NotesonMathematics-1021 - IITK

1 Notes on Mathematics - 1021 Peeyush Chandra, A. K. Lal, V. Raghavendra, G. Santhanam1 Supported by a grant from MHRD2 ContentsI Linear Algebra71 Definition of a Matrix .. Special Matrices .. Operations on Matrices .. Multiplication of Matrices .. Some More Special Matrices .. Submatrix of a Matrix .. Block Matrices .. Matrices over Complex Numbers .. 172 Linear System of Introduction .. Definition and a Solution Method .. A Solution Method .. Row Operations and Equivalent Systems .. Gauss Elimination Method .. Row Reduced Echelon Form of a Matrix .. Gauss-Jordan Elimination .. Elementary Matrices.

2 Rank of a Matrix .. Existence of Solution ofAx=b.. Example .. Main Theorem .. Exercises .. Invertible Matrices .. Inverse of a Matrix .. Equivalent conditions for Invertibility .. Inverse and Gauss-Jordan Method .. Determinant .. Adjoint of a Matrix .. Cramer s Rule .. Miscellaneous Exercises .. 463 Finite Dimensional Vector Vector Spaces .. Definition .. Examples .. Subspaces .. Linear Combinations .. Linear Independence .. Bases .. Important Results .. Ordered Bases .. 664 Linear Definitions and Basic Properties .. Matrix of a linear transformation .. Rank-Nullity Theorem.

3 Similarity of Matrices .. 805 Inner Product Definition and Basic Properties .. Gram-Schmidt Orthogonalisation Process .. Orthogonal Projections and Applications .. Matrix of the Orthogonal Projection .. 1036 Eigenvalues, Eigenvectors and Introduction and Definitions .. diagonalization .. Diagonalizable matrices .. Sylvester s Law of Inertia and Applications .. 121II Ordinary Differential Equation1297 Differential Introduction and Preliminaries .. Separable Equations .. Equations Reducible to Separable Form .. Exact Equations .. Integrating Factors .. Linear Equations .. Miscellaneous Remarks .. Initial Value Problems.

4 Orthogonal Trajectories .. Numerical Methods .. 1508 Second Order and Higher Order Introduction .. More on Second Order Equations .. Wronskian .. Method of Reduction of Order .. Second Order equations with Constant Coefficients .. Non Homogeneous Equations .. Variation of Parameters .. Higher Order Equations with Constant Coefficients .. Method of Undetermined Coefficients .. 1709 Solutions Based on Power Introduction .. Properties of Power Series .. Solutions in terms of Power Series .. Statement of Frobenius Theorem for Regular (Ordinary) Point .. Legendre Equations and Legendre Polynomials .. Introduction.

5 Legendre Polynomials .. 182 III Laplace Transform18910 Laplace Introduction .. Definitions and Examples .. Examples .. Properties of Laplace Transform .. Inverse Transforms of Rational Functions .. Transform of Unit Step Function .. Some Useful Results .. Limiting Theorems .. Application to Differential Equations .. Transform of the Unit-Impulse Function .. 204IV Numerical Applications20711 Newton s Interpolation Introduction .. Difference Operator .. Forward Difference Operator .. Backward Difference Operator .. Central Difference Operator .. Shift Operator .. Averaging Operator .. Relations between Difference operators.

6 Newton s Interpolation Formulae .. 21512 Lagrange s Interpolation Introduction .. Divided Differences .. Lagrange s Interpolation formula .. Gauss s and Stirling s Formulas .. 22613 Numerical Differentiation and Introduction .. Numerical Differentiation .. Numerical Integration .. A General Quadrature Formula .. Trapezoidal Rule .. Simpson s Rule .. 23514 System of Linear Equations .. Determinant .. Properties of Determinant .. Dimension ofM+N.. Proof of Rank-Nullity Theorem .. Condition for Exactness .. 252 Part ILinear Algebra7 Chapter Definition of a MatrixDefinition (Matrix)A rectangular array of numbers is called a shall mostly be concerned with matrices having real numbers as horizontal arrays of a matrix are called itsrowsand the vertical arrays are called matrix havingmrows andncolumns is said to have the orderm matrixAoforderm ncan be represented in the following form:A= a11a12 a1na21a22 amn ,whereaijis the entry at the intersection of theithrow a more concise manner, we also denote the matrixAby [aij] by suppressing its books also use a11a12 a1na21a22 amn to represent a "1 3 74 5 6#.

7 Thena11= 1, a12= 3, a13= 7, a21= 4, a22= 5,anda23= matrix having only one column is called acolumn vector; and a matrix with only one row iscalled arow a vector is used, it should be understood from the context whether it isa row vector or a column (Equality of two Matrices)Two matricesA= [aij]andB= [bij]having the same orderm nare equal ifaij=bijfor eachi= 1,2,..,mandj= 1,2,.., other words, two matrices are said to be equal if they have the same order and their correspondingentries are 1. MATRICESE xample linear system of equations2x+ 3y= 5and3x+ 2y= 5can be identified with thematrix"2 3 : 53 2 : 5#. Special MatricesDefinition A matrix in which each entry is zero is called a zero-matrix, denoted ,02 2="0 00 0#and02 3="0 0 00 0 0#.

8 2. A matrix having the number of rows equal to the number of columns is called a square matrix. Thus,its order ism m(for somem) and is represented In a square matrix,A= [aij],of ordern, the entriesa11,a22,..,annare called the diagonal entriesand form the principal diagonal A square matrixA= [aij]is said to be a diagonal matrix ifaij= 0fori6= other words, thenon-zero entries appear only on the principal diagonal. Forexample, the zero matrix0nand"4 00 1#are a few diagonal diagonal matrixDof ordernwith the diagonal entriesd1,d2,..,dnis denoted byD=diag(d1,..,dn).Ifdi=dfor alli= 1,2,..,nthen the diagonal matrixDis called ascalar A square matrixA= [aij]withaij=(1ifi=j0ifi6=jis called the identity matrix, denoted example,I2="1 00 1#,andI3= 1 0 00 1 00 0 1.)

9 The subscriptnis suppressed in case the order is clear from the context or ifno confusion A square matrixA= [aij]is said to be an upper triangular matrix ifaij= 0fori > square matrixA= [aij]is said to be an lower triangular matrix ifaij= 0fori < square matrixAis said to be triangular if it is an upper or a lower example 2 1 40 3 10 0 2 is an upper triangular matrix. An upper triangular matrix will be representedby a11a12 a1n0a22 0 ann . Operations on MatricesDefinition (Transpose of a Matrix)The transpose of anm nmatrixA= [aij]is defined as then mmatrixB= [bij],withbij=ajifor1 i mand1 j transpose ofAis denoted OPERATIONS ON MATRICES11 That is, by the transpose of anm nmatrixA,we mean a matrix of ordern mhaving the rowsofAas its columns and the columns ofAas its example, ifA="1 4 50 1 2#thenAt= 1 04 15 2.

10 Thus, the transpose of a row vector is a column vector and any matrixA,we have(At)t= [aij], At= [bij] and (At)t= [cij].Then, the definition of transpose givescij=bji=aijfor alli,jand the result follows. Definition (Addition of Matrices)letA= [aij]andB= [bij]be are twom nmatrices. Then thesumA+Bis defined to be the matrixC= [cij]withcij=aij+ that, we define the sum of two matrices only when the order ofthe two matrices are (Multiplying a Scalar to a Matrix)LetA= [aij]be anm nmatrix. Then for anyelementk R,we definekA= [kaij].For example, ifA="1 4 50 1 2#andk= 5,then 5A="5 20 250 5 10#.Theorem ,BandCbe matrices of orderm n,and letk, +B=B+A(commutativity).