LINEAR ALGEBRA
Course description
Linear
Algebra is a half-semester (12 weeks) class that is obligatory for the
curriculum of the second-year MIEF students. The course was originally designed as an instrumental
supplement to the principal quantitative block subjects such as “Methods of optimization”,
“Time series analysis”, and “Econometrics”. Linear Algebra shares many exam
topics with the program of London University, for instance in “Mathematics 1”,
“Mathematics 2” and “Further mathematics for economists”. At the same time, the
class of Linear Algebra in MIEF is taught on its own to deliver basic
principles of matrix calculus. From a broader prospective, the aim of the
course is to deliver one of the most general mathematical concepts - the idea
of linearity.
The course
splits naturally into the following three parts:
1.
Problems
related to systems of linear equations and to the extension of the 2D- and 3D-
intuition to linear spaces of higher dimensions. This part includes the
concepts of basis, rank, dimension, linear hull, linear subspace, etc.
2.
Problems
that involve antisymmetric polylinear forms (determinants) and also problems
from the geometry of linear operators such that eigenvectors and eigenvalues,
matrix diagonalization, etc.
3.
Problems
from the calculus of bilinear forms: quadratic forms, orthogonalization, and
other geometric problems in higher-dimensional Eucledian spaces.
Teaching objectives
In Linear
Algebra it is critically important to teach not only the technique of
manipulations with matrices and vectors, but also general algebraic concepts
that are used, for instance, in problems that involve linear differential
equations or linear differential difference equations
Teaching methods
- Lectures
- Discussion sections
- Homeworks (weekly)
- Self-study
Assessment
- Homeworks (weekly)
- Online home teat
- Mock exam
- Final exam
The mock exam that will be held approximately after
the 4th or 5th lecture. The final exam that will be held when the course is
completed. Both exams consist of two
parts, Multiple choice and Free response. The mock exam takes 90 minutes; the
final exam takes 120 minutes. The final exam is not cumulative, i.e., it covers
the part of the course that was not covered by the mock exam.
Grade determination
The weekly homeworks constitute 10% of the final
grade. The mock exam contributes 40% to the final grade. The final exam is 50%
of the final grade. If a student missed the mock exam without a valid excuse
(see school’s schedule for valid excuses), he or she will be given zero grade,
which contributes as zero with 40% weight to the total grade. A student who
missed the mock exam with a valid excuse will be graded based on the grade for
the final exam, taken with the weight of 90%. The weight of the grades on
retake will be decided based on the validity of excuse for missing the exam and
also taking into account the grade received earlier.
Main reading
1.
Pervouchine
DD. Lecture Notes on Linear Algebra. ICEF 2011 (Pervouchine)
2.
Chernyak V.
Lecture Notes on Linear Algebra. Introductory course. Dialog, MSU, 1998, 2000
(Chernyak)
3.
Carl P.
Simon and Lawrence Blume. Mathematics for Economists, W.W. Norton &
Company, 1994 (Simon, Blume)
Additional reading
Chiang, Fundamental Methods of Mathematical Economics,
McGraw-Hill, 3rd ed., 1984
R.O.Hill, Elementary Linear Algebra, Academic Press,
1986
Гельфанд
И.М. Лекции по линейной алгебре Москва, Наука, 1999.
Кострикин
А.И., Манин Ю.И., Линейная алгебра и геометрия, Москва, Наука 1986.
Проскуряков
И.В. Сборник задач по линейной алгебре, Москва, Наука, 1985.
Course outline
1.
Systems of linear equations in
matrix form. Basic
concepts and geometric interpretation. Consistency. Elementary transformations
of equations. Gauss and Gauss-Jordan methods. (Pervouchine, ch. 1; Chernyak, ch. 1
- 5; Simon & Blume, ch. 7)
2.
Linear space. Linear independence. Rank. Linear span. Bases and
dimension of a linear space. Ordered bases and coordinates. Transition from one
basis to another. Properties of linearly dependent and linearly independent
vectors. Examples.
(Pervouchine, ch. 2; Chernyak, ch.
9-11; Simon & Blume, ch. 7,11)
3.
Linear subspace. The set of solutions as a linear
subspace. General and particular solutions.
Fundamental set of solutions.
(Pervouchine, ch. 2-3; Chernyak, ch.
11; Simon & Blume, ch. 11)
4.
Matrix as a set of columns and as a
set of rows. Linear
operations on matrices. Transpose matrix and matrix algebra. Special types of
matrices. Matrices of elementary transformations. (Pervouchine,
ch. 5; Chernyak, ch. 2-3; Simon & Blume, ch. 8)
5.
Determinant of a set of vectors. Geometric interpretation.
Determinant of a matrix. Computation and basic properties of determinants.
Cramer's rule. Applications to rank computation. (Pervouchine, ch. 4; Chernyak, ch.
6-8; Simon & Blume, ch. 9)
6.
Inverse matrix. Degenerate matrices. Computation of
the inverse matrix by the extended Gauss algorithm and by using algebraic
complements.
(Pervouchine, ch. 4-5; Chernyak, ch.
12; Simon & Blume, ch. 8)
7.
Linear operator as a geometric
object. Matrix of a
linear operator. Examples, including linear operators in functional spaces.
Transformations of vectors and matrices of linear operators induced by a change
of coordinates. Conjugate matrices.
(Pervouchine, ch. 6; Chernyak, ch.
15)
8.
Eigenvalues, eigenvectors and their properties. Characteristic equation. Basis and
dimension of eigenspaces. Diagonalization and its applications.
(Pervouchine, ch. 6; Chernyak, ch.
13-14; Simon & Blume, ch. 23)
9.
Bilinear and quadratic forms. Canonical representation. Full
squares method. Symmetric matrices and quadratic forms. Definite, indefinite,
and semidefinite forms. Silvester's criterion. (Pervouchine, ch. 7; Simon & Blume, ch. 16)
10.
Dot product in linear spaces. Norm of a vector. Metric
properties: distances and angles. Projection onto a subspace. Orthogonal bases.
Orthogonalization. Equations of lines and planes. (Pervouchine, ch. 8; Chernyak, ch.
16, Simon & Blume, ch. 10)
Distribution of hours
|
№
|
Topic
|
Total
|
In-class
hours
|
Self-study
|
|
|
|
|
|
Lectures
|
Seminars
|
|
|
1.
|
Systems of linear equations in matrix form
|
16
|
2
|
2
|
12
|
|
2.
|
Linear
space. Linear independence
|
8
|
1
|
1
|
6
|
|
3.
|
Linear
subspace
|
12
|
2
|
2
|
8
|
|
4.
|
Matrix as a set of columns and as a set of rows
|
16
|
2
|
2
|
12
|
|
5.
|
Determinant of a set of vectors
|
4
|
2
|
2
|
0
|
|
6.
|
Inverse
matrix
|
4
|
2
|
2
|
0
|
|
7.
|
Linear operator as a geometric object
|
8
|
2
|
2
|
4
|
|
8.
|
Eigenvalues, eigenvectors and their properties
|
12
|
2
|
2
|
10
|
|
9.
|
Bilinear
and quadratic forms
|
8
|
2
|
2
|
6
|
|
10
|
Dot product in linear spaces
|
12
|
1
|
1
|
10
|
|
|
Total:
|
108
|
18
|
18
|
72
|
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