules. Kathmandu: Subhakamana Prakashan.
• Kunze, H.E. (1996). Linear Algebra. D.T. (1986). Introduction to Matrices and Linear Transformations. Delhi: CBS Publishers and Distributers.
• Hohn, F.E. (1971). Elementary Matrix Algebra. Delhi: Amerind Publishing Co.Pvt.Ltd
• Subedi, A. (2014). Linear Algebra. Kathmandu: Sunlight Publication.

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