BSc: Analytic Geometry And Linear Algebra I.f22

Analytical Geometry & Linear Algebra – I

• Course name: Analytical Geometry & Linear Algebra – I
• Code discipline: CSE202
• Subject area: Math. Computer Science

Short Description

This is an introductory course in analytical geometry and linear algebra. After having studied the course, students get to know fundamental principles of vector algebra and its applications in solving various geometry problems, different types of equations of lines and planes, conics and quadric surfaces, transformations in the plane and in the space. An introduction on matrices and determinants as a fundamental knowledge of linear algebra is also provided.

Course Topics

Course Sections and Topics

Section	Topics within the section
Vector algebra	Vector spaces Basic operations on vectors (summation, multiplication by scalar, dot product) Linear dependency and independency of the vectors. Basis in vector spaces. Introduction to matrices and determinants. The rank of a matrix. Inverse matrix. Systems of linear equations Changing basis and coordinates
Line and Plane	General equation of a line in the plane General parametric equation of a line in the space Line as intersection between planes. Vector equation of a line. Distance from a point to a line. Distance between lines General equation of a plane. Normalized linear equation of a plane. Vector equation of a plane. Parametric equation of a plane Inter-positioning of lines and planes Cross Product of two vectors. Triple Scalar Product
Quadratic curves and surfaces	Circle, Ellipse, Hyperbola, Parabola. Canonical equations Shift of coordinate system. Rotation of coordinate system. Parametrization General equation of the quadric surfaces. Canonical equations of a sphere, ellipsoid, hyperboloid and paraboloid Surfaces of revolution. Canonical equation of a cone and cylinder Vector equations of some quadric surfaces

Intended Learning Outcomes (ILOs)

ILOs defined at three levels

We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

• explain the geometrical interpretation of the basic operations of vector algebra,
• restate equations of lines and planes in different forms,
• interpret the geometrical meaning of the conic sections in the mathematical expression,
• give the examples of the surfaces of revolution,
• understand the value of geometry in various fields of science and techniques.

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

• perform the basic operations of vector algebra,
• use different types of equations of lines and planes to solve the plane and space problems,
• represent the conic section in canonical form,
• compose the equation of quadric surface.

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

By the end of the course, the students should be able to ...

• list basic notions of vector algebra,
• recite the base form of the equations of transformations in planes and spaces,
• recall equations of lines and planes,
• identify the type of conic section,
• recognize the kind of quadric surfaces.

Grading

Course grading range

Grade	Range	Description of performance
A. Excellent	85-100	-
B. Good	70-84	-
C. Satisfactory	55-70	-
D. Fail	0-54	-

Course activities and grading breakdown

Activity Type	Percentage of the overall course grade
Midterm	30
Tests	20 (10 for each)
Final exam	50
In-class participation	5 extras

Recommendations for students on how to succeed in the course

• Participation is important. Attending lectures is the key to success in this course.
• Review lecture materials before classes to do well.
• Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.

Resources, literature and reference materials

Open access resources

• V.V. Konev. Linear Algebra, Vector Algebra and Analytical Geometry. Textbook. Tomsk: TPU Press, 2009, 114 pp book1
• R.A.Sharipov. Course of Analytical Geometry Textbook, Ufa, BSU, 2013. 227pp book2
• P.R. Vital. Analytical Geometru 2D and 3D Analytical Geometry 2D and 3D book3

Software and tools used within the course

• No.

Activities and Teaching Methods

Teaching and Learning Methods within each section

Teaching Techniques	Section 1	Section 2	Section 3
Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution)	1	1	1
Project-based learning (students work on a project)	0	0	0
Modular learning (facilitated self-study)	0	0	0
Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level)	1	1	1
Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them)	0	0	0
Business game (learn by playing a game that incorporates the principles of the material covered within the course)	0	0	0
Inquiry-based learning	0	0	0
Just-in-time teaching	0	0	0
Process oriented guided inquiry learning (POGIL)	0	0	0
Studio-based learning	0	0	0
Universal design for learning	0	0	0
Task-based learning	0	0	0

Activities within each section

Learning Activities	Section 1	Section 2	Section 3
Lectures	1	1	1
Interactive Lectures	1	1	1
Lab exercises	1	1	1
Experiments	0	0	0
Modeling	0	0	0
Cases studies	0	0	0
Development of individual parts of software product code	0	0	0
Individual Projects	0	0	0
Group projects	0	0	0
Flipped classroom	0	0	0
Quizzes (written or computer based)	1	1	1
Peer Review	0	0	0
Discussions	1	1	1
Presentations by students	0	0	0
Written reports	0	0	0
Simulations and role-plays	0	0	0
Essays	0	0	0
Oral Reports	0	0	0

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

1. How to perform the shift of the vector?
2. What is the geometrical interpretation of the dot product?
3. How to determine whether the vectors are linearly dependent?
4. What is a vector basis?
5. What is the difference between matrices and determinants?
6. Matrices ${\textstyle A}$ and ${\textstyle C}$ have dimensions of ${\textstyle m\times n}$ and ${\textstyle p\times q}$ respectively, and it is known that the product ${\textstyle ABC}$ exists. What are possible dimensions of ${\textstyle B}$ and ${\textstyle ABC}$?
7. How to determine the rank of a matrix?
8. What is the meaning of the inverse matrix?
9. How to restate a system of linear equations in the matrix form?

Section 2

1. How to represent a line in the vector form?
2. What is the result of intersection of two planes in vector form?
3. How to derive the formula for the distance from a point to a line?
4. How to interpret geometrically the distance between lines?
5. List all possible inter-positions of lines in the space.
6. What is the difference between general and normalized forms of equations of a plane?
7. How to rewrite the equation of a plane in a vector form?
8. What is the normal to a plane?
9. How to interpret the cross products of two vectors?
10. What is the meaning of scalar triple product of three vectors?

Section 3

1. What is the characteristic property of trees?
2. How to find an Euler tour?
3. What is a Hamilton path?
4. Why are K3 and K5,5 not planar?
5. What is the difference between undirected and directed graphs?
6. Why do we consider weighted graphs?
7. What practical problems are solved using Dijkstra's algorithm?
8. What is the maximum flow problem?

Final assessment

Section 1

1. What is the difference between Categorical and Propositional Logic?
2. How does Predicate Logic differ from Categorical and Propositional Logic?
3. Why is Predicate Logic so important?
4. What are Truth-Functions and why do we use them?
5. Compute True Tables for Propositions
6. Compute True Tables for Arguments

Section 2

Section 3

1. Explain handshaking lemma.
2. Give necessary and sufficient conditions for the existence of an Euler tour.
3. Give sufficient conditions for the existence of a Hamilton path (theorems of Dirac and Ore).
4. Explain Kuratowski’s theorem.
5. Explain the difference between undirected and directed graphs.
6. Give the definition of weighted graphs?
7. Explain Dijkstra's algorithm?
8. What is the solution of the maximum flow problem (the Ford-Fulkerson algorithm)?

The retake exam

Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.