BSc: Analytic Geometry And Linear Algebra II.s23
================================================






Contents
--------


* [1 Analytical Geometry & Linear Algebra – II](#Analytical_Geometry_.26_Linear_Algebra_.E2.80.93_II)
	+ [1.1 Short Description](#Short_Description)
	+ [1.2 Course Topics](#Course_Topics)
	+ [1.3 Intended Learning Outcomes (ILOs)](#Intended_Learning_Outcomes_.28ILOs.29)
		- [1.3.1 ILOs defined at three levels](#ILOs_defined_at_three_levels)
			* [1.3.1.1 Level 1: What concepts should a student know/remember/explain?](#Level_1:_What_concepts_should_a_student_know.2Fremember.2Fexplain.3F)
			* [1.3.1.2 Level 2: What basic practical skills should a student be able to perform?](#Level_2:_What_basic_practical_skills_should_a_student_be_able_to_perform.3F)
			* [1.3.1.3 Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?](#Level_3:_What_complex_comprehensive_skills_should_a_student_be_able_to_apply_in_real-life_scenarios.3F)
	+ [1.4 Grading](#Grading)
		- [1.4.1 Course grading range](#Course_grading_range)
		- [1.4.2 Course activities and grading breakdown](#Course_activities_and_grading_breakdown)
		- [1.4.3 Recommendations for students on how to succeed in the course](#Recommendations_for_students_on_how_to_succeed_in_the_course)
	+ [1.5 Resources, literature and reference materials](#Resources.2C_literature_and_reference_materials)
		- [1.5.1 Open access resources](#Open_access_resources)
	+ [1.6 Activities and Teaching Methods](#Activities_and_Teaching_Methods)
	+ [1.7 Formative Assessment and Course Activities](#Formative_Assessment_and_Course_Activities)
		- [1.7.1 Ongoing performance assessment](#Ongoing_performance_assessment)
			* [1.7.1.1 Section 1](#Section_1)
			* [1.7.1.2 Section 2](#Section_2)
			* [1.7.1.3 Section 3](#Section_3)
			* [1.7.1.4 Section 4](#Section_4)
		- [1.7.2 Final assessment](#Final_assessment)
			* [1.7.2.1 Section 1](#Section_1_2)
			* [1.7.2.2 Section 2](#Section_2_2)
			* [1.7.2.3 Section 3](#Section_3_2)
			* [1.7.2.4 Section 4](#Section_4_2)
		- [1.7.3 The retake exam](#The_retake_exam)



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 | 1. The geometry of linear equations. Elimination with matrices
2. Matrix operations, including inverses
3. LU and LDU factorization
4. Transposes and permutations
5. Vector spaces and subspaces
6. The null space: Solving Ax=0 and Ax=b
7. Row reduced echelon form. Matrix rank
8. Numerical methods for solving systems of linear algebraic equations
 |
| Linear regression analysis, QR-decomposition | 1. Independence, basis and dimension
2. The four fundamental subspaces
3. Orthogonal vectors and subspaces
4. Projections onto subspaces Projection matrices
5. Least squares approximations
6. Gram-Schmidt orthogonalization and A = QR
 |
| Matrix Diagonalization | 1. Complex Numbers
2. Hermitian and Unitary Matrices
3. Eigenvalues and eigenvectors
4. Matrix diagonalization
 |
| Symmetric, positive definite and similar matrices. Singular value decomposition | 1. Linear differential equations.
2. Symmetric matrices.
3. Positive definite matrices
4. Similar matrices.
5. Left and right inverses, pseudoinverse
6. 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 | 90-110 | -
 |
| B. Good | 75-89 | -
 |
| C. Satisfactory | 60-74 | -
 |
| D. Fail | 0-59 | -
 |


### Course activities and grading breakdown





| Activity Type | Percentage of the overall course grade
 |
| --- | --- |
| Midterm | 30
 |
| Two intermediate tests | 30 (15 for each)
 |
| Final exam | 30
 |
| Five programming tasks | 20
 |


### 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


Activities and Teaching Methods
-------------------------------




Teaching and Learning Methods within each section
| Teaching Techniques | Section 1 | Section 2 | Section 3 | Section 4
 |
| --- | --- | --- | --- | --- |
| Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) | 0 | 0 | 0 | 0
 |
| Project-based learning (students work on a project) | 0 | 0 | 0 | 0
 |
| Modular learning (facilitated self-study) | 0 | 0 | 0 | 0
 |
| Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level) | 1 | 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 | 0
 |
| Business game (learn by playing a game that incorporates the principles of the material covered within the course) | 0 | 0 | 0 | 0
 |
| Inquiry-based learning | 0 | 0 | 0 | 0
 |
| Just-in-time teaching | 0 | 0 | 0 | 0
 |
| Process oriented guided inquiry learning (POGIL) | 0 | 0 | 0 | 0
 |
| Studio-based learning | 0 | 0 | 0 | 0
 |
| Universal design for learning | 0 | 0 | 0 | 0
 |
| Task-based learning | 0 | 0 | 0 | 0
 |




Activities within each section
| Learning Activities | Section 1 | Section 2 | Section 3 | Section 4
 |
| --- | --- | --- | --- | --- |
| Lectures | 1 | 1 | 1 | 1
 |
| Interactive Lectures | 1 | 1 | 1 | 1
 |
| Lab exercises | 1 | 1 | 1 | 1
 |
| Experiments | 0 | 0 | 0 | 0
 |
| Modeling | 0 | 0 | 0 | 0
 |
| Cases studies | 0 | 0 | 0 | 0
 |
| Development of individual parts of software product code | 0 | 0 | 0 | 0
 |
| Individual Projects | 0 | 0 | 0 | 0
 |
| Group projects | 0 | 0 | 0 | 0
 |
| Flipped classroom | 0 | 0 | 0 | 0
 |
| Quizzes (written or computer based) | 1 | 1 | 1 | 1
 |
| Peer Review | 0 | 0 | 0 | 0
 |
| Discussions | 1 | 1 | 1 | 1
 |
| Presentations by students | 0 | 0 | 0 | 0
 |
| Written reports | 0 | 0 | 0 | 0
 |
| Simulations and role-plays | 0 | 0 | 0 | 0
 |
| Essays | 0 | 0 | 0 | 0
 |
| Oral Reports | 0 | 0 | 0 | 0
 |


Formative Assessment and Course Activities
------------------------------------------


### Ongoing performance assessment


#### Section 1


1. How to perform Gauss elimination?
2. How to perform matrices multiplication?
3. How to perform LU factorization?
4. How to find complete solution for any linear equation system Ax=b?


#### Section 2


1. What is linear independence of vectors?
2. Define the four fundamental subspaces of a matrix?
3. How to define orthogonal vectors and subspaces?
4. How to define orthogonal complements of the space?
5. How to find vector projection on a subspace?
6. How to perform linear regression for the given measurements?
7. How to find an orthonormal basis for the subspace spanned by the given vectors?


#### Section 3


1. Give the definition of Hermitian Matrix.
2. Give the definition of Unitary Matrix.
3. How to find matrix for the Fourier transform?
4. When we can make fast Fourier transform?
5. How to find eigenvalues and eigenvectors of a matrix?
6. How to diagonalize a square matrix?


#### Section 4


1. How to solve linear differential equations?
2. Make the definition of symmetric matrix?
3. Make the definition of positive definite matrix?
4. Make the definition of similar matrices?
5. How to find left and right inverses matrices, pseudoinverse matrix?
6. How to make singular value decomposition of the matrix?


### Final assessment


#### Section 1


1. Find linear independent vectors (exclude dependent): 





a
→


=
[
4
,
0
,
3
,
2

]

T




{\textstyle {\overrightarrow {a}}=[4,0,3,2]^{T}}

![{\textstyle {\overrightarrow {a}}=[4,0,3,2]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b573acd686425e4bb95dcc7342a0c8cdff33e546), 





b
→


=
[
1
,
−
7
,
4
,
5

]

T




{\textstyle {\overrightarrow {b}}=[1,-7,4,5]^{T}}

![{\textstyle {\overrightarrow {b}}=[1,-7,4,5]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f97f34707d7d042d0080df772a5cc5bf030c784), 





c
→


=
[
7
,
1
,
5
,
3

]

T




{\textstyle {\overrightarrow {c}}=[7,1,5,3]^{T}}

![{\textstyle {\overrightarrow {c}}=[7,1,5,3]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77c5a21e300ddbdd0f7fecdd617b910827046006), 





d
→


=
[
−
5
,
−
3
,
−
3
,
−
1

]

T




{\textstyle {\overrightarrow {d}}=[-5,-3,-3,-1]^{T}}

![{\textstyle {\overrightarrow {d}}=[-5,-3,-3,-1]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/424ab1d843887cea7abfa3dca20106545d8a8a0b), 





e
→


=
[
1
,
−
5
,
2
,
3

]

T




{\textstyle {\overrightarrow {e}}=[1,-5,2,3]^{T}}

![{\textstyle {\overrightarrow {e}}=[1,-5,2,3]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b9b7357251158c6b1954e17b912f77a1e38c4b). Find 



r
a
n
k
(
A
)


{\textstyle rank(A)}

![{\textstyle rank(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af25f6717bd9130cfded69f3333e47486b7d71cc) if 



A


{\textstyle A}

![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31) is a composition of this vectors. Find 



r
a
n
k
(

A

T


)


{\textstyle rank(A^{T})}

![{\textstyle rank(A^{T})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/671f71d97f14e03b5a9b0e0fb5724ef62720a731).
2. Find 



E


{\textstyle E}

![{\textstyle E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d934f67126f64e9c061b598b8941b8767a8d343): 



E
A
=
U


{\textstyle EA=U}

![{\textstyle EA=U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6a98a37022c82117c658535da16aecb68ac071) (



U


{\textstyle U}

![{\textstyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/087a7d8a39fe35012cbf2f561879f9e975cb4555) – upper-triangular matrix). Find 



L
=

E

−


1


{\textstyle L=E^{-}1}

![{\textstyle L=E^{-}1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e415e968cd0834b022ca5585654bf2ba668c5954), if 



A
=

(




2


5


7




6


4


9




4


1


8




)



{\textstyle A=\left({\begin{array}{ccc}2&5&7\\6&4&9\\4&1&8\\\end{array}}\right)}

![{\textstyle A=\left({\begin{array}{ccc}2&5&7\\6&4&9\\4&1&8\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/270385e97a68b305e3c25e81e8ef90cc0b8dc9cc).
3. Find complete solution for the system 



A
x
=
b


{\textstyle Ax=b}

![{\textstyle Ax=b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18892a5ddb5ea2e30e8a4a9488b35d67a3fdbf6d), if 



b
=
[
7
,
18
,
5

]

T




{\textstyle b=[7,18,5]^{T}}

![{\textstyle b=[7,18,5]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4faa204bdd0b26fb09cd8abebf3124e7a6b8ca1e) and 



A
=

(




6


−
2


1


−
4




4


2


14


−
31




2


−
1


3


−
7




)



{\textstyle A=\left({\begin{array}{cccc}6&-2&1&-4\\4&2&14&-31\\2&-1&3&-7\\\end{array}}\right)}

![{\textstyle A=\left({\begin{array}{cccc}6&-2&1&-4\\4&2&14&-31\\2&-1&3&-7\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a27c93084b6e998864553331b01fc5eb2339f12). Provide an example of vector b that makes this system unsolvable.


#### Section 2


1. Find the dimensions of the four fundamental subspaces associated with 



A


{\textstyle A}

![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31), depending on the parameters 



a


{\textstyle a}

![{\textstyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a503f107a7c104e40e484cee9e1f5993d28ffd8) and 



b


{\textstyle b}

![{\textstyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a780b69dfc55238880ef18a134dc65260877e2): 



A
=

(




7


8


5


3




4


a


3


2




6


8


4


b




3


4


2


1




)



{\textstyle A=\left({\begin{array}{cccc}7&8&5&3\\4&a&3&2\\6&8&4&b\\3&4&2&1\\\end{array}}\right)}

![{\textstyle A=\left({\begin{array}{cccc}7&8&5&3\\4&a&3&2\\6&8&4&b\\3&4&2&1\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc803202bc5e0460954845a195e4b92208ac2a54).
2. Find a vector 



x


{\textstyle x}

![{\textstyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781) orthogonal to the Row space of matrix 



A


{\textstyle A}

![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31), and a vector 



y


{\textstyle y}

![{\textstyle y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db9936ddb2761b76fa640fb275cb5d1fa4d6fa23) orthogonal to the 



C
(
A
)


{\textstyle C(A)}

![{\textstyle C(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50e890f8d67ec1b4b4fd8b7a6fbbfb8172f20866), and a vector 



z


{\textstyle z}

![{\textstyle z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a33e37010e3acdeeb80fdb95df9bfe411fd79e6) orthogonal to the 



N
(
A
)


{\textstyle N(A)}

![{\textstyle N(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/766235dfa8f2a0cbce4df704eb334785c7094ca4): 



A
=

(




1


2


2




3


4


2




4


6


4




)



{\textstyle A=\left({\begin{array}{ccc}1&2&2\\3&4&2\\4&6&4\\\end{array}}\right)}

![{\textstyle A=\left({\begin{array}{ccc}1&2&2\\3&4&2\\4&6&4\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c65bde22a0e3936c85821df45484f3216a7dd74e).
3. Find the best straight-line 



y
(
x
)


{\textstyle y(x)}

![{\textstyle y(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4639e7d86a9d2274f64a48570e7fe4ef17f7efa) fit to the measurements: 



y
(
−
2
)
=
4


{\textstyle y(-2)=4}

![{\textstyle y(-2)=4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75edda9fdb1aa1cab6ac772869d7441bdf5539ef), 



y
(
−
1
)
=
3


{\textstyle y(-1)=3}

![{\textstyle y(-1)=3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39365fc4327d63e6b3f4568f085c953942a32bcc), 



y
(
0
)
=
2


{\textstyle y(0)=2}

![{\textstyle y(0)=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4321858864c6ee38bc8ea6a44b43c1fb7b1b7bdb), 



y
(
1
)
−
0


{\textstyle y(1)-0}

![{\textstyle y(1)-0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ac0d6451cf2f604f303ed26f3e1b26941b680a).
4. Find the projection matrix 



P


{\textstyle P}

![{\textstyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038590207af1024a629c1a08c855e9ac46bf5610) of vector 



[
4
,
3
,
2
,
0

]

T




{\textstyle [4,3,2,0]^{T}}

![{\textstyle [4,3,2,0]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1407743b577f1595b88b73ce535c1b1c6810d14) onto the 



C
(
A
)


{\textstyle C(A)}

![{\textstyle C(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50e890f8d67ec1b4b4fd8b7a6fbbfb8172f20866): 



A
=

(




1


−
2




1


−
1




1


0




1


1




)



{\textstyle A=\left({\begin{array}{cc}1&-2\\1&-1\\1&0\\1&1\\\end{array}}\right)}

![{\textstyle A=\left({\begin{array}{cc}1&-2\\1&-1\\1&0\\1&1\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42f35bb9ce7dd23b55e504ef145789134642fd56).
5. Find an orthonormal basis for the subspace spanned by the vectors: 





a
→


=
[
−
2
,
2
,
0
,
0

]

T




{\textstyle {\overrightarrow {a}}=[-2,2,0,0]^{T}}

![{\textstyle {\overrightarrow {a}}=[-2,2,0,0]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d10ddb57c762601e62967544737ea022841f1c7e), 





b
→


=
[
0
,
1
,
−
1
,
0

]

T




{\textstyle {\overrightarrow {b}}=[0,1,-1,0]^{T}}

![{\textstyle {\overrightarrow {b}}=[0,1,-1,0]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a91f7a5cfe5477b27b926bf40a2ab8579777b444), 





c
→


=
[
0
,
1
,
0
,
−
1

]

T




{\textstyle {\overrightarrow {c}}=[0,1,0,-1]^{T}}

![{\textstyle {\overrightarrow {c}}=[0,1,0,-1]^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6018717796286d4e371639685094a4653c67237b). Then express 



A
=
[
a
,
b
,
c
]


{\textstyle A=[a,b,c]}

![{\textstyle A=[a,b,c]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/473b98961e85b063ef9a834f6f8a2caeeeb8b199) in the form of 



A
=
Q
R


{\textstyle A=QR}

![{\textstyle A=QR}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7960350ee376840d36b9f7d47d761d8112fbfe)


#### Section 3


1. Find eigenvector of the circulant matrix 



C


{\textstyle C}

![{\textstyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dca76d9ff4b48256b6a4a99bcb234b64b2fa72b) for the eigenvalue = 





c


1




{\textstyle {c}\_{1}}

![{\textstyle {c}_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0405b69bc9082831ceeeb8fa7936239466318b4e)+





c


2




{\textstyle {c}\_{2}}

![{\textstyle {c}_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a63a7c192e428b5b247dedd7a2dc5d9d2bc4ade)+





c


3




{\textstyle {c}\_{3}}

![{\textstyle {c}_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd3760c114db3e5fbe4f814e7755ce35617c2b2)+





c


4




{\textstyle {c}\_{4}}

![{\textstyle {c}_{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54dd52f52cb3ef9c6a5591247f6625340fd561ef): 



C
=

(






c


1






c


2






c


3






c


4








c


4






c


1






c


2






c


3








c


3






c


4






c


1






c


2








c


2






c


3






c


4






c


1






)



{\textstyle C=\left({\begin{array}{cccc}{c}\_{1}&{c}\_{2}&{c}\_{3}&{c}\_{4}\\{c}\_{4}&{c}\_{1}&{c}\_{2}&{c}\_{3}\\{c}\_{3}&{c}\_{4}&{c}\_{1}&{c}\_{2}\\{c}\_{2}&{c}\_{3}&{c}\_{4}&{c}\_{1}\\\end{array}}\right)}

![{\textstyle C=\left({\begin{array}{cccc}{c}_{1}&{c}_{2}&{c}_{3}&{c}_{4}\\{c}_{4}&{c}_{1}&{c}_{2}&{c}_{3}\\{c}_{3}&{c}_{4}&{c}_{1}&{c}_{2}\\{c}_{2}&{c}_{3}&{c}_{4}&{c}_{1}\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69efe5a0d7045da18110c47c68ba00309abb5323).
2. Diagonalize this matrix: 



A
=

(




2


1
−
i




1
+
i


3




)



{\textstyle A=\left({\begin{array}{cc}2&1-i\\1+i&3\\\end{array}}\right)}

![{\textstyle A=\left({\begin{array}{cc}2&1-i\\1+i&3\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de9882c4748ca263e22135e69cccce208a645a47).
3. A


{\textstyle A}

![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31) is the matrix with full set of orthonormal eigenvectors. Prove that 



A
A
=

A

H



A

H




{\textstyle AA=A^{H}A^{H}}

![{\textstyle AA=A^{H}A^{H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79272abd7682c09ed733b7808f9b5a49b522957d).
4. Find all eigenvalues and eigenvectors of the cyclic permutation matrix 



P
=

(




0


1


0


0




0


0


1


0




0


0


0


1




1


0


0


0




)



{\textstyle P=\left({\begin{array}{cccc}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\\\end{array}}\right)}

![{\textstyle P=\left({\begin{array}{cccc}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf3bc0ef61513dc72ccbe9d43301d8f22ac731f).


#### Section 4


1. Find 



d
e
t
(

e

A


)


{\textstyle det(e^{A})}

![{\textstyle det(e^{A})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe0bc4f6b8861e3358b76828c70affbe91b5546) for 



A
=

(




2


1




2


3




)



{\textstyle A=\left({\begin{array}{cc}2&1\\2&3\\\end{array}}\right)}

![{\textstyle A=\left({\begin{array}{cc}2&1\\2&3\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b79dfa2ad1a2c18b276ba62242becf7749a0b4).
2. Write down the first order equation system for the following differential equation and solve it:







d

3


y

/

d
x
+

d

2


y

/

d
x
−
2
d
y

/

d
x
=
0


{\textstyle d^{3}y/dx+d^{2}y/dx-2dy/dx=0}

![{\textstyle d^{3}y/dx+d^{2}y/dx-2dy/dx=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e608e72541239be7801b94dd8fed3e355050fc)







y
″

(
0
)
=
6


{\textstyle y''(0)=6}

![{\textstyle y''(0)=6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92039ff6d21382f32158e6eef4642cb55688b549), 




y
′

(
0
)
=
0


{\textstyle y'(0)=0}

![{\textstyle y'(0)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8839484b7380ec6af6ed1f232536ae98cf093a), 



y
(
0
)
=
3


{\textstyle y(0)=3}

![{\textstyle y(0)=3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7459acffb9257b7b7f64ba0c5d24c4ca651280f5).


Is the solution of this system will be stable?
3. For which 



a


{\textstyle a}

![{\textstyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a503f107a7c104e40e484cee9e1f5993d28ffd8) and 



b


{\textstyle b}

![{\textstyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a780b69dfc55238880ef18a134dc65260877e2) quadratic form 



Q
(
x
,
y
,
z
)


{\textstyle Q(x,y,z)}

![{\textstyle Q(x,y,z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/723d35bcca34d49404fcc5d0e7cca10bbaf9ec8c) is positive definite:






Q
(
x
,
y
,
z
)
=
a

x

2


+

y

2


+
2

z

2


+
2
b
x
y
+
4
x
z


{\textstyle Q(x,y,z)=ax^{2}+y^{2}+2z^{2}+2bxy+4xz}

![{\textstyle Q(x,y,z)=ax^{2}+y^{2}+2z^{2}+2bxy+4xz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f831bc4412b380c9a2c2940d4cd3dd6f46d7f3dc)
4. Find the SVD and the pseudoinverse of the matrix 



A
=

(




1


0


0




0


1


1




)



{\textstyle A=\left({\begin{array}{ccc}1&0&0\\0&1&1\\\end{array}}\right)}

![{\textstyle A=\left({\begin{array}{ccc}1&0&0\\0&1&1\\\end{array}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/152a05619788dda3e990b8af6b737277189c507d).


### The retake exam


Exams will be paper-based and will be conducted in a form of problem solving, where the problems will be similar to those mentioned above. Students will be given 1-2 hours to complete the exam.
First retake will be conducted in the same form as the midterm and final exams. The weight of the retake exam will be the same as the all course.
Second retake will be conducted in the same form as the midterm and final exams. The weight of the retake exam will be the same as the all course.