Announcements

Lecture No	Date	Content	Corresponding Reading material
1	Sep. 27	Basic structure of mathematics and natural sciences, abstract versus applied mathematics, mathematical theories, basics of logic, sets and their elementary properties, relations	Pages 3-12 of "A First Course in Abstract Math," Pages 1-7 of "A First Course in Linear Algebra"
2	Sep. 29	Equivalence relations and classification schemes. Functions: Image and inverse image of subsets under a function, domain and range, one-to-one and onto functions, bijections, composition of functions, invertible functions, equivalent sets	Pages 1-11 of "A First Course in Linear Algebra"
3	Oct. 04	Plane vectors, componentwise addition and multiplication, complex multiplication and complex number system, algebraic properties of complex numbers; modulus, argument, principal argument, and complex-conjugate of a complex number, polynomial and rational functions of a complex variable; real exponential function	Pages 13-17 of "A First Course in Linear Algebra" & pages 602-608 of "Kreyszig"
4	Oct. 06	Complex sequences and series and their convergence, The exponential of a complex number, Euler's formula and polar representation of complex numbers; Linear Algebra, plane vectors and their algebraic properties, generalization to R^n, Real and complex abstract vector spaces, notation and some basic properties, examples: Function spaces, space of sequences.	Pages 18-21 of "A First Course in Linear Algebra"
5	Oct. 11	Vector space of polynomials, Subspaces of a vector space, linear combination, span of a subet, characterization theorem for the span, Linear independence and the characterization theorem for linear independence	Pages 21-25 of "A First Course in Linear Algebra"
6	Oct. 13	Examples of linearly independent and dependent subsets, Unique Expansion Theorem, Basis and the dimension of a vector space, properties and examples	Pages 25-31 of "A First Course in Linear Algebra"
Quiz 1		Covers the material of the Homework Assignments #1 and #2.
7	Oct. 20	Linear Operators and their examples, Thm: The image and inverse image of subspaces under a linear operator are subspaces, Null space, the dimension theorem.	Pages 37-42 of "A First Course in Linear Algebra"
8	Oct. 22	The most general linear equations and the existence and uniqueness of their solutions. The matrix representation of linear operators acting in finite-dimensional vector spaces.	Pages 42-47 of "A First Course in Linear Algebra"
9	Oct. 25	Algebra of linear operators, application to linear second order ordinary differential equations (method of factorization for homogeneous equations with constant coefficients.)	Pages 47-51 of "A First Course in Linear Algebra"
Quiz 2	Oct. 27	Covers the material of the Homework Assignments #3 and #4.
10	Nov. 01	Invertible operators, linearity of the inverse of a linear operator, ontoness of invertible operators acting in a finite-dimensional vector space, isomorphisms and their role in classifying vector spaces, various properties of the isomorphisms; the dual space of a vector space and the dual basis	Pages 51-57 of "A First Course in Linear Algebra"
11	Nov. 03	Addition and scalar multiplications of matrices, isomorphism between the vector space of linear operators defined between finite-dimensional vector spaces and the space of matrices, matrix multiplication, invertible matrices, determinant of a square matrix.	Pages 59-67 of "A First Course in Linear Algebra"
Midterm Exam 1	Nov. 08	Covers the material of of Lectures 1-11.
12	Nov. 10	Properties of the determinant. A simple method of calculating the determinant of 3x3 matrices, calculating the inverse of an invertible matrix; Basis transformations: Transformation rule for the representation of the vectors and linear operators (similarity transformations)	Pages 67-72 of "A First Course in Linear Algebra"
Kurban Bayramı	Nov. 15-19
13	Nov. 22	Systems of linear algebraic equations, existence and uniqueness of solutions, Cramer's rule; Eigenvalue problem	Pages 72-76 of "A First Course in Linear Algebra"
14	Nov. 24	Eigenvalues and eigenvectors of a linear operator and a square matrix, the case of 2x2 matrices, an example with a single linearly-independent eigenvector, linear-independence of eigenvectors with different eigenvalues, diagonalizable operators and matrices, diagonalization procedure, example.	Pages 334-339 of Kreyszig 9th Ed.
15	Nov. 29	Inner product spaces: inner product, its associated norm and metric, examples: Orthonormal subsets and bases, orthogonal projection operators associated with the elements of an orthonormal basis.	-
16	Dec. 01	Gram-Schmidt orthonormalization sheme; matrix representation of linear operators in an orthonormal basis; self-adjoint operators and their matrix representation in orthonormal bases.	-
Quiz 3		Covers the material of the Homework Assignments #5 and #6.
17	Dec. 06	Eigenvalues and eigenvectors of a self-adjoint operator, Spectral resolution of a self-adjoint operator in finite-dimensions; Linear differential equations: Generalities and the solution of a first order linear ODEs	Pages 26-29 of Kreyszig 9th Ed.
18	Dec. 08	Solution of non-homogeneous first order linear ODEs and the notion of a Green's function; Second order linear ODEs: Wronskien and Abel's theorem, the form of the general solution, reduction of order, equations with constant coefficients	Pages 45- 47, 50-56, 73-78 of Kreyszig 9th Ed.
19	Dec. 13	Solution of non-homogeneous second order linear ODEs (using variation of parameters) and the derivation of the formula for the Green's function; Power series and their convergence	78-79, 98-104 of Kreyszig 9th Ed.
20	Dec. 15	Ratio test, (real) analytic functions. Power series solution of 2nd order linear ODE about an ordinary point, Hermite's eqn and Hermite polynomials	Kreyszig: 166-182
21	Dec. 20	Frobenius method, an application	Kreyszig: 182-203
Midterm Exam 2	Dec. 22	Covers the material of of Lectures 12-20.
22	Dec. 29	Boundary-value problems: Dirichlet, Neumann, and Sturm-Liouville boundary conditions, Sturm-Liouville problem, orthogonal functions	Kreyszig: 203-209
23	Dec. 31	Examples of Sturm-Liouville problems; Systems of first order linear ODE's: Basic results, solution of homogeneous systems with constant coefficients.	Kreyszig: 136-140
24	Jan. 03	Fundamental matrix and the solution of the non-homogeneous first order linear ODE's. An Example.	Kreyszig: 159-161
25	Jan. 05	Examples of systems of first order linear ODE's	Kreyszig: 159-165
Quiz 4		Covers the material of the Homework Assignments #9

Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.