**IIT Kanpur Mathematics-II Syllabus**

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**MTH 102 MATHEMATICS – II Prereq. : MTH 101**

Matrices; Matrix Operations (Addition, Scalar Multiplication, Multiplication, Transpose, Adjoint) and their properties; Special types of matrices (Null, Identity, Diagonal, Triangular, Symmetric, Skew-Symmetric, Hermitian, Skew-Hermitian, Orthogonal, Unitary, Normal), Solution of the matrix EquationAx=b; Row-reduced Echelon Form; Determinants and their properties, Vector Space R^{n} (R); Subspaces; Linear Dependence / Independence; Basis; Standard Basis of R^{n} ; Dimension; Co-ordinates with respect to a basis; Complementary Subspaces; Standard Inner product; Norm; Gram-Schmidt Orthogonalisation Process; Generalisation to the vector space C^{n} (C), Linear Transformation from R^{n} to Rm (motivation, X – AX); Image of a basis identifies the linear transformation; Range Space and Rank; Null Space and Nullity; Matrix Representation of a linear transformation; Structure of the solutions of the matrix equation Ax = b, Linear Operators on R^{n} and their representation as square matrices; Similar Matrices and linear operators; Invertible linear operators; Inverse of a non-singular matrix; Cramer’s method to solve the matrix equation Ax=b, Eigenvalues and eigenvectors of a linear operator; Characteristic Equation; Bounds on eigenvalues; Diagonalisability of a linear operator; Properties of eigenvalues and eigenvectors of Hermitian, skewHermitian, Unitary, and Normal matrices (including Symmetric, Skew-Symmetric, and Orthogonal matrices), Implication of diagonalisability of the matrix A + A^{ r} in the real Quadratic form X^{ T} AX; Positive Definite and Semi-Positive Definite Matrices, Complex Numbers, geometric representation, powers and roots of complex numbers, Functions of a complex variable, Analytic functions, CauchyRiemann equations; elementary functions, Conformal mapping (for linear transformation) Contours and contour integration, Cauchy’s theorem, Cauchy integral formula, Power Series, term by term differentiation, Taylor series, Laurent series, Zeros, singularities, poles, essential singularities, Residue theorem, Evaluation of real integrals and improper integrals.