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= Analytical Geometry & Linear Algebra – II =
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= MathematicalAnalysis II =
  
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* '''Course name''': MathematicalAnalysis II
 +
* '''Course name''': Analytical Geometry & Linear Algebra – II
 
* '''Code discipline''':
 
* '''Code discipline''':
  +
* '''Subject area''': fundamental principles of linear algebra,; concepts of linear algebra objects and their representation in vector-matrix form
−
* '''Subject area''': Multivariate calculus: derivatives, differentials, maxima and minima, Multivariate integration, Functional series. Fourier series, Integrals with parameters
  
   
 
== Short Description ==
  
== Short Description ==
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! Section !! Topics within the section
  
! Section !! Topics within the section
 
|-
  
|-
  +
| Linear equation system solving by using the vector-matrix approach ||
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| Differential Analysis of Functions of Several Variables ||
 
  +
# The geometry of linear equations. Elimination with matrices.
−
# Limits of functions of several variables
  
  +
# Matrix operations, including inverses.
−
# Partial and directional derivatives of functions of several variables. Gradient
  
  +
# L
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# Differentials of functions of several variables. Taylor formula
  
  +
# U
−
# Maxima and minima for functions of several variables
  
  +
# {\textstyle LU}
−
# Maxima and minima for functions of several variables subject to a constraint
  
  +
# and
  +
# L
  +
# D
  +
# U
  +
# {\textstyle LDU}
  +
# factorization.
  +
# Transposes and permutations. Vector spaces and subspaces.
  +
# The null space: Solving
  +
# A
  +
# x
  +
# =
  +
# 0
  +
# {\textstyle Ax=0}
  +
# and
  +
# A
  +
# x
  +
# =
  +
# b
  +
# {\textstyle Ax=b}
  +
# . Row reduced echelon form. Matrix rank.
 
|-
  
|-
  +
| Linear regression analysis and decompositionA=QR{\textstyle A=QR}. ||
−
| Integration of Functions of Several Variables ||
 
  +
# Independence, basis and dimension. The four fundamental subspaces.
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# Z-test
  
  +
# Orthogonal vectors and subspaces. Projections onto subspaces
−
# Double integrals. Fubini’s theorem and iterated integrals
  
  +
# Projection matrices. Least squares approximations. Gram-Schmidt and A = QR.
−
# Substituting variables in double integrals. Polar coordinates
  
−
# Triple integrals. Use of Fubini’s theorem
  
−
# Spherical and cylindrical coordinates
  
−
# Path integrals
  
−
# Area of a surface
  
−
# Surface integrals
  
 
|-
  
|-
  +
| Fast Fourier Transform. Matrix Diagonalization. ||
−
| Uniform Convergence of Functional Series. Fourier Series ||
 
  +
# Complex Numbers. Hermitian and Unitary Matrices.
−
# Uniform and point wise convergence of functional series
  
  +
# Fourier Series. The Fast Fourier Transform
−
# Properties of uniformly convergent series
  
  +
# Eigenvalues and eigenvectors. Matrix diagonalization.
−
# Fourier series. Sufficient conditions of convergence and uniform convergence
  
−
# Bessel’s inequality and Parseval’s identity.
  
 
|-
  
|-
  +
| Symmetric, positive definite and similar matrices. Singular value decomposition. ||
−
| Integrals with Parameter(s) ||
 
  +
# Linear differential equations.
−
# Definite integrals with parameters
  
  +
# Symmetric matrices. Positive definite matrices.
−
# Improper integrals with parameters. Uniform convergence
  
  +
# Similar matrices. Left and right inverses, pseudoinverse. Singular value decomposition (SVD).
−
# Properties of uniformly convergent integrals
  
−
# Beta-function and gamma-function
  
−
# Fourier transform
  
 
|}
 
|}
 
== Intended Learning Outcomes (ILOs) ==
  
== Intended Learning Outcomes (ILOs) ==
   
 
=== What is the main purpose of this course? ===
  
=== What is the main purpose of this course? ===
  +
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.
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The goal of the course is to study basic mathematical concepts that will be required in further studies. The course is based on Mathematical Analysis I, and the concepts studied there are widely used in this course. The course covers differentiation and integration of functions of several variables. Some more advanced concepts, as uniform convergence of series and integrals, are also considered, since they are important for understanding applicability of many theorems of mathematical analysis. In the end of the course some useful applications are covered, such as gamma-function, beta-function, and Fourier transform.
  
   
 
=== ILOs defined at three levels ===
  
=== ILOs defined at three levels ===
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==== Level 1: What concepts should a student know/remember/explain? ====
  
==== Level 1: What concepts should a student know/remember/explain? ====
 
By the end of the course, the students should be able to ...
  
By the end of the course, the students should be able to ...
  +
* List basic notions of linear algebra
−
* find partial and directional derivatives of functions of several variables;
  
  +
* Understand key principles involved in solution of linear equation systems and the properties of matrices
−
* find maxima and minima for a function of several variables
  
  +
* Linear regression analysis
−
* use Fubini’s theorem for calculating multiple integrals
  
  +
* Fast Fourier Transform
−
* calculate line and path integrals
  
  +
* How to find eigenvalues and eigenvectors for matrix diagonalization and single value decomposition
−
* distinguish between point wise and uniform convergence of series and improper integrals
  
−
* decompose a function into Fourier series
  
−
* calculate Fourier transform of a function
  
   
 
==== 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? ====
 
By the end of the course, the students should be able to ...
  
By the end of the course, the students should be able to ...
  +
* Key principles involved in solution of linear equation systems and the properties of matrices
−
* how to find minima and maxima of a function subject to a constraint
  
  +
* Become familiar with the four fundamental subspaces
−
* how to represent double integrals as iterated integrals and vice versa
  
  +
* Linear regression analysis
−
* what the length of a curve and the area of a surface is
  
  +
* Fast Fourier Transform
−
* properties of uniformly convergent series and improper integrals
  
  +
* How to find eigenvalues and eigenvectors for matrix diagonalization and single value decomposition
−
* beta-function, gamma-function and their properties
  
−
* how to find Fourier transform of a function
  
   
 
==== 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? ====
 
By the end of the course, the students should be able to ...
  
By the end of the course, the students should be able to ...
  +
* Linear equation system solving by using the vector-matrix approach
−
* find multiple, path, surface integrals
  
  +
* Make linear regression analysis
−
* find the range of a function in a given domain
  
−
* decompose a function into Fourier series
+
* Fast Fourier Transform
  +
* To find eigenvalues and eigenvectors for matrix diagonalization and single value decomposition
 
== Grading ==
  
== Grading ==
   
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| B. Good || 65-84 || -
  
| B. Good || 65-84 || -
 
|-
  
|-
−
| C. Satisfactory || 45-64 || -
 +
| C. Satisfactory || 50-64 || -
 
|-
  
|-
−
| D. Poor || 0-44 || -
 +
| D. Poor || 0-49 || -
 
|}
  
|}
   
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! Activity Type !! Percentage of the overall course grade
  
! Activity Type !! Percentage of the overall course grade
 
|-
  
|-
−
| Test 1 || 10
 +
| Labs/seminar classes || 20
 
|-
  
|-
  +
| Interim performance assessment || 30
−
| Midterm || 25
  
 
|-
  
|-
−
| Test 2 || 10
 +
| Exams || 50
−
|-
  
−
| Participation || 5
  
−
|-
  
−
| Final exam || 50
  
 
|}
  
|}
   
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=== Open access resources ===
  
=== Open access resources ===
  +
−
* Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson
  
−
* Jerrold Marsden, Alan Weinstein (1985) Calculus (in three volumes; volumes 2 and 3), Springer
  
   
 
=== Closed access resources ===
  
=== Closed access resources ===
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|-
  
|-
 
! Learning Activities !! Section 1 !! Section 2 !! Section 3 !! Section 4
  
! Learning Activities !! Section 1 !! Section 2 !! Section 3 !! Section 4
  +
|-
  +
| Development of individual parts of software product code || 1 || 1 || 1 || 1
 
|-
  
|-
 
| Homework and group projects || 1 || 1 || 1 || 1
  
| Homework and group projects || 1 || 1 || 1 || 1
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! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
  +
| Question || How to perform Gauss elimination? || 1
−
| Question || Find <math>{\textstyle \lim \limits _{x\to 0}\lim \limits _{y\to 0}u(x;y)}</math> , <math>{\textstyle \lim \limits _{y\to 0}\lim \limits _{x\to 0}u(x;y)}</math> and <math>{\textstyle \lim \limits _{(x;y)\to (0;0)}u(x;y)}</math> if <math>{\textstyle u(x;y)={\frac {x^{2}y+xy^{2}}{x^{2}-xy+y^{2}}}}</math> . || 1
  
 
|-
  
|-
  +
| Question || How to perform matrices multiplication? || 1
−
| Question || Find the differential of a function: (a) <math>{\textstyle u(x;y)=\ln \left(x+{\sqrt {x^{2}+y^{2}}}\right)}</math> ; (b) <math>{\textstyle u(x;y)=\ln \sin {\frac {x+1}{\sqrt {y}}}}</math> . || 1
  
 
|-
  
|-
  +
| Question || How to perform LU factorization? || 1
−
| Question || Find the differential of <math>{\textstyle u(x;y)}</math> given implicitly by an equation <math>{\textstyle x^{3}+2y^{3}+u^{3}-3xyu+2y-3=0}</math> at points <math>{\textstyle M(1;1;1)}</math> and <math>{\textstyle N(1;1;-2)}</math> . || 1
  
 
|-
  
|-
  +
| Question || How to find complete solution for any linear equation system Ax=b? || 1
−
| Question || Find maxima and minima of a function subject to a constraint (or several constraints):<br><math>{\textstyle u=x^{2}y^{3}z^{4}}</math> , <math>{\textstyle 2x+3y+4z=18}</math> , <math>{\textstyle x>0}</math> , <math>{\textstyle y>0}</math> , <math>{\textstyle z>0}</math> ;<br><math>{\textstyle u=x-y+2z}</math> , <math>{\textstyle x^{2}+y^{2}+2z^{2}=16}</math> ;<br><math>{\textstyle u=\sum \limits _{i=1}^{k}a_{i}x_{i}^{2}}</math> , <math>{\textstyle \sum \limits _{i=1}^{k}x_{i}=1}</math> , <math>{\textstyle a_{i}>0}</math> ; || 1
  
 
|-
  
|-
  +
| Question || Find the solution for the given linear equation system <math>{\textstyle Ax=b}</math> by using Gauss elimination. || 0
−
| Question || Let us consider <math>{\textstyle u(x;y)={\begin{cases}1,&x=y^{2},\\0,&x\neq y^{2}.\end{cases}}}</math> Show that this function has a limit at the origin along any straight line that passes through it (and all these limits are equal to each other), yet this function does not have limit as <math>{\textstyle (x;y)\to (0;0)}</math> . || 0
  
 
|-
  
|-
−
| Question || Find the largest possible value of directional derivative at point <math>{\textstyle M(1;-2;-3)}</math> of function <math>{\textstyle f=\ln xyz}</math> . || 0
 +
| Question || Perform <math>{\textstyle A=LU}</math> factorization for the given matrix <math>{\textstyle A}</math> . || 0
 
|-
  
|-
−
| Question || Find maxima and minima of functions <math>{\textstyle u(x,y)}</math> given implicitly by the equations:<br><math>{\textstyle x^{2}+y^{2}+u^{2}-4x-6y-4u+8=0}</math> , <math>{\textstyle u>2}</math> ;<br><math>{\textstyle x^{3}-y^{2}+u^{2}-3x+4y+u-8=0}</math> . || 0
 +
| Question || Factor the given symmetric matrix <math>{\textstyle A}</math> into <math>{\textstyle A=LDL^{T}}</math> with the diagonal pivot matrix <math>{\textstyle D}</math> . || 0
 
|-
  
|-
−
| Question || Find maxima and minima of functions subject to constraints:<br><math>{\textstyle u=xy^{2}}</math> , <math>{\textstyle x+2y-1=0}</math> ;<br><math>{\textstyle u=xy+yz}</math> , <math>{\textstyle x^{2}+y^{2}=2}</math> , <math>{\textstyle y+z=2}</math> , <math>{\textstyle y>0}</math> . || 0
 +
| Question || Find inverse matrix <math>{\textstyle A^{-}1}</math> for the given matrix <math>{\textstyle A}</math> . || 0
 
|}
 
|}
 
==== Section 2 ====
  
==== Section 2 ====
Line 179: Line 186:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
  +
| Question || What is linear independence of vectors? || 1
−
| Question || Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: <math>{\textstyle \iint \limits _{D}f(x;y)\,dx\,dy}</math> where <math>{\textstyle D=\left\{(x;y)\left|x^{2}+y^{2}\leq 9,\,x^{2}+(y+4)^{2}\geq 25\right.\right\}}</math> . || 1
  
 
|-
  
|-
  +
| Question || Define the four fundamental subspaces of a matrix? || 1
−
| Question || Represent integral <math>{\textstyle I=\displaystyle \iiint \limits _{D}f(x;y;z)\,dx\,dy\,dz}</math> as iterated integrals with all possible (i.e. 6) orders of integration; <math>{\textstyle D}</math> is bounded by <math>{\textstyle x=0}</math> , <math>{\textstyle x=a}</math> , <math>{\textstyle y=0}</math> , <math>{\textstyle y={\sqrt {ax}}}</math> , <math>{\textstyle z=0}</math> , <math>{\textstyle z=x+y}</math> . || 1
  
 
|-
  
|-
  +
| Question || How to define orthogonal vectors and subspaces? || 1
−
| Question || Find line integrals of a scalar fields <math>{\textstyle \displaystyle \int \limits _{\Gamma }(x+y)\,ds}</math> where <math>{\textstyle \Gamma }</math> is boundary of a triangle with vertices <math>{\textstyle (0;0)}</math> , <math>{\textstyle (1;0)}</math> and <math>{\textstyle (0;1)}</math> . || 1
  
 
|-
  
|-
  +
| Question || How to define orthogonal complements of the space? || 1
−
| Question || Change order of integration in the iterated integral <math>{\textstyle \int \limits _{0}^{\sqrt {2}}dy\int \limits _{y}^{\sqrt {4-y^{2}}}f(x;y)\,dx}</math> . || 0
  
 
|-
  
|-
  +
| Question || How to find vector projection on a subspace? || 1
−
| Question || Find the volume of a solid given by <math>{\textstyle 0\leq z\leq x^{2}}</math> , <math>{\textstyle x+y\leq 5}</math> , <math>{\textstyle x-2y\geq 2}</math> , <math>{\textstyle y\geq 0}</math> . || 0
  
 
|-
  
|-
  +
| Question || How to perform linear regression for the given measurements? || 1
−
| Question || Change into polar coordinates and rewrite the integral as a single integral: <math>{\textstyle \displaystyle \iint \limits _{G}f\left({\sqrt {x^{2}+y^{2}}}\right)\,dx\,dy}</math> , <math>{\textstyle G=\left\{(x;y)\left|x^{2}+y^{2}\leq x;\,x^{2}+y^{2}\leq y\right.\right\}}</math> . || 0
  
 
|-
  
|-
  +
| Question || How to find an orthonormal basis for the subspace spanned by the given vectors? || 1
−
| Question || Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at <math>{\textstyle A}</math> and finishes at <math>{\textstyle B}</math> : <math>{\textstyle \displaystyle \int \limits _{\Gamma }\left(x^{4}+4xy^{3}\right)\,dx+\left(6x^{2}y^{2}-5y^{4}\right)\,dy}</math> , <math>{\textstyle A(-2;-1)}</math> , <math>{\textstyle B(0;3)}</math> ; || 0
  
  +
|-
  +
| Question || Check out linear independence of the given vectors || 0
  +
|-
  +
| Question || Find four fundamental subspaces of the given matrix. || 0
  +
|-
  +
| Question || Check out orthogonality of the given subspaces. || 0
  +
|-
  +
| Question || Find orthogonal complement for the given subspace. || 0
  +
|-
  +
| Question || Find vector projection on the given subspace. || 0
  +
|-
  +
| Question || Perform linear regression for the given measurements. || 0
  +
|-
  +
| Question || Find an orthonormal basis for the subspace spanned by the given vectors. || 0
 
|}
 
|}
 
==== Section 3 ====
  
==== Section 3 ====
Line 199: Line 220:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
  +
| Question || Make the definition of Hermitian Matrix. || 1
−
| Question || Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math>{\textstyle \sum \limits _{n=1}^{\infty }e^{-n\left(x^{2}+2\sin x\right)}}</math> , <math>{\textstyle \Delta _{1}=(0;1]}</math> , <math>{\textstyle \Delta _{2}=[1;+\infty )}</math> ; || 1
  
 
|-
  
|-
  +
| Question || Make the definition of Unitary Matrix. || 1
−
| Question || <math>{\textstyle \sum \limits _{n=1}^{\infty }{\frac {\sqrt {nx^{3}}}{x^{2}+n^{2}}}}</math> , <math>{\textstyle \Delta _{1}=(0;1)}</math> , <math>{\textstyle \Delta _{2}=(1;+\infty )}</math> || 1
  
 
|-
  
|-
  +
| Question || How to find matrix for the Fourier transform? || 1
−
| Question || Show that sequence <math>{\textstyle f_{n}(x)=nx\left(1-x\right)^{n}}</math> converges non-uniformly on <math>{\textstyle [0;1]}</math> to a continuous function <math>{\textstyle f(x)}</math> , but <math>{\textstyle \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x)\,dx=\lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x)\,dx}</math> . || 1
  
 
|-
  
|-
  +
| Question || When we can make fast Fourier transform? || 1
−
| Question || Decompose the following function determined on <math>{\textstyle [-\pi ;\pi ]}</math> into Fourier series using the standard trigonometric system <math>{\textstyle \left.\left\{1;\cos kx;\sin kx\right\}\right|_{k=1}^{\infty }}</math> . Draw the graph of the sum of Fourier series obtained. <math>{\textstyle f(x)={\begin{cases}1,\;0\leq x\leq \pi ,\\0,\;-\pi \leq x<0.\end{cases}}}</math> || 1
  
 
|-
  
|-
  +
| Question || How to find eigenvalues and eigenvectors of a matrix? || 1
−
| Question || Prove that if for an absolutely integrable function <math>{\textstyle f(x)}</math> on <math>{\textstyle [-\pi ;\pi ]}</math> <br><math>{\textstyle f(x+\pi )=f(x)}</math> then <math>{\textstyle a_{2k-1}=b_{2k-1}=0}</math> , <math>{\textstyle k\in \mathbb {N} }</math> ;<br><math>{\textstyle f(x+\pi )=-f(x)}</math> then <math>{\textstyle a_{0}=0}</math> , <math>{\textstyle a_{2k}=b_{2k}=0}</math> , <math>{\textstyle k\in \mathbb {N} }</math> . || 1
  
 
|-
  
|-
  +
| Question || How to diagonalize a square matrix? || 1
−
| Question || Show that sequence <math>{\textstyle f_{n}(x)=nx\left(1-x^{2}\right)^{n}}</math> converges on <math>{\textstyle [0;1]}</math> to a continuous function <math>{\textstyle f(x)}</math> , and at that <math>{\textstyle \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x)\,dx\neq \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x)\,dx}</math> . || 0
  
 
|-
  
|-
  +
| Question || Check out is the given matrix Hermitian. || 0
−
| Question || Show that sequence <math>{\textstyle f_{n}(x)=x^{3}+{\frac {1}{n}}\sin \left(nx+{\frac {n\pi }{2}}\right)}</math> converges uniformly on <math>{\textstyle \mathbb {R} }</math> , but <math>{\textstyle \left(\lim \limits _{n\rightarrow +\infty }f_{n}(x)\right)'\neq \lim \limits _{n\rightarrow +\infty }f'_{n}(x)}</math> . || 0
  
 
|-
  
|-
  +
| Question || Check out is the given matrix Unitary. || 0
−
| Question || Decompose <math>{\textstyle \cos \alpha x}</math> , <math>{\textstyle \alpha \notin \mathbb {Z} }</math> into Fourier series on <math>{\textstyle [-\pi ;\pi ]}</math> . Using this decomposition prove that <math>{\textstyle \cot y={\frac {1}{y}}+\sum \limits _{k=1}^{\infty }{\frac {2y}{y^{2}-\pi ^{2}k^{2}}}}</math> . || 0
  
 
|-
  
|-
  +
| Question || Find the matrix for the given Fourier transform. || 0
−
| Question || Function <math>{\textstyle f(x)}</math> is absolutely integrable on <math>{\textstyle [0;\pi ]}</math> , and <math>{\textstyle f(\pi -x)=f(x)}</math> . Prove that<br>if it is decomposed into Fourier series of sines then <math>{\textstyle b_{2k}=0}</math> , <math>{\textstyle k\in \mathbb {N} }</math> ;<br>if it is decomposed into Fourier series of cosines then <math>{\textstyle a_{2k-1}=0}</math> , <math>{\textstyle k\in \mathbb {N} }</math> . || 0
  
 
|-
  
|-
  +
| Question || Find eigenvalues and eigenvectors for the given matrix. || 0
−
| Question || ## Decompose <math>{\textstyle f(x)={\begin{cases}1,\;|x|<\alpha ,\\0,\;\alpha \leqslant |x|<\pi \end{cases}}}</math> into Fourier series using the standard trigonometric system.<br>Using Parseval’s identity find <math>{\textstyle \sigma _{1}=\sum \limits _{k=1}^{\infty }{\frac {\sin ^{2}k\alpha }{k^{2}}}}</math> and <math>{\textstyle \sigma _{2}=\sum \limits _{k=1}^{\infty }{\frac {\cos ^{2}k\alpha }{k^{2}}}}</math> . || 0
  
  +
|-
  +
| Question || Find diagonalize form for the given matrix. || 0
 
|}
 
|}
 
==== Section 4 ====
  
==== Section 4 ====
Line 225: Line 248:
 
! Activity Type !! Content !! Is Graded?
  
! Activity Type !! Content !! Is Graded?
 
|-
  
|-
  +
| Question || How to solve linear differential equations? || 1
−
| Question || Find out if <math>{\textstyle \displaystyle \int \limits _{0}^{1}\left(\lim \limits _{\alpha \to 0}{\frac {2x\alpha ^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}\right)\,dx=\lim \limits _{\alpha \to 0}\int \limits _{0}^{1}{\frac {2x\alpha ^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}\,dx}</math> . || 1
  
  +
|-
  +
| Question || Make the definition of symmetric matrix? || 1
  +
|-
  +
| Question || Make the definition of positive definite matrix? || 1
  +
|-
  +
| Question || Make the definition of similar matrices? || 1
 
|-
  
|-
  +
| Question || How to find left and right inverses matrices, pseudoinverse matrix? || 1
−
| Question || Differentiating the integrals with respect to parameter <math>{\textstyle \varphi }</math> , find it: <math>{\textstyle I(\alpha )=\int \limits _{0}^{\pi /2}\ln \left(\alpha ^{2}-\sin ^{2}\varphi \right)\,d\varphi }</math> , <math>{\textstyle \alpha >1}</math> . || 1
  
 
|-
  
|-
  +
| Question || How to make singular value decomposition of the matrix? || 1
−
| Question || Prove that the following integral converges uniformly on the indicated set. <math>{\textstyle \displaystyle \int \limits _{0}^{+\infty }e^{-\alpha x}\cos 2x\,dx}</math> , <math>{\textstyle \Delta =[1;+\infty )}</math> ; || 1
  
 
|-
  
|-
  +
| Question || Find solution of the linear differential equation. || 0
−
| Question || It is known that Dirichlet’s integral <math>{\textstyle \int \limits _{0}^{+\infty }{\frac {\sin x}{x}}\,dx}</math> is equal to <math>{\textstyle {\frac {\pi }{2}}}</math> . Find the values of the following integrals using Dirichlet’s integral<br><math>{\textstyle \int \limits _{0}^{+\infty }{\frac {\sin }{\alpha x}}x\,dx}</math> , <math>{\textstyle \alpha \neq 0}</math> ;<br><math>{\textstyle \int \limits _{0}^{+\infty }{\frac {\sin x-x\cos x}{x^{3}}}\,dx}</math> . || 1
  
 
|-
  
|-
  +
| Question || Make the definition of symmetric matrix. || 0
−
| Question || Find out if <math>{\textstyle \displaystyle \int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha )\,d\alpha \right)\,dx=\int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha )\,dx\right)\,d\alpha }</math> if <math>{\textstyle f(x;\alpha )={\frac {\alpha -x}{(\alpha +x)^{3}}}}</math> . || 0
  
 
|-
  
|-
  +
| Question || Check out the given matrix on positive definess || 0
−
| Question || Find <math>{\textstyle \Phi '(\alpha )}</math> if <math>{\textstyle \Phi (\alpha )=\int \limits _{1}^{2}{\frac {e^{\alpha x^{2}}}{x}}\,dx}</math> . || 0
  
 
|-
  
|-
  +
| Question || Check out the given matrices on similarity. || 0
−
| Question || Differentiating the integral with respect to parameter <math>{\textstyle \alpha }</math> , find it: <math>{\textstyle I(\alpha )=\int \limits _{0}^{\pi }{\frac {1}{\cos x}}\ln {\frac {1+\alpha \cos x}{1-\alpha \cos x}}\,dx}</math> , <math>{\textstyle |\alpha |<1}</math> . || 0
  
 
|-
  
|-
  +
| Question || For the given matrix find left and right inverse matrices, pseudoinverse matrix. || 0
−
| Question || Find Fourier transform of the following functions:<br><math>{\textstyle f(x)={\begin{cases}1,&|x|\leq 1,\\0,&|x|>1;\end{cases}}}</math> || 0
  
 
|-
  
|-
  +
| Question || Make the singular value decomposition of the given matrix. || 0
−
| Question || Let <math>{\textstyle {\widehat {f}}(y)}</math> be Fourier transform of <math>{\textstyle f(x)}</math> . Prove that Fourier transform of <math>{\textstyle e^{i\alpha x}f(x)}</math> is equal to <math>{\textstyle {\widehat {f}}(y-\alpha )}</math> , <math>{\textstyle \alpha \in \mathbb {R} }</math> . || 0
  
 
|}
 
|}
 
=== Final assessment ===
  
=== Final assessment ===
 
'''Section 1'''
  
'''Section 1'''
  +
# Find linear independent vectors (exclude dependent): <math>{\textstyle {\overrightarrow {a}}=[4,0,3,2]^{T}}</math> , <math>{\textstyle {\overrightarrow {b}}=[1,-7,4,5]^{T}}</math> , <math>{\textstyle {\overrightarrow {c}}=[7,1,5,3]^{T}}</math> , <math>{\textstyle {\overrightarrow {d}}=[-5,-3,-3,-1]^{T}}</math> , <math>{\textstyle {\overrightarrow {e}}=[1,-5,2,3]^{T}}</math> . Find <math>{\textstyle rank(A)}</math> if <math>{\textstyle A}</math> is a composition of this vectors. Find <math>{\textstyle rank(A^{T})}</math> .
−
# Find all points where the differential of a function <math>{\textstyle f(x;y)=(5x+7y-25)e^{-x^{2}-xy-y^{2}}}</math> is equal to zero.
  
−
# Show that function <math>{\textstyle \varphi =f\left({\frac {x}{y}};x^{2}+y-z^{2}\right)}</math> satisfies the equation <math>{\textstyle 2xz\varphi _{x}+2yz\varphi _{y}+\left(2x^{2}+y\right)\varphi _{z}=0}</math> .
 +
# Find <math>{\textstyle E}</math> : <math>{\textstyle EA=U}</math> (<math>{\textstyle U}</math> – upper-triangular matrix). Find <math>{\textstyle L=E^{-}1}</math> , if <math>{\textstyle A=\left({\begin{array}{ccc}2&5&7\\6&4&9\\4&1&8\\\end{array}}\right)}</math> .
−
# Find maxima and minima of function <math>{\textstyle u=2x^{2}+12xy+y^{2}}</math> under condition that <math>{\textstyle x^{2}+4y^{2}=25}</math> . Find the maximum and minimum value of a function
 +
# Find complete solution for the system <math>{\textstyle Ax=b}</math> , if <math>{\textstyle b=[7,18,5]^{T}}</math> and <math>{\textstyle A=\left({\begin{array}{cccc}6&-2&1&-4\\4&2&14&-31\\2&-1&3&-7\\\end{array}}\right)}</math> . Provide an example of vector b that makes this system unsolvable.
−
# <math>{\textstyle u=\left(y^{2}-x^{2}\right)e^{1-x^{2}+y^{2}}}</math> on a domain given by inequality <math>{\textstyle x^{2}+y^{2}\leq 4}</math> ;
  
 
'''Section 2'''
  
'''Section 2'''
−
# Domain <math>{\textstyle G}</math> is bounded by lines <math>{\textstyle y=2x}</math> , <math>{\textstyle y=x}</math> and <math>{\textstyle y=2}</math> . Rewrite integral <math>{\textstyle \iint \limits _{G}f(x)\,dx\,dy}</math> as a single integral.
 +
# Find the dimensions of the four fundamental subspaces associated with <math>{\textstyle A}</math> , depending on the parameters <math>{\textstyle a}</math> and <math>{\textstyle b}</math> : <math>{\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)}</math> .
−
# Represent the integral <math>{\textstyle \displaystyle \iint \limits _{G}f(x;y)\,dx\,dy}</math> as iterated integrals with different order of integration in polar coordinates if <math>{\textstyle G=\left\{(x;y)\left|a^{2}\leq x^{2}+y^{2}\leq 4a^{2};\,|x|-y\geq 0\right.\right\}}</math> .
 +
# Find a vector <math>{\textstyle x}</math> orthogonal to the Row space of matrix <math>{\textstyle A}</math> , and a vector <math>{\textstyle y}</math> orthogonal to the <math>{\textstyle C(A)}</math> , and a vector <math>{\textstyle z}</math> orthogonal to the <math>{\textstyle N(A)}</math> : <math>{\textstyle A=\left({\begin{array}{ccc}1&2&2\\3&4&2\\4&6&4\\\end{array}}\right)}</math> .
−
# Find the integral making an appropriate substitution: <math>{\textstyle \displaystyle \iiint \limits _{G}\left(x^{2}-y^{2}\right)\left(z+x^{2}-y^{2}\right)\,dx\,dy\,dz}</math> , <math>{\textstyle G=\left\{(x;y;z)\left|x-1<y<x;\,1-x<y<2-x;\,1-x^{2}+y^{2}<z<y^{2}-x^{2}+2x\right.\right\}}</math> .
 +
# Find the best straight-line <math>{\textstyle y(x)}</math> fit to the measurements: <math>{\textstyle y(-2)=4}</math> , <math>{\textstyle y(-1)=3}</math> , <math>{\textstyle y(0)=2}</math> , <math>{\textstyle y(1)-0}</math> .
−
# Use divergence theorem to find the following integrals <math>{\textstyle \displaystyle \iint \limits _{S}(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy}</math> where <math>{\textstyle S}</math> is the outer surface of a tetrahedron <math>{\textstyle {\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}\leq 1}</math> , <math>{\textstyle x\geq 0}</math> , <math>{\textstyle y\geq 0}</math> , <math>{\textstyle z\geq 0}</math> ;
 +
# Find the projection matrix <math>{\textstyle P}</math> of vector <math>{\textstyle [4,3,2,0]^{T}}</math> onto the <math>{\textstyle C(A)}</math> : <math>{\textstyle A=\left({\begin{array}{cc}1&-2\\1&-1\\1&0\\1&1\\\end{array}}\right)}</math> .
  +
# Find an orthonormal basis for the subspace spanned by the vectors: <math>{\textstyle {\overrightarrow {a}}=[-2,2,0,0]^{T}}</math> , <math>{\textstyle {\overrightarrow {b}}=[0,1,-1,0]^{T}}</math> , <math>{\textstyle {\overrightarrow {c}}=[0,1,0,-1]^{T}}</math> . Then express <math>{\textstyle A=[a,b,c]}</math> in the form of <math>{\textstyle A=QR}</math>
 
'''Section 3'''
  
'''Section 3'''
−
# Find out whether the following functional series converge uniformly on the indicated intervals. Justify your answer. <math>{\textstyle \sum \limits _{n=1}^{\infty }{\frac {xn+{\sqrt {n}}}{n+x}}\ln \left(1+{\frac {x}{n{\sqrt {n}}}}\right)}</math> , <math>{\textstyle \Delta _{1}=(0;1)}</math> , <math>{\textstyle \Delta _{2}=(1;+\infty )}</math> ;
 +
# Find eigenvector of the circulant matrix <math>{\textstyle C}</math> for the eigenvalue = <math>{\textstyle {c}_{1}}</math> +<math>{\textstyle {c}_{2}}</math> +<math>{\textstyle {c}_{3}}</math> +<math>{\textstyle {c}_{4}}</math> : <math>{\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)}</math> .
  +
# Diagonalize this matrix: <math>{\textstyle A=\left({\begin{array}{cc}2&1-i\\1+i&3\\\end{array}}\right)}</math> .
−
# Show that sequence <math>{\textstyle f_{n}(x)={\frac {\sin nx}{\sqrt {n}}}}</math> converges uniformly on <math>{\textstyle \mathbb {R} }</math> to a differentiable function <math>{\textstyle f(x)}</math> , and at that <math>{\textstyle \lim \limits _{n\rightarrow +\infty }f'_{n}(0)\neq f'(0)}</math> .
  
  +
# <math>{\textstyle A}</math> is the matrix with full set of orthonormal eigenvectors. Prove that <math>{\textstyle AA=A^{H}A^{H}}</math> .
  +
# Find all eigenvalues and eigenvectors of the cyclic permutation matrix <math>{\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)}</math> .
 
'''Section 4'''
  
'''Section 4'''
−
# Find out if <math>{\textstyle \displaystyle \int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha )\,d\alpha \right)\,dx=\int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha )\,dx\right)\,d\alpha }</math> if <math>{\textstyle f(x;\alpha )={\frac {\alpha ^{2}-x^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}}</math> .
 +
# Find <math>{\textstyle det(e^{A})}</math> for <math>{\textstyle A=\left({\begin{array}{cc}2&1\\2&3\\\end{array}}\right)}</math> .
  +
# Write down the first order equation system for the following differential equation and solve it:
−
# Find <math>{\textstyle \Phi '(\alpha )}</math> if <math>{\textstyle \Phi (\alpha )=\int \limits _{0}^{\alpha }{\frac {\ln(1+\alpha x)}{x}}\,dx}</math> .
  
  +
<math>{\textstyle d^{3}y/dx+d^{2}y/dx-2dy/dx=0}</math>
−
# Prove that the following integral converges uniformly on the indicated set. <math>{\textstyle \displaystyle \int \limits _{-\infty }^{+\infty }{\frac {\cos \alpha x}{4+x^{2}}}\,dx}</math> , <math>{\textstyle \Delta =\mathbb {R} }</math> ;
  
−
# Find Fourier integral for <math>{\textstyle f(x)={\begin{cases}1,&|x|\leq \tau ,\\0,&|x|>\tau ;\end{cases}}}</math>
 +
<math>{\textstyle y''(0)=6}</math> , <math>{\textstyle y'(0)=0}</math> , <math>{\textstyle y(0)=3}</math> .
  +
Is the solution of this system will be stable?
  +
# For which <math>{\textstyle a}</math> and <math>{\textstyle b}</math> quadratic form <math>{\textstyle Q(x,y,z)}</math> is positive definite:
  +
<math>{\textstyle Q(x,y,z)=ax^{2}+y^{2}+2z^{2}+2bxy+4xz}</math>
  +
# Find the SVD and the pseudoinverse of the matrix <math>{\textstyle A=\left({\begin{array}{ccc}1&0&0\\0&1&1\\\end{array}}\right)}</math> .
   
 
=== The retake exam ===
  
=== The retake exam ===

Revision as of 12:27, 20 April 2022

Analytical Geometry & Linear Algebra – II

  • Course name: Analytical Geometry & Linear Algebra – II
  • Code discipline:
  • Subject area: fundamental principles of linear algebra,; concepts of linear algebra objects and their representation in vector-matrix form

Short Description

Prerequisites

Prerequisite subjects

Prerequisite topics

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. L
  4. U
  5. {\textstyle LU}
  6. and
  7. L
  8. D
  9. U
  10. {\textstyle LDU}
  11. factorization.
  12. Transposes and permutations. Vector spaces and subspaces.
  13. The null space: Solving
  14. A
  15. x
  16. =
  17. 0
  18. {\textstyle Ax=0}
  19. and
  20. A
  21. x
  22. =
  23. b
  24. {\textstyle Ax=b}
  25. . Row reduced echelon form. Matrix rank.
Linear regression analysis and decompositionA=QR{\textstyle A=QR}.
  1. Independence, basis and dimension. The four fundamental subspaces.
  2. Orthogonal vectors and subspaces. Projections onto subspaces
  3. Projection matrices. Least squares approximations. Gram-Schmidt and A = QR.
Fast Fourier Transform. Matrix Diagonalization.
  1. Complex Numbers. Hermitian and Unitary Matrices.
  2. Fourier Series. The Fast Fourier Transform
  3. Eigenvalues and eigenvectors. Matrix diagonalization.
Symmetric, positive definite and similar matrices. Singular value decomposition.
  1. Linear differential equations.
  2. Symmetric matrices. Positive definite matrices.
  3. Similar matrices. Left and right inverses, pseudoinverse. Singular value decomposition (SVD).

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

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.

ILOs defined at three levels

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

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

  • List basic notions of linear algebra
  • Understand key principles involved in solution of linear equation systems and the properties of matrices
  • Linear regression analysis
  • Fast Fourier Transform
  • How to find eigenvalues and eigenvectors for matrix diagonalization and single value decomposition

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 ...

  • Key principles involved in solution of linear equation systems and the properties of matrices
  • Become familiar with the four fundamental subspaces
  • Linear regression analysis
  • Fast Fourier Transform
  • How to find eigenvalues and eigenvectors for matrix diagonalization and single value decomposition

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 ...

  • Linear equation system solving by using the vector-matrix approach
  • Make linear regression analysis
  • Fast Fourier Transform
  • To find eigenvalues and eigenvectors for matrix diagonalization and single value decomposition

Grading

Course grading range

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

Course activities and grading breakdown

Activity Type Percentage of the overall course grade
Labs/seminar classes 20
Interim performance assessment 30
Exams 50

Recommendations for students on how to succeed in the course

Resources, literature and reference materials

Open access resources

Closed access resources

Software and tools used within the course

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section
Learning Activities Section 1 Section 2 Section 3 Section 4
Development of individual parts of software product code 1 1 1 1
Homework and group projects 1 1 1 1
Midterm evaluation 1 1 1 1
Testing (written or computer based) 1 1 1 1
Discussions 1 1 1 1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type Content Is Graded?
Question How to perform Gauss elimination? 1
Question How to perform matrices multiplication? 1
Question How to perform LU factorization? 1
Question How to find complete solution for any linear equation system Ax=b? 1
Question Find the solution for the given linear equation system by using Gauss elimination. 0
Question Perform factorization for the given matrix . 0
Question Factor the given symmetric matrix into with the diagonal pivot matrix . 0
Question Find inverse matrix for the given matrix . 0

Section 2

Activity Type Content Is Graded?
Question What is linear independence of vectors? 1
Question Define the four fundamental subspaces of a matrix? 1
Question How to define orthogonal vectors and subspaces? 1
Question How to define orthogonal complements of the space? 1
Question How to find vector projection on a subspace? 1
Question How to perform linear regression for the given measurements? 1
Question How to find an orthonormal basis for the subspace spanned by the given vectors? 1
Question Check out linear independence of the given vectors 0
Question Find four fundamental subspaces of the given matrix. 0
Question Check out orthogonality of the given subspaces. 0
Question Find orthogonal complement for the given subspace. 0
Question Find vector projection on the given subspace. 0
Question Perform linear regression for the given measurements. 0
Question Find an orthonormal basis for the subspace spanned by the given vectors. 0

Section 3

Activity Type Content Is Graded?
Question Make the definition of Hermitian Matrix. 1
Question Make the definition of Unitary Matrix. 1
Question How to find matrix for the Fourier transform? 1
Question When we can make fast Fourier transform? 1
Question How to find eigenvalues and eigenvectors of a matrix? 1
Question How to diagonalize a square matrix? 1
Question Check out is the given matrix Hermitian. 0
Question Check out is the given matrix Unitary. 0
Question Find the matrix for the given Fourier transform. 0
Question Find eigenvalues and eigenvectors for the given matrix. 0
Question Find diagonalize form for the given matrix. 0

Section 4

Activity Type Content Is Graded?
Question How to solve linear differential equations? 1
Question Make the definition of symmetric matrix? 1
Question Make the definition of positive definite matrix? 1
Question Make the definition of similar matrices? 1
Question How to find left and right inverses matrices, pseudoinverse matrix? 1
Question How to make singular value decomposition of the matrix? 1
Question Find solution of the linear differential equation. 0
Question Make the definition of symmetric matrix. 0
Question Check out the given matrix on positive definess 0
Question Check out the given matrices on similarity. 0
Question For the given matrix find left and right inverse matrices, pseudoinverse matrix. 0
Question Make the singular value decomposition of the given matrix. 0

Final assessment

Section 1

  1. Find linear independent vectors (exclude dependent): , , , , . Find if is a composition of this vectors. Find .
  2. Find  : ( – upper-triangular matrix). Find , if .
  3. Find complete solution for the system , if and . Provide an example of vector b that makes this system unsolvable.

Section 2

  1. Find the dimensions of the four fundamental subspaces associated with , depending on the parameters and  : .
  2. Find a vector orthogonal to the Row space of matrix , and a vector orthogonal to the , and a vector orthogonal to the  : .
  3. Find the best straight-line fit to the measurements: , , , .
  4. Find the projection matrix of vector onto the  : .
  5. Find an orthonormal basis for the subspace spanned by the vectors: , , . Then express in the form of

Section 3

  1. Find eigenvector of the circulant matrix for the eigenvalue = + + + : .
  2. Diagonalize this matrix: .
  3. is the matrix with full set of orthonormal eigenvectors. Prove that .
  4. Find all eigenvalues and eigenvectors of the cyclic permutation matrix .

Section 4

  1. Find for .
  2. Write down the first order equation system for the following differential equation and solve it:

, , . Is the solution of this system will be stable?

  1. For which and quadratic form is positive definite:

  1. Find the SVD and the pseudoinverse of the matrix .

The retake exam

Section 1

Section 2

Section 3

Section 4

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