BSc: Analytic Geometry And Linear Algebra II.s23

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

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

Grade	Range	Description of performance
A. Excellent	85-100	-
B. Good	65-84	-
C. Satisfactory	50-64	-
D. Fail	0-49	-

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

• Gilbert Strang. Linear Algebra and Its Applications, 4th Edition, Brooks Cole, 2006. ISBN: 9780030105678
• Gilbert Strang. Introduction to Linear Algebra, 4th Edition, Wellesley, MA: Wellesley-Cambridge Press, 2009. ISBN: 9780980232714
• Gilbert Strang, Brett Coonley, Andrew Bulman-Fleming. Student Solutions Manual for Strang’s Linear Algebra and Its Applications, 4th Edition, Thomson Brooks, 2005. ISBN-13: 9780495013259

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

Final assessment

Section 1

1. Evaluate ${\textstyle |{\textbf {a}}|^{2}-2{\sqrt {3}}{\textbf {a}}\cdot {\textbf {b}}-7|{\textbf {b}}|^{2}}$ given that ${\textstyle |{\textbf {a}}|=4}$, ${\textstyle |{\textbf {b}}|=1}$, ${\textstyle \angle ({\textbf {a}},\,{\textbf {b}})=150^{\circ }}$.
2. Prove that vectors ${\textstyle {\textbf {b}}({\textbf {a}}\cdot {\textbf {c}})-{\textbf {c}}({\textbf {a}}\cdot {\textbf {b}})}$ and ${\textstyle {\textbf {a}}}$ are perpendicular to each other.
3. Bases ${\textstyle AD}$ and ${\textstyle BC}$ of trapezoid ${\textstyle ABCD}$ are in the ratio of ${\textstyle 4:1}$. The diagonals of the trapezoid intersect at point ${\textstyle M}$ and the extensions of sides ${\textstyle AB}$ and ${\textstyle CD}$ intersect at point ${\textstyle P}$. Let us consider the basis with ${\textstyle A}$ as the origin, ${\textstyle {\overrightarrow {AD}}}$ and ${\textstyle {\overrightarrow {AB}}}$ as basis vectors. Find the coordinates of points ${\textstyle M}$ and ${\textstyle P}$ in this basis.
4. 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 ${\textstyle 3:1}$, the longer segments being on the side of the vertex of the tetrahedron.
5. Find ${\textstyle A+B}$ and ${\textstyle 2A-3B+I}$.
6. Find the products ${\textstyle AB}$ and ${\textstyle BA}$ (and so make sure that, in general, ${\textstyle AB\neq BA}$ for matrices).
7. Find the inverse matrices for the given ones.
8. Find the determinants of the given matrices.
9. Point ${\textstyle M}$ is the centroid of face ${\textstyle BCD}$ of tetrahedron ${\textstyle ABCD}$. The old coordinate system is given by ${\textstyle A}$, ${\textstyle {\overrightarrow {AB}}}$, ${\textstyle {\overrightarrow {AC}}}$, ${\textstyle {\overrightarrow {AD}}}$, and the new coordinate system is given by ${\textstyle M}$, ${\textstyle {\overrightarrow {MB}}}$, ${\textstyle {\overrightarrow {MC}}}$, ${\textstyle {\overrightarrow {MA}}}$. Find the coordinates of a point in the old coordinate system given its coordinates ${\textstyle x'}$, ${\textstyle y'}$, ${\textstyle z'}$ in the new one.

Section 2

1. Two lines are given by the equations ${\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}$ and ${\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}$, and at that ${\textstyle {\textbf {a}}\cdot {\textbf {n}}\neq 0}$. Find the position vector of the intersection point of these lines.
2. Find the distance from point ${\textstyle M_{0}}$ with the position vector ${\textstyle {\textbf {r}}_{0}}$ to the line defined by the equation (a) ${\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}$; (b) ${\textstyle {\textbf {r}}\cdot {\textbf {n}}=A}$.
3. Diagonals of a rhombus intersect at point ${\textstyle M(1;\,2)}$, the longest of them being parallel to a horizontal axis. The side of the rhombus equals 2 and its obtuse angle is ${\textstyle 120^{\circ }}$. Compose the equations of the sides of this rhombus.
4. Compose the equations of lines passing through point ${\textstyle A(2;-4)}$ and forming angles of ${\textstyle 60^{\circ }}$ with the line ${\textstyle {\frac {1-2x}{3}}={\frac {3+2y}{-2}}}$.
5. Find the cross product of (a) vectors ${\textstyle {\textbf {a}}(3;-2;\,1)}$ and ${\textstyle {\textbf {b}}(2;-5;-3)}$; (b) vectors ${\textstyle {\textbf {a}}(3;-2;\,1)}$ and ${\textstyle {\textbf {c}}(-18;\,12;-6)}$.
6. A triangle is constructed on vectors ${\textstyle {\textbf {a}}(2;4;-1)}$ and ${\textstyle {\textbf {b}}(-2;1;1)}$. (a) Find the area of this triangle. (b) Find the altitudes of this triangle.
7. Find the scalar triple product of ${\textstyle {\textbf {a}}(1;\,2;-1)}$, ${\textstyle {\textbf {b}}(7;3;-5)}$, ${\textstyle {\textbf {c}}(3;\,4;-3)}$.
8. It is known that basis vectors ${\textstyle {\textbf {e}}_{1}}$, ${\textstyle {\textbf {e}}_{2}}$, ${\textstyle {\textbf {e}}_{3}}$ have lengths of ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 2{\sqrt {2}}}$ respectively, and ${\textstyle \angle ({\textbf {e}}_{1},{\textbf {e}}_{2})=120^{\circ }}$, ${\textstyle \angle ({\textbf {e}}_{1},{\textbf {e}}_{3})=135^{\circ }}$, ${\textstyle \angle ({\textbf {e}}_{2},{\textbf {e}}_{3})=45^{\circ }}$. Find the volume of a parallelepiped constructed on vectors with coordinates ${\textstyle (-1;\,0;\,2)}$, ${\textstyle (1;\,1\,4)}$ and ${\textstyle (-2;\,1;\,1)}$ in this basis.

Section 3

1. Prove that a curve given by ${\textstyle 34x^{2}+24xy+41y^{2}-44x+58y+1=0}$ 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.
2. 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) ${\textstyle 9x^{2}-16y^{2}-6x+8y-144=0}$; (b) ${\textstyle 9x^{2}+4y^{2}+6x-4y-2=0}$; (c) ${\textstyle 12x^{2}-12x-32y-29=0}$; (d) ${\textstyle xy+2x+y=0}$;
3. Find the equations of lines tangent to curve ${\textstyle 6xy+8y^{2}-12x-26y+11=0}$ that are (a) parallel to line ${\textstyle 6x+17y-4=0}$; (b) perpendicular to line ${\textstyle 41x-24y+3=0}$; (c) parallel to line ${\textstyle y=2}$.
4. For each value of parameter ${\textstyle a}$ determine types of surfaces given by the equations: (a) ${\textstyle x^{2}+y^{2}-z^{2}=a}$; (b) ${\textstyle x^{2}+a\left(y^{2}+z^{2}\right)=1}$; (c) ${\textstyle x^{2}+ay^{2}=az}$; (d) ${\textstyle x^{2}+ay^{2}=az+1}$.
5. Find a vector equation of a right circular cone with apex ${\textstyle M_{0}\left({\textbf {r}}_{0}\right)}$ and axis ${\textstyle {\textbf {r}}={\textbf {r}}_{0}+{\textbf {a}}t}$ if it is known that generatrices of this cone form the angle of ${\textstyle \alpha }$ with its axis.
6. Find the equation of a cylinder with radius ${\textstyle {\sqrt {2}}}$ that has an axis ${\textstyle x=1+t}$, ${\textstyle y=2+t}$, ${\textstyle z=3+t}$.
7. An ellipsoid is symmetric with respect to coordinate planes, passes through point ${\textstyle M(3;\,1;\,1)}$ and circle ${\textstyle x^{2}+y^{2}+z^{2}=9}$, ${\textstyle x-z=0}$. 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.