Matrices: matrix operations (Addition, Scalar Multiplication, Matrix 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 Equation Ax = b; Row-reduced Echelon form, Determinants and their properties, Vector Space Rⁿ(R); Subspaces; Linear Dependence/Independence; Basis; Standard Basis of Rⁿ; Dimension; Coordinates with respect to a basis; Complementary Subspaces; Standard Inner product; Norm; Gram- Schmidt Orthogonalization Process; Generalization to the Vector Space Cⁿ (C), Linear Transformation from Rⁿ to R^m (motivation); 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 Rⁿ 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; Diagonalizability of a linear operator; Properties of eigenvalues and eigenvectors of Hermitian, skew-Hermitian, Unitary, and Normal matrices (including Symmetric, Skew-Symmetric, and Orthogonal matrices), Implication of diagonalaizability of the matrix A + A^T in the real Quadratic form X^TAX; Positive Define and Semi-Positive Definite matrices,

Complex Numbers, Geometric Representation, Powers and Roots of Complex Numbers, Functions of a complex variable, Analytic functions, Cauchy-Riemann 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.