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=== Open access resources ===
 
=== 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
 
   
 
=== Closed access resources ===
 
=== Closed access resources ===

Revision as of 12:35, 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. and factorization.
  3. Transposes and permutations. Vector spaces and subspaces.
  4. The null space: Solving and . Row reduced echelon form. Matrix rank.
Linear regression analysis and decomposition .
  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

  • 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

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