Linear algebra, 2019/2020 spring

Wednesday 11:00-14:00, Room KO 410 online course

REQUIREMENTS

Seminar: There will be two tests (50 + 50 points): A midterm on 25th March, and a final test on 13th May. Additionally, bonus points can be earned during the semester. Make-up for the mid-term and final tests will only be given for unavoidable and documented absences. The modified requirements (adjusted to online teaching) can be found on CooSpace's newsboard.

Lecture: First the seminar requirements have to be passed. The offered grade for the lecture course is the grade earned on the seminar course. The students can accept this offered grade until the last day of term-time. The students who reject the offered grade, can take the (written) lecture exam on every week of the exam period. The lecture exam covers the whole course, it contains both theoretical questions and practice exercises. Sample lecture exams will be available on the course's homepage. The exam will be held on my Jitsi channel, see CooSpace's newsboard for the details.

Grades (for both courses):
0% – 50%: fail (1)
51% – 62%: pass (2)
63% – 75%: satisfactory (3)
76% – 87%: good (4)
88% – 100%: excellent (5).

LECTURE SLIDES, SYLLABUS

1. Matrices. Definition of matrix. Equality of matrices. Matrices of special shape. Matrix operations (addition, scalar multiplication, transpose, matrix multiplication) and their properties. Special matrices.
2. Determinants. Definition of determinant. Basic properties of determinants. Determinant of matrix product. Determinant of transpose. Duality principle. Determinant expansion according to arbitrary row or column.
3-4. Applications of determinants. Efficient determinant evaluation. Inverse of a matrix. Linear systems. Regular linear systems, Cramer's rule.
5. Solving linear systems using Gaussian elimination. The matrix and augmented matrix of a linear system. Elementary row operations. Row-echelon form, Gaussian elimination. The number of solutions of a linear system (and how to read it off from the row-echelon form).
6-7. Vector spaces. The Rn space. The definition of vector spaces. Subspaces. Linear combination. Finding coefficients of linear combinations. Spanned subspace. Linear independence. Generator systems. Basis, coordinate vector. The standard basis in Rn. Dimension of a vector space. Basic properties of linear independence and generator systems. Rank of vectors and its applications. Submatrix. Rank of matrices.
8. Homogeneous linear systems. Eigenvalues and eigenvectors. Homogeneous linear systems. Solution space. Finding a basis in the solution space. The dimension of the solution space. Eigenvalues and eigenvectors. Characteristic polynomial, and its connection with eigenvalues. Eigenspaces. Linear independence of eigenvectors associated with distinct eigenvalues.
9. Elementary basis transformation. The main theorem on replacing a vector in the basis. Basis table. The process of elementary basis transformation. Applications: Calculating the inverse of a matrix, calculating the rank of vectors/matrix.
10. Leontief input/output model. The input-output matrix. Final demand vector, total output vector. Leontief inverse: calculating the total output vector, productivity of an economy. Price vector, determining the profit/loss vector, connection with productivity.
11-12. Introduction to linear programming. Solving linear systems using basis transformation. (And the three cases on the number of solutions.) The standard LP problem. The general LP problem. Simplex algorithm.
13. Powers of matrices. Powers of a matrix. Powers of diagonal matrices. Powers of diagonalizable matrices. n×n matrices with n distinct eigenvalues are diagonalizable.

WORKED-OUT EXAMPLES

Finding coordinate vector: Example #1, Example #2.
Determining rank of vectors, testing linear independence, generator system, basis: Example #1, Example #2.
Finding a basis in the solution space of a homogeneous linear system: Example #1, Example #2.
Finding eigenvalues of a matrix: Examples #1-#3.
Solving a standard LP problem: Example #1, Example #2.

SAMPLE TESTS AND EXAMS

Midterm #1 (SAMPLE) + solution.

USEFUL LINKS

Matrix calculator (Matrix operations, determinants, linear systems, rank, etc.)

TEXTBOOKS

Dormán Milkós: LINEAR ALGEBRA (Lecture Notes for the Course Linear Algebra). See CooSpace/Documents.
David McMahon: Linear Algebra Demystified (McGraw-Hill)

Main page