SEFI comparative study: Course - Algebra and Geometry

1. NATIONAL RESEARCH OGAREV MORDOVIA STATE UNIVERSITY
Course: Algebra and Geometry
Major profiles:
Informatics and computer science (ICS), Programming Engineering (PE)
SEFI comparative study

2. SEFI – Algebra and Geometry
OMSU – PE & ICS
Content of the course – 1st year, 1st (autumn) semester:
• “Matrices and vectors” (=“Introduction to linear and vector algebra”)
• “Analytic geometry in plane”
• “Analytic geometry in space”
Note. So the correct name is “Linear algebra and analytic geometry”.
Classification of SEFI competences:
• Formed in secondary school (not in high school)
• (Italics) Formed in secondary school and partially duplicated in high school
• Formed in high school (competences from basic areas)
• Formed in high school (competences from other areas)
• Not formed in high school by the course on the discussed profiles
Basic areas for the course according to SEFI:
• (Linear) Algebra
• Geometry (& Trigonometry)
Outline

3. SEFI – Algebra and Geometry
OMSU – PE & ICSCore 0
Algebra
Arithmetic of real numbers:
• All the competencies of this section are formed in secondary school. The students
entering high school must have these competencies already.
Algebraic expressions and formulae:
• All the competencies of this section are formed in secondary school. The students
entering high school must have these competencies already.
Linear laws:
• All the competencies of this section are formed in secondary school and partially
duplicated in high school while studying straight lines in plane (section “Analytic
geometry in plane”).
Quadratics, cubics and polynomials:
• All the competencies of this section are formed in secondary school. The students
entering high school must have these competencies already.

4. SEFI – Algebra and Geometry
OMSU – PE & ICSCore 0
Analysis and Calculus
Functions and their inverses:
• understand how a graphical translation can alter a functional description;
• understand how a reflection in either axis can alter a functional description;
• understand how a scaling transformation can alter a functional description.
Note. All this is for section “Analytic geometry in plane”.
Rates of change and differentiation:
• obtain the equation of the tangent and normal to the graph of a function (section
“Analytic geometry in plane”).
Proof (through the course):
• understand how a theorem is deduced from a set of assumptions;
• appreciate how a corollary is developed from a theorem;
• follow proofs of theorems.

5. SEFI – Algebra and Geometry
OMSU – PE & ICSCore 0
Geometry and Trigonometry
Geometry
• understand the basic concept of a geometric transformation in the plane (section
“Analytic geometry in plane”);
• recognize examples of a metric transformation (isometry) and affine transformation
(similitude) (section “Analytic geometry in plane”);
• obtain the image of a plane figure in a defined geometric transformation: a
translation in a given direction, a rotation about a given centre, a symmetry with
respect to the centre or to the axis, scaling to a centre by a given ratio (section
“Analytic geometry in plane”);
• All other competencies of this section are formed in secondary school. The students
entering high school must have these competencies already.
Trigonometry
• All the competencies of this section are formed in secondary school. The students
entering high school must have these competencies already.

6. SEFI – Algebra and Geometry
OMSU – PE & ICSCore 0
Co-ordinate geometry
• calculate the distance between two points (section “Matrices and vectors”);
• find the position of a point which divides a line segment in a given ratio (section
“Matrices and vectors”);
• find the angle between two straight lines (section “Analytic geometry in plane”);
• calculate the distance of a given point from a given line (section “Analytic geometry
in plane”);
• calculate the area of a triangle knowing the co-ordinates of its vertices (section
“Analytic geometry in plane”);
• give simple example of a locus;
• recognize and interpret the equation of a circle in standard form and state its radius
and centre;
• convert the general equation of a circle to standard form;
• recognize the parametric equations of a circle (section “Analytic geometry in
plane”);
• derive the main properties of a circle, including the equation of the tangent at a
point (section “Analytic geometry in plane”);
…to be continued

7. SEFI – Algebra and Geometry
OMSU – PE & ICSCore 0
…the beginning is on the previous slide
• define a parabola as a locus (section “Analytic geometry in plane”);
• recognize and interpret the equation of a parabola in standard form and state its
vertex, focus, axis, parameter and directrix (section “Analytic geometry in plane”);
• recognize the parametric equation of a parabola (section “Analytic geometry in
plane”);
• derive the main properties of a parabola, including the equation of a tangent at a
point (section “Analytic geometry in plane”);
• understand the concept of parametric representation of a curve (section “Analytic
geometry in plane”);
• use polar co-ordinates and convert to and from Cartesian co-ordinates (section
“Analytic geometry in plane”);
Trigonometric functions and applications
• All the competencies of this section are formed in secondary school. The students
entering high school must have these competencies already.
Trigonometric identities
• All the competencies of this section are formed in secondary school. The students
entering high school must have these competencies already.

8. SEFI – Algebra and Geometry
OMSU – PE & ICSLevel 1
Geometry
Conic sections
• All the competencies of this section are formed in section “Analytic geometry in
plane” of the course.
3D co-ordinate geometry
• All the competencies of this section are formed in section “Analytic geometry in
space” of the course.
Linear algebra
Vector arithmetic
• distinguish between vector and scalar quantities;
• understand and use vector notation;
• represent a vector pictorially;
• carry out addition and scalar multiplication and represent them pictorially;
• determine the unit vector in a specified direction (section “Matrices and vectors”);
• represent a vector in component form (two and three components only) (section
“Matrices and vectors”);

9. SEFI – Algebra and Geometry
OMSU – PE & ICSLevel 1
Vector algebra and applications
• All the competencies of this section are formed in section “Matrices and vectors” of
the course.
Matrices and determinants
• All the competencies of this section are formed in section “Matrices and vectors” of
the course.
Solution of simultaneous linear equations
• All the competencies of this section are formed in section “Matrices and vectors” of
the course.
Least squares curve fitting
• All the competencies of this section are not formed in the discussed course;
corresponding material is studied at the course of Probability and Statistics and,
perhaps, Numerical methods.

10. SEFI – Algebra and Geometry
OMSU – PE & ICSLevel 1
Linear spaces and transformations
• define a linear space (section “Matrices and vectors”);
• define and recognize linear independence (section “Matrices and vectors”);
• define and obtain a basis for a linear space (section “Matrices and vectors”);
• define a subspace of a linear space and find a basis for it (it’s not the point for the
profiles discussed);
• define scalar product in a linear space (it’s not the point for the profiles discussed,
this competence is formed only while studying special math courses for major in
maths);
• understand the concept of norm (it’s not the point for the profiles discussed, this
competence is formed only while studying special math courses for major in maths);
• define the Euclidean norm (section “Matrices and vectors”);
• define a linear transformation between two spaces; define the image space and the
null space for the transformation (section “Matrices and vectors”);
• derive the matrix representation of a linear transformations (section “Matrices and
vectors”);
…to be continued

11. SEFI – Algebra and Geometry
OMSU – PE & ICSLevel 1
…the beginning is on the previous slide
• understand how to carry out a change of basis (section “Analytic geometry in
plane”);
• define an orthogonal transformation (section “Analytic geometry in plane”);
• apply the above matrices of linear transformations in the Euclidean plane and
Euclidean space (section “Analytic geometry in plane”);
• recognise matrices of Euclidean and affine transformations: identity, translation,
symmetry, rotation and scaling (section “Analytic geometry in plane”).

12. SEFI – Algebra and Geometry
OMSU – PE & ICSLevel 2
Analysis and calculus
Linear optimization
• recognize a linear programming problem in words and formulate it mathematically;
• represent the feasible region graphically;
• solve a maximization problem graphically by superimposing lines of equal profit.
Note. All this is for section “Analytic geometry in plane”. The simplest optimization
problem is treated as an application of topic “Straight line in plane”.
Discrete Mathematics
Algorithms
• understand the 'big O' notation for functions.
Note. The idea of algorithm’s complexity is needed when the determinants are
calculated, i.e. in the section “Matrices and vectors”.

13. SEFI – Algebra and Geometry
OMSU – PE & ICSLevel 2
Geometry
Helix
• recognize the parametric equation of a helix (section “Analytic geometry in plane”);
• derive the main properties of a helix, including the equation of the tangent at a
point, slope and pitch.
Geometric spaces and transformations
• define Euclidean space and state its general properties (section “Matrices and
vectors”);
• understand the Cartesian co-ordinate system in the space (section “Analytic
geometry in space”);
• apply the Euler transformations of the co-ordinate system (section “Analytic
geometry in space”);
• understand the polar co-ordinate system in the plane (section “Analytic geometry in
plane”);
• understand the cylindrical co-ordinate system in the space (section “Analytic
geometry in space”);
…to be continued

14. SEFI – Algebra and Geometry
OMSU – PE & ICSLevel 2
…the beginning is on the previous slide
• understand the spherical co-ordinate system in the space (section “Analytic
geometry in space”);
• define Affine space and state its general properties;
• understand the general concept of a geometric transformation on a set of points
(section “Analytic geometry in plane”);
• understand the term 'invariants' and 'invariant properties‘;
• know and use the non-commutativity of the composition of transformations (section
“Analytic geometry in plane”);
• understand the group representation of geometric transformations (later the group
of transformations is an example of an algebraic group in Discrete maths);
• classify specific groups of geometric transformations with respect to invariants;
• derive the matrix form of basic Euclidean transformations (section “Analytic
geometry in plane”);
• derive the matrix form of an affine transformation (“Analytic geometry in plane”);
• calculate coordinates of an image of a point in a geometric transformation (section
“Analytic geometry in plane”);
• apply a geometric transformation to a plane figure (“Analytic geometry in plane”);

15. SEFI – Algebra and Geometry
OMSU – PE & ICSLevel 2
Linear algebra
Matrix methods
• The competencies of this section (“define a banded matrix”, “find the inverse of a
matrix in partitioned form”) are formed during studying of Numerical methods.
Eigenvalue problems
• interpret eigenvectors and eigenvalues of a matrix in terms of the transformation it
represents
• convert a transformation into a matrix eigenvalue problem
• find the eigenvalues and eigenvectors of 2x2 and 3x3 matrices algebraically
• determine the modal matrix for a given matrix
• reduce a matrix to diagonal form
• Other competencies of this section aren’t covered. Either it’s not the point for the
profiles discussed and this competence is formed when student has major in maths
or this material is studied at Calculus (solving differential equations, describing
oscillatory motion).

16. SEFI – Algebra and Geometry
OMSU – PE & ICSLevel 3
These competencies are optional, they are formed only if there is enough time for them
Geometry
Geometric core of Computer Graphics
• write a computer program that plots a curve which is described by explicit or
parametric equations in Cartesian or polar coordinates (section “Analytic geometry
in plane”);
• know Bresenham's algorithm and Xiaolin Wu's algorithm of drawing lines on the
display monitor (section “Analytic geometry in plane”);
Linear algebra
Matrix decomposition
• know Strassen's algorithm for quick multiplying of matrices (section “Matrices and
vectors”).

17. SEFI – Algebra and Geometry
OMSU – PE & ICSConclusions
• The greater part of competencies in areas “(Linear) Algebra” and “Geometry
(& Trigonometry)” is formed by the course “Algebra and geometry” on ICS and
PE majors.
• Some competencies of SEFI Level 3 are formed too.
• For the Core 0 and Level 1 many of the competencies are formed at secondary
school (not at high school). Some part of the material is studied twice: at
secondary school and then at high school (at greater depth).
• Some of the competencies in the areas “(Linear) Algebra” and “Geometry (&
Trigonometry)” are formed by other courses such as Discrete maths and
Numerical methods.
• Some of the competencies in the areas listed are not formed because of their
less necessity.
• Some competencies from other areas are formed by the course “Algebra and
geometry”.