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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

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. ${\textstyle LU}$ and ${\textstyle LDU}$ factorization. Transposes and permutations. Vector spaces and subspaces. The null space: Solving ${\textstyle Ax=0}$ and ${\textstyle Ax=b}$ . Row reduced echelon form. Matrix rank.
Linear regression analysis and decomposition ${\textstyle A=QR}$ .	Independence, basis and dimension. The four fundamental subspaces. Orthogonal vectors and subspaces. Projections onto subspaces Projection matrices. Least squares approximations. Gram-Schmidt and A = QR.
Fast Fourier Transform. Matrix Diagonalization.	Complex Numbers. Hermitian and Unitary Matrices. Fourier Series. The Fast Fourier Transform 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)

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

Resources, literature and reference materials

Open access resources

• 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

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 ${\textstyle Ax=b}$ by using Gauss elimination.	0
Question	Perform ${\textstyle A=LU}$ factorization for the given matrix ${\textstyle A}$ .	0
Question	Factor the given symmetric matrix ${\textstyle A}$ into ${\textstyle A=LDL^{T}}$ with the diagonal pivot matrix ${\textstyle D}$ .	0
Question	Find inverse matrix ${\textstyle A^{-}1}$ for the given matrix ${\textstyle A}$ .	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): ${\textstyle {\overrightarrow {a}}=[4,0,3,2]^{T}}$ , ${\textstyle {\overrightarrow {b}}=[1,-7,4,5]^{T}}$ , ${\textstyle {\overrightarrow {c}}=[7,1,5,3]^{T}}$ , ${\textstyle {\overrightarrow {d}}=[-5,-3,-3,-1]^{T}}$ , ${\textstyle {\overrightarrow {e}}=[1,-5,2,3]^{T}}$ . Find ${\textstyle rank(A)}$ if ${\textstyle A}$ is a composition of this vectors. Find ${\textstyle rank(A^{T})}$ .
2. Find ${\textstyle E}$ : ${\textstyle EA=U}$ (${\textstyle U}$ – upper-triangular matrix). Find ${\textstyle L=E^{-}1}$ , if ${\textstyle A=\left({\begin{array}{ccc}2&5&7\\6&4&9\\4&1&8\\\end{array}}\right)}$ .
3. Find complete solution for the system ${\textstyle Ax=b}$ , if ${\textstyle b=[7,18,5]^{T}}$ and ${\textstyle A=\left({\begin{array}{cccc}6&-2&1&-4\\4&2&14&-31\\2&-1&3&-7\\\end{array}}\right)}$ . Provide an example of vector b that makes this system unsolvable.

Section 2

1. Find the dimensions of the four fundamental subspaces associated with ${\textstyle A}$ , depending on the parameters ${\textstyle a}$ and ${\textstyle b}$ : ${\textstyle A=\left({\begin{array}{cccc}7&8&5&3\\4&a&3&2\\6&8&4&b\\3&4&2&1\\\end{array}}\right)}$ .
2. Find a vector ${\textstyle x}$ orthogonal to the Row space of matrix ${\textstyle A}$ , and a vector ${\textstyle y}$ orthogonal to the ${\textstyle C(A)}$ , and a vector ${\textstyle z}$ orthogonal to the ${\textstyle N(A)}$ : ${\textstyle A=\left({\begin{array}{ccc}1&2&2\\3&4&2\\4&6&4\\\end{array}}\right)}$ .
3. Find the best straight-line ${\textstyle y(x)}$ fit to the measurements: ${\textstyle y(-2)=4}$ , ${\textstyle y(-1)=3}$ , ${\textstyle y(0)=2}$ , ${\textstyle y(1)-0}$ .
4. Find the projection matrix ${\textstyle P}$ of vector ${\textstyle [4,3,2,0]^{T}}$ onto the ${\textstyle C(A)}$ : ${\textstyle A=\left({\begin{array}{cc}1&-2\\1&-1\\1&0\\1&1\\\end{array}}\right)}$ .
5. Find an orthonormal basis for the subspace spanned by the vectors: ${\textstyle {\overrightarrow {a}}=[-2,2,0,0]^{T}}$ , ${\textstyle {\overrightarrow {b}}=[0,1,-1,0]^{T}}$ , ${\textstyle {\overrightarrow {c}}=[0,1,0,-1]^{T}}$ . Then express ${\textstyle A=[a,b,c]}$ in the form of ${\textstyle A=QR}$

Section 3

1. Find eigenvector of the circulant matrix ${\textstyle C}$ for the eigenvalue = ${\textstyle {c}_{1}}$ +${\textstyle {c}_{2}}$ +${\textstyle {c}_{3}}$ +${\textstyle {c}_{4}}$ : ${\textstyle C=\left({\begin{array}{cccc}{c}_{1}&{c}_{2}&{c}_{3}&{c}_{4}\\{c}_{4}&{c}_{1}&{c}_{2}&{c}_{3}\\{c}_{3}&{c}_{4}&{c}_{1}&{c}_{2}\\{c}_{2}&{c}_{3}&{c}_{4}&{c}_{1}\\\end{array}}\right)}$ .
2. Diagonalize this matrix: ${\textstyle A=\left({\begin{array}{cc}2&1-i\\1+i&3\\\end{array}}\right)}$ .
3. ${\textstyle A}$ is the matrix with full set of orthonormal eigenvectors. Prove that ${\textstyle AA=A^{H}A^{H}}$ .
4. Find all eigenvalues and eigenvectors of the cyclic permutation matrix ${\textstyle P=\left({\begin{array}{cccc}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\\\end{array}}\right)}$ .

Section 4

1. Find ${\textstyle det(e^{A})}$ for ${\textstyle A=\left({\begin{array}{cc}2&1\\2&3\\\end{array}}\right)}$ .
2. Write down the first order equation system for the following differential equation and solve it:

${\textstyle d^{3}y/dx+d^{2}y/dx-2dy/dx=0}$ ${\textstyle y''(0)=6}$ , ${\textstyle y'(0)=0}$ , ${\textstyle y(0)=3}$ . Is the solution of this system will be stable?

1. For which ${\textstyle a}$ and ${\textstyle b}$ quadratic form ${\textstyle Q(x,y,z)}$ is positive definite:

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

1. Find the SVD and the pseudoinverse of the matrix ${\textstyle A=\left({\begin{array}{ccc}1&0&0\\0&1&1\\\end{array}}\right)}$ .

The retake exam

Section 1

Section 2

Section 3

Section 4