Notes on Mathematics - 1021 Peeyush Chandra, A. K. Lal, V. Raghavendra, G. Santhanam 1Supported by a grant from MHRD 2 Contents I Linear Algebra 7 1 Matrices 9 1.1 DefinitionofaMatrix .................................. .... 9 1.1.1 SpecialMatrices ................................... .. 10 1.2 OperationsonMatrices ............................... ...... 10 1.2.1 Multiplication of Matrices . .. 12 1.3 SomeMoreSpecialMatrices.. .. .. .. .. .. .. .. .. .. .. .. ....... 13 1.3.1 SubmatrixofaMatrix................................ .. 14 1.3.1 BlockMatrices ..................................... 15 1.4 MatricesoverComplexNumbers . ....... 17 2 Linear System of Equations 19 2.1 Introduction...................................... ...... 19 2.2 Definition and a Solution Method . ...... 20 2.2.1 ASolutionMethod................................... 21 2.3 RowOperationsandEquivalentSystems. ......... 21 2.3.1 Gauss Elimination Method . 24 2.4 RowReducedEchelonFormofaMatrix . ....... 26 2.4.1 Gauss-Jordan Elimination . ... 27 2.4.2 ElementaryMatrices................................ ... 29 2.5 RankofaMatrix..................................... .... 30 2.6 Existence of Solution of Ax = b ................................. 33 2.6.1 Example.......................................... 33 2.6.2 MainTheorem ...................................... 34 2.6.3 Exercises ........................................ 35 2.7 InvertibleMatrices .................................. ...... 35 2.7.1 InverseofaMatrix................................. ... 35 2.7.2 Equivalent conditions for Invertibility . ..... 37 2.7.3 InverseandGauss-JordanMethod . ....... 39 2.8 Determinant....................................... ..... 40 2.8.1 AdjointofaMatrix .................................. 43 2.8.2 Cramer’sRule ...................................... 45 2.9 Miscellaneous Exercises . ...... 46 3 Finite Dimensional Vector Spaces 49 3.1 VectorSpaces ..................................... ..... 49 3.1.1 Definition......................................... 49 3.1.2 Examples ......................................... 51 3 4 CONTENTS 3.1.3 Subspaces....................................... .. 53 3.1.4 Linear Combinations . .. 54 3.2 LinearIndependence ................................ ....... 57 3.3 Bases ............................................ ... 58 3.3.1 ImportantResults ................................. ... 60 3.4 OrderedBases ..................................... ..... 66 4 Linear Transformations 69 4.1 Definitions and Basic Properties . ....... 69 4.2 Matrix of a linear transformation . ........ 72 4.3 Rank-NullityTheorem................................. ..... 75 4.4 Similarity of Matrices . .... 80 5 Inner Product Spaces 87 5.1 Definition and Basic Properties . ....... 87 5.2 Gram-Schmidt Orthogonalisation Process . .......... 92 5.3 Orthogonal Projections and Applications . ...........100 5.3.1 MatrixoftheOrthogonalProjection . .......103 6 Eigenvalues, Eigenvectors and Diagonalization 107 6.1 IntroductionandDefinitions. ........107 6.2 diagonalization . .113 6.3 Diagonalizable matrices . .116 6.4 Sylvester’s Law of Inertia and Applications . .........121 II Ordinary Differential Equation 129 7 Differential Equations 131 7.1 Introduction and Preliminaries . .......131 7.2 SeparableEquations ................................. .. ..134 7.2.1 Equations Reducible to Separable Form . .134 7.3 ExactEquations .................................... .136 7.3.1 IntegratingFactors............................... .138 7.4 LinearEquations .................................... .141 7.5 MiscellaneousRemarks................................ .. ..143 7.6 Initial Value Problems . .145 7.6.1 OrthogonalTrajectories . .. ..149 7.7 NumericalMethods.................................. .. ..150 8 Second Order and Higher Order Equations 153 8.1 Introduction...................................... .. ..153 8.2 MoreonSecondOrderEquations . ........156 8.2.1 Wronskian........................................ 156 8.2.2 MethodofReductionofOrder . .159 8.3 Second Order equations with Constant Coefficients . ............160 8.4 NonHomogeneousEquations . .......162 8.5 VariationofParameters .............................. .......164 8.6 Higher Order Equations with Constant Coefficients . ...........166 CONTENTS 5 8.7 MethodofUndeterminedCoefficients. .........170 9 Solutions Based on Power Series 175 9.1 Introduction...................................... .. ..175 9.1.1 PropertiesofPowerSeries. .177 9.2 Solutions in terms of Power Series . .......179 9.3 Statement of Frobenius Theorem for Regular (Ordinary) Point . ..............180 9.4 Legendre Equations and Legendre Polynomials . ..........181 9.4.1 Introduction ..................................... 181 9.4.2 Legendre Polynomials . 182 III Laplace Transform 189 10 Laplace Transform 191 10.1Introduction..................................... .......191 10.2DefinitionsandExamples .............................. .. ..191 10.2.1 Examples ........................................ 192 10.3 PropertiesofLaplaceTransform . ..........194 10.3.1 InverseTransformsof RationalFunctions . ..........199 10.3.2 TransformofUnitStepFunction . .. ..199 10.4SomeUsefulResults ................................ .......200 10.4.1 Limiting Theorems . 200 10.5 Application to Differential Equations . .........202 10.6 TransformoftheUnit-ImpulseFunction . ...........204 IV Numerical Applications 207 11 Newton’s Interpolation Formulae 209 11.1Introduction..................................... .......209 11.2DifferenceOperator ................................ .......209 11.2.1 ForwardDifferenceOperator . .. ..209 11.2.2 BackwardDifferenceOperator . .. ..211 11.2.3 CentralDifferenceOperator. .......213 11.2.4 ShiftOperator ................................... .214 11.2.5 AveragingOperator ............................... .214 11.3 RelationsbetweenDifferenceoperators . ............214 11.4 Newton’s Interpolation Formulae . .........215 12 Lagrange’s Interpolation Formula 221 12.1Introduction..................................... .......221 12.2DividedDifferences.................................. .. ..221 12.3 Lagrange’s Interpolation formula . ..........224 12.4 Gauss’s and Stirling’s Formulas . .......226 13 Numerical Differentiation and Integration 229 13.1Introduction..................................... .......229 13.2 Numerical Differentiation . .......229 13.3NumericalIntegration . .. .. .. .. .. .. .. .. .. .. .. .. ........233 6 CONTENTS 13.3.1 AGeneralQuadratureFormula. .. ..233 13.3.2 TrapezoidalRule .................................. .234 13.3.3 Simpson’s Rule . 235 14 Appendix 239 14.1 SystemofLinearEquations . ........239 14.2Determinant...................................... .. ..242 14.3 PropertiesofDeterminant . .........246 14.4 Dimension of M + N .......................................250 14.5 ProofofRank-NullityTheorem . ........251 14.6 ConditionforExactness .. .. .. .. .. .. .. .. .. .. .. .. ........252 Part I Linear Algebra 7 Chapter 1 Matrices 1.1 Definition of a Matrix Definition 1.1.1 (Matrix) A rectangular array of numbers is called a matrix. We shall mostly be concerned with matrices having real numbers as entries. The horizontal arrays of a matrix are called its rows and the vertical arrays are called its columns. A matrix having m rows and n columns is said to have the order m n. × A matrix A of order m n can be represented in the following form: × a a a 11 12 ··· 1n a21 a22 a2n A = ··· , . .. . a a a m1 m2 ··· mn th th where aij is the entry at the intersection of the i row and j column. In a more concise manner, we also denote the matrix A by [aij ] by suppressing its order. a a a 11 12 ··· 1n a21 a22 a2n Remark 1.1.2 Some books also use ··· to represent a matrix. .. . a a a m1 m2 ··· mn 1 3 7 Let A = . Then a11 =1, a12 =3, a13 =7, a21 =4, a22 =5, and a23 =6. "4 5 6# A matrix having only one column is called a column vector; and a matrix with only one row is called a row vector. Whenever a vector is used, it should be understood from the context whether it is a row vector or a column vector. Definition 1.1.3 (Equality of two Matrices) Two matrices A = [aij ] and B = [bij ] having the same order m n are equal if a = b for each i =1, 2,...,m and j =1, 2, . , n. × ij ij In other words, two matrices are said to be equal if they have the same order and their corresponding entries are equal. 9 10 CHAPTER 1. MATRICES Example 1.1.4 The linear system of equations 2x +3y = 5 and 3x +2y = 5 can be identified with the 2 3: 5 matrix . "3 2: 5# 1.1.1 Special Matrices Definition 1.1.5 1. A matrix in which each entry is zero is called a zero-matrix, denoted by 0. For example, 0 0 0 0 0 02 2 = and 02 3 = . × "0 0# × "0 0 0# 2. A matrix having the number of rows equal to the number of columns is called a square matrix. Thus, its order is m m (for some m) and is represented by m only. × 3. In a square matrix, A = [aij ], of order n, the entries a11,a22,...,ann are called the diagonal entries and form the principal diagonal of A. 4. A square matrix A = [a ] is said to be a diagonal matrix if a = 0 for i = j. In other words, the ij ij 6 4 0 non-zero entries appear only on the principal diagonal. For example, the zero matrix 0n and "0 1# are a few diagonal matrices. A diagonal matrix D of order n with the diagonal entries d1, d2,...,dn is denoted by D = diag(d1,...,dn). If di = d for all i =1, 2,...,n then the diagonal matrix D is called a scalar matrix. 1 if i = j 5. A square matrix A = [aij ] with aij = 0 if i = j ( 6 is called the identity matrix, denoted by In. 1 0 0 1 0 For example, I2 = , and I3 = 0 1 0 . 0