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= Analytical Geometry & Linear Algebra – II =
* '''Course name''': Analytical Geometry & Linear Algebra – II
* '''Code discipline''': CSE204
* '''Subject area''': Math
== Short Description ==
This course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines. Linear algebra is a branch of mathematics that studies systems of linear equations and the properties of matrices. The concepts of linear algebra are extremely useful in physics, data sciences, and robotics. Due to its broad range of applications, linear algebra is one of the most widely used subjects in mathematics.

== Course Topics ==
{| class="wikitable"
|+ Course Sections and Topics
|-
! Section !! Topics within the section
|-
| Linear equation system solving by using the vector-matrix approach || 
# The geometry of linear equations. Elimination with matrices
# Matrix operations, including inverses
# LU and LDU  factorization
# Transposes and permutations
# Vector spaces and subspaces
# The null space: Solving Ax=0  and Ax=b
# Row reduced echelon form. Matrix rank
# Numerical methods for solving systems of linear algebraic equations
|-
| Linear regression analysis, QR-decomposition ||
# Independence, basis and dimension
# The four fundamental subspaces
# Orthogonal vectors and subspaces 
# Projections onto subspaces Projection matrices
# Least squares approximations
# Gram-Schmidt orthogonalization and A = QR
|-
|  Matrix Diagonalization || 
# Complex Numbers
# Hermitian and Unitary Matrices
# Eigenvalues and eigenvectors
# Matrix diagonalization
|-
| Symmetric, positive definite and similar matrices. Singular value decomposition || 
#  Linear differential equations.
# Symmetric matrices. 
# Positive definite matrices
# Similar matrices.
# Left and right inverses, pseudoinverse
# Singular value decomposition (SVD)


|}

== 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 ===
{| class="wikitable"
|+ 
|-
! 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 ===
{| class="wikitable"
|+ 
|-
! 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 [https://portal.tpu.ru/SHARED/k/KONVAL/Textbooks/Tab1/Konev-Linear_Algebra_Vector_Algebra_and_Analytical_Geome.pdf book1]
* R.A.Sharipov. Course of Analytical Geometry Textbook, Ufa, BSU, 2013. 227pp [https://arxiv.org/pdf/1111.6521.pdf book2]
* P.R. Vital. Analytical Geometru 2D and 3D Analytical Geometry 2D and 3D [https://www.amazon.com/Analytical-Geometry-2D-3D-Vittal/dp/8131773604 book3]


=== Software and tools used within the course ===
* No.

== Activities and Teaching Methods ==
{| class="wikitable"
|+ 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
|}
{| class="wikitable"
|+ 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 ====
# How to perform the shift of the vector?
# What is the geometrical interpretation of the dot product?
# How to determine whether the vectors are linearly dependent?
# What is a vector basis?
# What is the difference between matrices and determinants?
# Matrices <math display="inline">A</math> and <math display="inline">C</math> have dimensions of <math display="inline">m\times n</math> and <math display="inline">p\times q</math> respectively, and it is known that the product <math display="inline">ABC</math> exists. What are possible dimensions of <math display="inline">B</math> and <math display="inline">ABC</math>?
# How to determine the rank of a matrix?
# What is the meaning of the inverse matrix?
# How to restate a system of linear equations in the matrix form?

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

==== Section 3 ====

# Formulate the canonical equation of the given quadratic curve.
# Which orthogonal transformations of coordinates do you know?
# How to perform a transformation of the coordinate system?
# How to represent a curve in the space?
# What is the type of a quadric surface given by a certain equation?
# How to compose the equation of a surface of revolution?
# What is the difference between a directrix and generatrix?
# How to represent a quadric surface in the vector form?

=== Final assessment ===
'''Section 1'''
# Evaluate <math display="inline">|\textbf{a}|^2-2\sqrt3\textbf{a}\cdot\textbf{b}-7|\textbf{b}|^2</math> given that <math display="inline">|\textbf{a}|=4</math>, <math display="inline">|\textbf{b}|=1</math>, <math display="inline">\angle(\textbf{a},\,\textbf{b})=150^{\circ}</math>.
# Prove that vectors <math display="inline">\textbf{b}(\textbf{a}\cdot\textbf{c})-\textbf{c}(\textbf{a}\cdot\textbf{b})</math> and <math display="inline">\textbf{a}</math> are perpendicular to each other.
# Bases <math display="inline">AD</math> and <math display="inline">BC</math> of trapezoid <math display="inline">ABCD</math> are in the ratio of <math display="inline">4:1</math>. The diagonals of the trapezoid intersect at point <math display="inline">M</math> and the extensions of sides <math display="inline">AB</math> and <math display="inline">CD</math> intersect at point <math display="inline">P</math>. Let us consider the basis with <math display="inline">A</math> as the origin, <math display="inline">\overrightarrow{AD}</math> and <math display="inline">\overrightarrow{AB}</math> as basis vectors. Find the coordinates of points <math display="inline">M</math> and <math display="inline">P</math> in this basis.
# A line segment joining a vertex of a tetrahedron with the centroid of the opposite face (the centroid of a triangle is an intersection point of all its medians) is called a median of this tetrahedron. Using vector algebra prove that all the four medians of any tetrahedron concur in a point that divides these medians in the ratio of <math display="inline">3:1</math>, the longer segments being on the side of the vertex of the tetrahedron.
# Find <math display="inline">A+B</math> and <math display="inline">2A-3B+I</math>.
# Find the products <math display="inline">AB</math> and <math display="inline">BA</math> (and so make sure that, in general, <math display="inline">AB\neq BA</math> for matrices).
# Find the inverse matrices for the given ones.
# Find the determinants of the given matrices.
# Point <math display="inline">M</math> is the centroid of face <math display="inline">BCD</math> of tetrahedron <math display="inline">ABCD</math>. The old coordinate system is given by <math display="inline">A</math>, <math display="inline">\overrightarrow{AB}</math>, <math display="inline">\overrightarrow{AC}</math>, <math display="inline">\overrightarrow{AD}</math>, and the new coordinate system is given by <math display="inline">M</math>, <math display="inline">\overrightarrow{MB}</math>, <math display="inline">\overrightarrow{MC}</math>, <math display="inline">\overrightarrow{MA}</math>. Find the coordinates of a point in the old coordinate system given its coordinates <math display="inline">x'</math>, <math display="inline">y'</math>, <math display="inline">z'</math> in the new one.

'''Section 2'''
# Two lines are given by the equations <math display="inline">\textbf{r}\cdot\textbf{n}=A</math> and <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math>, and at that <math display="inline">\textbf{a}\cdot\textbf{n}\neq0</math>. Find the position vector of the intersection point of these lines.
# Find the distance from point <math display="inline">M_0</math> with the position vector <math display="inline">\textbf{r}_0</math> to the line defined by the equation (a) <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math>; (b) <math display="inline">\textbf{r}\cdot\textbf{n}=A</math>.
# Diagonals of a rhombus intersect at point <math display="inline">M(1;\,2)</math>, the longest of them being parallel to a horizontal axis. The side of the rhombus equals 2 and its obtuse angle is <math display="inline">120^{\circ}</math>. Compose the equations of the sides of this rhombus.
# Compose the equations of lines passing through point <math display="inline">A(2;-4)</math> and forming angles of <math display="inline">60^{\circ}</math> with the line <math display="inline">\frac{1-2x}3=\frac{3+2y}{-2}</math>.
# Find the cross product of (a) vectors <math display="inline">\textbf{a}(3;-2;\,1)</math> and <math display="inline">\textbf{b}(2;-5;-3)</math>; (b) vectors <math display="inline">\textbf{a}(3;-2;\,1)</math> and <math display="inline">\textbf{c}(-18;\,12;-6)</math>.
# A triangle is constructed on vectors <math display="inline">\textbf{a}(2;4;-1)</math> and <math display="inline">\textbf{b}(-2;1;1)</math>. (a) Find the area of this triangle. (b) Find the altitudes of this triangle.
# Find the scalar triple product of <math display="inline">\textbf{a}(1;\,2;-1)</math>, <math display="inline">\textbf{b}(7;3;-5)</math>, <math display="inline">\textbf{c}(3;\,4;-3)</math>.
# It is known that basis vectors <math display="inline">\textbf{e}_1</math>, <math display="inline">\textbf{e}_2</math>, <math display="inline">\textbf{e}_3</math> have lengths of <math display="inline">1</math>, <math display="inline">2</math>, <math display="inline">2\sqrt2</math> respectively, and <math display="inline">\angle(\textbf{e}_1,\textbf{e}_2)=120^{\circ}</math>, <math display="inline">\angle(\textbf{e}_1,\textbf{e}_3)=135^{\circ}</math>, <math display="inline">\angle(\textbf{e}_2,\textbf{e}_3)=45^{\circ}</math>. Find the volume of a parallelepiped constructed on vectors with coordinates <math display="inline">(-1;\,0;\,2)</math>, <math display="inline">(1;\,1\,4)</math> and <math display="inline">(-2;\,1;\,1)</math> in this basis.

'''Section 3'''
# Prove that a curve given by <math display="inline">34x^2+24xy+41y^2-44x+58y+1=0</math> is an ellipse. Find the major and minor axes of this ellipse, its eccentricity, coordinates of its center and foci. Find the equations of axes and directrices of this ellipse.
# Determine types of curves given by the following equations. For each of the curves, find its canonical coordinate system (i.e. indicate the coordinates of origin and new basis vectors in the initial coordinate system) and its canonical equation. (a) <math display="inline">9x^2-16y^2-6x+8y-144=0</math>; (b) <math display="inline">9x^2+4y^2+6x-4y-2=0</math>; (c) <math display="inline">12x^2-12x-32y-29=0</math>; (d) <math display="inline">xy+2x+y=0</math>;
# Find the equations of lines tangent to curve <math display="inline">6xy+8y^2-12x-26y+11=0</math> that are (a) parallel to line <math display="inline">6x+17y-4=0</math>; (b) perpendicular to line <math display="inline">41x-24y+3=0</math>; (c) parallel to line <math display="inline">y=2</math>.
# For each value of parameter <math display="inline">a</math> determine types of surfaces given by the equations: (a) <math display="inline">x^2+y^2-z^2=a</math>; (b) <math display="inline">x^2+a\left(y^2+z^2\right)=1</math>; (c) <math display="inline">x^2+ay^2=az</math>; (d) <math display="inline">x^2+ay^2=az+1</math>.
# Find a vector equation of a right circular cone with apex <math display="inline">M_0\left(\textbf{r}_0\right)</math> and axis <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math> if it is known that generatrices of this cone form the angle of <math display="inline">\alpha</math> with its axis.
# Find the equation of a cylinder with radius <math display="inline">\sqrt2</math> that has an axis <math display="inline">x=1+t</math>, <math display="inline">y=2+t</math>, <math display="inline">z=3+t</math>.
# An ellipsoid is symmetric with respect to coordinate planes, passes through point <math display="inline">M(3;\,1;\,1)</math> and circle <math display="inline">x^2+y^2+z^2=9</math>, <math display="inline">x-z=0</math>. Find the equation of this ellipsoid.

=== 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.