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Analytical Geometry \& Linear Algebra -- II
- Course name: Analytical Geometry \& Linear Algebra -- II
- Course number: XYZ
Course Characteristics
Key concepts of the class
- fundamental principles of linear algebra,
- concepts of linear algebra objects and their representation in vector-matrix form
What is the purpose of this course?
Course objectives based on Bloom’s taxonomy
- What should a student remember at the end of the course?
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
- What should a student be able to understand at the end of the course?
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
- What should a student be able to apply at the end of the course?
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
Course evaluation
Type | Points |
---|---|
Labs/seminar classes | 20 |
Interim performance assessment | 30 |
Exams | 50 |
Grades range
Grade | Points |
---|---|
A | [85, 100] |
B | [65, 84] |
C | [50, 64] |
D | [0, 49] |
Resources and reference material
- 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
Course Sections
The main sections of the course and approximate hour distribution between them is as follows:
Section 1
Section title
Linear equation system solving by using the vector-matrix approach
Topics covered in this section
- The geometry of linear equations. Elimination with matrices.
- Matrix operations, including inverses. and factorization.
- Transposes and permutations. Vector spaces and subspaces.
- The null space: Solving and . Row reduced echelon form. Matrix rank.
What forms of evaluation were used to test students’ performance in this section?
Form | Yes/No |
---|---|
Development of individual parts of software product code | 1 |
Homework and group projects | 1 |
Midterm evaluation | 1 |
Testing (written or computer based) | 1 |
Reports | 0 |
Essays | 0 |
Oral polls | 0 |
Discussions | 1 |
Typical questions for ongoing performance evaluation within this section
- How to perform Gauss elimination?
- How to perform matrices multiplication?
- How to perform LU factorization?
- How to find complete solution for any linear equation system Ax=b?
Typical questions for seminar classes (labs) within this section
- Find the solution for the given linear equation system by using Gauss elimination.
- Perform factorization for the given matrix .
- Factor the given symmetric matrix into with the diagonal pivot matrix .
- Find inverse matrix for the given matrix .
Tasks for midterm assessment within this section
Test questions for final assessment in this section
- Find linear independent vectors (exclude dependent): , , , , . Find if is a composition of this vectors. Find .
- Find : ( – upper-triangular matrix). Find , if
.
- Find complete solution for the system , if and
.
Provide an example of vector b that makes this system unsolvable.
Section 2
Section title
Linear regression analysis and decomposition .
Topics covered in this section
- 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.
What forms of evaluation were used to test students’ performance in this section?
Form | Yes/No |
---|---|
Development of individual parts of software product code | 1 |
Homework and group projects | 1 |
Midterm evaluation | 1 |
Testing (written or computer based) | 1 |
Reports | 0 |
Essays | 0 |
Oral polls | 0 |
Discussions | 1 |
Typical questions for ongoing performance evaluation within this section
- What is linear independence of vectors?
- Define the four fundamental subspaces of a matrix?
- How to define orthogonal vectors and subspaces?
- How to define orthogonal complements of the space?
- How to find vector projection on a subspace?
- How to perform linear regression for the given measurements?
- How to find an orthonormal basis for the subspace spanned by the given vectors?
Typical questions for seminar classes (labs) within this section
- Check out linear independence of the given vectors
- Find four fundamental subspaces of the given matrix.
- Check out orthogonality of the given subspaces.
- Find orthogonal complement for the given subspace.
- Find vector projection on the given subspace.
- Perform linear regression for the given measurements.
- Find an orthonormal basis for the subspace spanned by the given vectors.
Tasks for midterm assessment within this section
Test questions for final assessment in this section
- Find the dimensions of the four fundamental subspaces associated with , depending on the parameters and :
.
- Find a vector orthogonal to the Row space of matrix , and a vector orthogonal to the , and a vector orthogonal to the :
.
- Find the best straight-line fit to the measurements: , , , .
- Find the projection matrix of vector onto the :
.
- Find an orthonormal basis for the subspace spanned by the vectors: , , . Then express in the form of
Section 3
Section title
Fast Fourier Transform. Matrix Diagonalization.
Topics covered in this section
- Complex Numbers. Hermitian and Unitary Matrices.
- Fourier Series. The Fast Fourier Transform
- Eigenvalues and eigenvectors. Matrix diagonalization.
What forms of evaluation were used to test students’ performance in this section?
Form | Yes/No |
---|---|
Development of individual parts of software product code | 1 |
Homework and group projects | 1 |
Midterm evaluation | 1 |
Testing (written or computer based) | 1 |
Reports | 0 |
Essays | 0 |
Oral polls | 0 |
Discussions | 1 |
Typical questions for ongoing performance evaluation within this section
- Make the definition of Hermitian Matrix.
- Make the definition of Unitary Matrix.
- How to find matrix for the Fourier transform?
- When we can make fast Fourier transform?
- How to find eigenvalues and eigenvectors of a matrix?
- How to diagonalize a square matrix?
Typical questions for seminar classes (labs) within this section
- Check out is the given matrix Hermitian.
- Check out is the given matrix Unitary.
- Find the matrix for the given Fourier transform.
- Find eigenvalues and eigenvectors for the given matrix.
- Find diagonalize form for the given matrix.
Tasks for midterm assessment within this section
Test questions for final assessment in this section
- Find eigenvector of the circulant matrix for the eigenvalue = +++:
.
- Diagonalize this matrix:
.
- is the matrix with full set of orthonormal eigenvectors. Prove that .
- Find all eigenvalues and eigenvectors of the cyclic permutation matrix
.