Difference between revisions of "IU:TestPage"

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# Find linear independent vectors (exclude dependent): $\overrightarrow{a}=[4,0,3,2]^T$, $\overrightarrow{b}=[1,-7,4,5]^T$, $\overrightarrow{c}=[7,1,5,3]^T$, $\overrightarrow{d}=[-5,-3,-3,-1]^T$, $\overrightarrow{e}=[1,-5,2,3]^T$. Find $rank(A)$ if $A$ is a composition of this vectors. Find $rank(A^T)$.
  
# Find linear independent vectors (exclude dependent): $\overrightarrow{a}=[4,0,3,2]^T$, $\overrightarrow{b}=[1,-7,4,5]^T$, $\overrightarrow{c}=[7,1,5,3]^T$, $\overrightarrow{d}=[-5,-3,-3,-1]^T$, $\overrightarrow{e}=[1,-5,2,3]^T$. Find $rank(A)$ if $A$ is a composition of this vectors. Find $rank(A^T)$.
 
# Find $E$: $EA=U$ ($U$ – upper-triangular matrix). Find $L=E^-1$, if
  
# Find $E$: $EA=U$ ($U$ – upper-triangular matrix). Find $L=E^-1$, if
  +
=== Section 2 ===
  +
  +
==== Section title ====
  +
Linear regression analysis and decomposition $A=QR$.
  +
  +
==== 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? ====
  +
{| class="wikitable"
  +
|+
  +
|-
  +
! 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 $A$, depending on the parameters $a$ and $b$:

Revision as of 14:59, 1 December 2021

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?

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 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

Course grade breakdown
Type Points
Labs/seminar classes 20
Interim performance assessment 30
Exams 50

Grades range

Course grading 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
  • 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

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. $LU$ and $LDU$ factorization.
  • Transposes and permutations. Vector spaces and subspaces.
  • The null space: Solving $Ax = 0$ and $Ax = b$. 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

  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?

Typical questions for seminar classes (labs) within this section

  1. Find the solution for the given linear equation system $Ax=b$ by using Gauss elimination.
  2. Perform $A=LU$ factorization for the given matrix $A$.
  3. Factor the given symmetric matrix $A$ into $A=LDL^T$ with the diagonal pivot matrix $D$.
  4. Find inverse matrix $A^-1$ for the given matrix $A$.

Tasks for midterm assessment within this section

Test questions for final assessment in this section

  1. Find linear independent vectors (exclude dependent): $\overrightarrow{a}=[4,0,3,2]^T$, $\overrightarrow{b}=[1,-7,4,5]^T$, $\overrightarrow{c}=[7,1,5,3]^T$, $\overrightarrow{d}=[-5,-3,-3,-1]^T$, $\overrightarrow{e}=[1,-5,2,3]^T$. Find $rank(A)$ if $A$ is a composition of this vectors. Find $rank(A^T)$.
  2. Find $E$: $EA=U$ ($U$ – upper-triangular matrix). Find $L=E^-1$, if

Section 2

Section title

Linear regression analysis and decomposition $A=QR$.

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

  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?

Typical questions for seminar classes (labs) within this section

  1. Check out linear independence of the given vectors
  2. Find four fundamental subspaces of the given matrix.
  3. Check out orthogonality of the given subspaces.
  4. Find orthogonal complement for the given subspace.
  5. Find vector projection on the given subspace.
  6. Perform linear regression for the given measurements.
  7. 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

  1. Find the dimensions of the four fundamental subspaces associated with $A$, depending on the parameters $a$ and $b$:
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