Math 107: Calculus II, Fall 2012
Topics Covered in Each Lecture
|
Lecture Number |
Date |
Content |
Corresponding Reading material |
|
1 |
Sep. 18 |
Review of the content of the syllabus;
Elementary discussion of Sets, subsets, equal sets, ordered pairs, Cartesian
product of two sets, and relations. |
|
|
2 |
Sep. 20 |
Functions: Definition of a function, domain and range of a
function, everywhere-defined and onto functions, sequences, convergence of
real sequences, sequences defined iteratively |
Stewart 6th
edition: Pages 674-678, 7th edition: Pages 689-695 |
|
3 |
Sep. 21 |
Bounded and monotonic sequences, monotone sequence theorem,
examples (taught by Mehmet Sarýdereli) |
Stewart 6th
edition: Pages 678-685, 7th edition: Pages 695-700 |
|
4 |
Sep. 25 |
Properties of the limit of convergent sequences, Convergence of
image of a convergent sequence under a continuous function, bounded
sequences, boundedness of convergent sequences, monotone sequence theorem and
its applications. |
Stewart 6th
edition: Pages 678-685, 7th edition: Pages 695-700 |
|
5 |
Sep. 27 |
Numerical series and their convergence, geometric and harmonic
series, testing for convergence by checking the limit of the sequence, sum of
two convergent series |
Stewart 6th
edition: Pages 687-694, 7th edition: Pages 703-710 |
|
6 |
Sep. 28 |
Relabeling and shifting dummy variables, extracting terms of a
series, integral test with proof, applications of the integral test,
approximating series with finite sums and the corresponding error bounds
(remainder estimate) |
Stewart 6th
edition: Pages 697-703, 7th edition: Pages 714-720 |
|
7 |
Oct. 02 |
Comparison and Limit Comparison tests with proofs, alternating
series test with proof |
Stewart 6th
edition: Pages 705-712, 7th edition: Pages 722-729 |
|
8 |
Oct. 04 |
Derivation and application of the error estimate for convergent
alternating series, absolute convergence, ratio test (without proof) and its
applications. |
Stewart 6th
edition: Pages 712-718, 7th edition: Pages 729-736 |
|
9 |
Oct. 05 |
Root test (without proof), rearranging terms of convergent
series, strategy for deciding convergence or divergence of a given series;
Power series: definition, examples, and the convergence behavior, radius of
convergence |
Stewart 6th
edition: Pages 718-725, 7th edition: Pages 736-743 |
|
10 |
Oct. 09 |
Application of the ratio test to determine the radius of
convergence of power series, differentiation and integration of power series |
Stewart 6th
edition: Pages 725-731, 7th edition: Pages 743-750 |
|
11 |
Oct. 11 |
Power series expansion of ln(1+x) and arctan(x) and approximate
calculation of ln(2) and pi, Taylor and Maclaurin series, analytic functions,
examples |
Stewart 6th
edition: Pages 731-736, 7th edition: Pages 750-755 |
|
12 |
Oct. 12 |
Taylor’s inequality and proof that the MacLaurin series for e^x
converges to e^x for all real numbers x, MacLaurian series for sin(x) and
cos(x), Binomial series, multiplication of power series |
Stewart 6th
edition: Pages 736-745, 7th edition: Pages 755-764 |
|
Exam 1 |
Oct. 15 |
Click on Midterm
Exam 1 for detailed information about this exam. |
|
|
13 |
Oct. 16 |
Binary operations, basic properties of addition and multiplication of real numbers, plane vectors, componentwise addition, scalar multiplication, and complex multiplication of plane vectors. |
A First Course in Linear Algebra: Pages
27-31 |
|
14 |
Oct. 18 |
Complex numbers: real and imaginary parts, modulus and argument, the principal argument, inverse of a nonzero complex number, polynomials and rational functions of a complex variable, e^x for real x and e^z for a complex z. |
A First Course in Linear Algebra: Pages
31-33 |
|
15 |
Oct. 19 |
Real and imaginary parts of e^z for a complex number z (Euler's formula), roots of a complex number and their multivalued-ness, the componentwise addition and scalar multiplication in R^n and their properties |
A First Course in Linear Algebra: Pages
33-35 |
|
16 |
Oct. 23 |
Real and complex vector spaces, trivial vector space, uniqueness of the zero vector, vector space of real-valued functions defined on a nonempty set, complex vector spaces, motivation for the notion of a subspace of a vector space |
A First Course in Linear Algebra: Pages
35-38 |
|
Kurban Bayramý |
Oct. 24-28 |
|
|
|
17 |
Oct. 30 |
Subspaces of a vector space, linear combination of elements of a vector space, and span of a subset of a vector space |
A First Course in Linear Algebra: Pages
38-40 |
|
18 |
Nov. 01 |
Span of a subset S as the smallest subspace containing S, linear dependence and independence for a finite subset of a vector space |
A First Course in Linear Algebra: Pages
41-46 |
|
19 |
Nov. 02 |
Linearly-dependent and -independent subsets of a vector space |
A First Course in Linear Algebra: Pages
42-46 |
|
20 |
Nov. 06 |
Review of the topics covered in the preceding three lectures: Subspaces of a vector space, linear combination and span, linear dependence and independence; Unique Expansion Theorem |
A First Course in Linear Algebra: Pages
38-47 |
|
21 |
Nov. 08 |
Implications of the Unique Expansion Theorem, the concept of a basis, existence and uniqueness of basis, standard basis for F^n, basis of the space of polynomials P(R,R), finite- and infinite-dimensional vector spaces, dimension |
A First Course in Linear Algebra: Pages
47-51 |
|
22 |
Nov. 09 |
Examples of two different bases of C^2 and the polynomial space P_1(R,R), matrix representation of vectors in a given basis |
A First Course in Linear Algebra: Pages
49-52 |
|
23 |
Nov. 13 |
Linear operators, zero and identity operators, linear operators mapping R to R, linear operators mapping R^n to R, linear operators mapping R^n to R^m and matrices. |
A First Course in Linear Algebra: Pages 63-65 |
|
24 |
Nov. 15 |
The differentiation and integral transforms as examples of linear operators; Theorem: Images and inverse images of subspaces under a linear operator are subspaces (without proof); Null space of a linear operator. The proof that it is a subspace. Range and null space of the zero and identity operators. |
A First Course in Linear Algebra: Pages
66-70 |
|
25 |
Nov. 16 |
Determination of the range and null space of linear operators acting in finite-dimensional vector spaces. The relation between dimension of the range, null space, and domain of a linear operator if the latter is finite-dimensional. |
A First Course in Linear Algebra: Pages 70-72 |
|
26 |
Nov. 20 |
Construction of a basis for the range of a linear operator with a finite-dimensional domain, linear equations defined by a linear operator, and the existence and uniqueness of their solution(s). |
A First Course in Linear Algebra: Pages 72-76 |
|
27 |
Nov. 22 |
Addressing the existence and uniqueness of the solution(s) of general linear equations, and their general form whenever they exist. |
A First Course in Linear Algebra: Pages
75-78
|
|
28 |
Nov. 23 |
Matrix representation of linear operators relating finite-dimensional vector spaces. |
A First Course in Linear Algebra: Pages
79-84
|
|
29 |
Nov. 27 |
Matrix representation of the derivative operator acting in the space of polynomials of degree not greater than 2, vector space of linear operators mapping a vector space to another, composition of linear operators, algebra of linear operators defined on a vector space. |
A First Course in Linear Algebra: Pages
86-89 |
|
30 |
Nov. 29 |
Invertible operators, domain and range of the inverse function, the linearity of the inverse of an invertible linear operator, isomorphisms |
A First Course in Linear Algebra: Pages 90-97 |
|
31 |
Nov. 30 |
Matrices, equal matrices, the connection with linear operators, use of the relationship between matrices and linear operators to introduce notions of addition and scalar multiplication for matrices of the same size, the vector space of m x n matrices and its standard basis |
A First Course in Linear Algebra: Pages
123-126 |
|
32 |
Dec. 04 |
Derivation of the multiplication rule for matrices from the composition rule for linear operators, properties of matrix multiplication, transpose and conjugate-transpose (adjoint) of a matrix, symmetric, Hermitian, upper-triangular, lower-triangular, and diagonal matrices |
A First Course in Linear Algebra: Pages 127-132 |
|
33 |
Dec. 06 |
Invertible matrices, determinant as means to test the invertibility of a 2 x 2 matrix, Levi Civita symbol, definition of the determinant of a square matrix, basic properties of the determinant. |
A First Course in Linear Algebra: Pages 132-136 |
|
34 |
Dec. 07 |
Calculation of the determinant of a square matrix and the inverse of an invertible matrix using the cofactors of its entries, a simple way of computing the determinant of 3 x 3 matrices; Effect of a basis transformation on the representation of vectors and linear operators, similarity transformations, invariants of a matrix, determinant and trace as examples of invariants of a square matrix. |
A First Course in Linear Algebra: Pages 137-146 |
|
Exam 2 |
Dec. 11 |
Click on Midterm
Exam 2 for detailed information about this exam. |
|
|
35 |
Dec. 13 |
Rank and Nullity of a matrix M as the dimension of the range and null space of a canonical linear operator M defined by the matrix; equality of the rank of M with the maximum number of linearly independent columns of M, augmented matrix and rank criterion for the existence of a column vector in the range of M, equality of the rank of M and its transpose. |
A First Course in Linear Algebra: Pages 147-154 |
|
36 |
Dec. 14 |
Systems of linear algebraic equations, the corresponding matrix equation and the linear equation defined by the linear operator of the matrix of coefficients, the rank criterion for the existence and uniqueness of the solution, the general form of the solution for the case that there are solutions. |
A First Course in Linear Algebra: Pages 156-160 |
|
37 |
Dec. 18 |
Equivalent systems of linear algebraic equations and the method of Gaussian Elimination: Basic row operations, application to systems of linear equations, application for computing the determinant of a square matrix, matrix realization of the basic row operations. |
A First Course in Linear Algebra: Pages 162-173 |
|
38 |
Dec. 20 |
Class was cancelled because of unfavorable weather conditions. |
|
|
|
Dec. 21 |
Class was cancelled because of unfavorable weather conditions. |
|
|
40 |
Dec. 25 |
Fixed points and invariant subspaces of a function, invariant subspaces of a linear operator, eigenvalue problem, eigenvalues, eigenvectors, and point spectrum, common eigenvectors of linear operators. |
A First Course in Linear Algebra: Pages 265-270 |
|
41 |
Dec. 27 |
Eigenvalue problem for matrices: Eigenvalues, eigenvectors, the spectrum of a square matrix, and the standard method for calculation of them, the existence of eigenvalues, algebraic multiplicity of eigenvalues, examples of matrices for which there is a basis consisting of their eigenvectors and matrices lacking this property. |
A First Course in Linear Algebra: Pages 271-276 |
|
42 |
Dec. 28 |
Diagonalization of diagonalizable matrices: Bases consisting of the eigenvectors of a linear operator and transformation to such a basis, similarity transformation achieving the diagonalization, diagonalizability of n x n matrices with n distinct eigenvalues. |
A First Course in Linear Algebra: Pages 277-281 |
Note: The pages from the textbook listed above may not include some of the material covered in the lectures.