Difference between revisions of "IU:TestPage"
Jump to navigation
Jump to search
R.sirgalina (talk | contribs) |
R.sirgalina (talk | contribs) |
||
| Line 1: | Line 1: | ||
| + | = Calculus I = |
||
| − | = Analytical Geometry \& Linear Algebra -- II = |
||
| − | * Course name: |
+ | * Course name: Calculus I |
* Course number: XYZ |
* Course number: XYZ |
||
| Line 7: | Line 7: | ||
=== Key concepts of the class === |
=== Key concepts of the class === |
||
| + | * Calculus for the functions of one variable: differentiation |
||
| − | * fundamental principles of linear algebra, |
||
| + | * Calculus for the functions of one variable: integration |
||
| − | * concepts of linear algebra objects and their representation in vector-matrix form |
||
| + | * Basics of series |
||
| + | * Multivariate calculus: derivatives, differentials, maxima and minima |
||
| + | * Multivariate integration |
||
| + | * Functional series. Fourier series |
||
| + | * Integrals with parameters |
||
=== What is the purpose of this course? === |
=== What is the purpose of this course? === |
||
| + | The course is designed to provide Software Engineers the knowledge of basic (core) concepts, definitions, theoretical results and techniques of calculus for the functions of one and several variables. The goal of the course is to study basic mathematical concepts that will be required in further studies. |
||
| − | 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 === |
||
| + | This calculus course will provide an opportunity for participants to understand key principles involved in differentiation and integration of functions: solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities, |
||
| − | |||
| + | get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation. |
||
| − | ==== - What should a student remember at the end of the course? ==== |
||
| + | |||
| − | By the end of the course, the students should be able to |
||
| + | All definitions and theorem statements (that will be given in lectures and that are needed to explain the keywords listed above) will be formal, but just few of these theorems will be proven formally. Instead (in the tutorial and practice classes) we will try these definitions and theorems on work with routine exercises and applied problems. |
||
| − | * 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 === |
||
| − | {| class="wikitable" |
||
| − | |+ Course grade breakdown |
||
| − | |- |
||
| − | ! Type !! Points |
||
| − | |- |
||
| − | | Labs/seminar classes || 20 |
||
| − | |- |
||
| − | | Interim performance assessment || 30 |
||
| − | |- |
||
| − | | Exams || 50 |
||
| − | |} |
||
| − | |||
| − | === Grades range === |
||
| − | {| class="wikitable" |
||
| − | |+ 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? ==== |
||
| − | {| 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 ==== |
||
| − | # 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 $Ax=b$ by using Gauss elimination. |
||
| − | # Perform $A=LU$ factorization for the given matrix $A$. |
||
| − | # Factor the given symmetric matrix $A$ into $A=LDL^T$ with the diagonal pivot matrix $D$. |
||
| − | # 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 ==== |
||
| − | # 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 |
||
| − | === 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 $b$: |
||
| − | === 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? ==== |
||
| − | {| 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 ==== |
||
| − | # 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 $C$ for the eigenvalue = ${c}_1$+${c}_2$+${c}_3$+${c}_4$: |
||
| − | === Section 4 === |
||
| − | |||
| − | ==== Section title ==== |
||
| − | Symmetric, positive definite and similar matrices. Singular value decomposition. |
||
| − | |||
| − | ==== Topics covered in this section ==== |
||
| − | * Linear differential equations. |
||
| − | * Symmetric matrices. Positive definite matrices. |
||
| − | * Similar matrices. Left and right inverses, pseudoinverse. Singular value decomposition (SVD). |
||
| − | |||
| − | ==== 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 ==== |
||
| − | # How to solve linear differential equations? |
||
| − | # Make the definition of symmetric matrix? |
||
| − | # Make the definition of positive definite matrix? |
||
| − | # Make the definition of similar matrices? |
||
| − | # How to find left and right inverses matrices, pseudoinverse matrix? |
||
| − | # How to make singular value decomposition of the matrix? |
||
| − | |||
| − | ==== Typical questions for seminar classes (labs) within this section ==== |
||
| − | # Find solution of the linear differential equation. |
||
| − | # Make the definition of symmetric matrix. |
||
| − | # Check out the given matrix on positive definess |
||
| − | # Check out the given matrices on similarity. |
||
| − | # For the given matrix find left and right inverse matrices, pseudoinverse matrix. |
||
| − | # Make the singular value decomposition of the given matrix. |
||
| − | |||
| − | ==== Tasks for midterm assessment within this section ==== |
||
| − | |||
| − | |||
| − | ==== Test questions for final assessment in this section ==== |
||
| − | # Find $det(e^A)$ for |
||
Revision as of 10:50, 2 December 2021
Calculus I
- Course name: Calculus I
- Course number: XYZ
Course Characteristics
Key concepts of the class
- Calculus for the functions of one variable: differentiation
- Calculus for the functions of one variable: integration
- Basics of series
- Multivariate calculus: derivatives, differentials, maxima and minima
- Multivariate integration
- Functional series. Fourier series
- Integrals with parameters
What is the purpose of this course?
The course is designed to provide Software Engineers the knowledge of basic (core) concepts, definitions, theoretical results and techniques of calculus for the functions of one and several variables. The goal of the course is to study basic mathematical concepts that will be required in further studies.
This calculus course will provide an opportunity for participants to understand key principles involved in differentiation and integration of functions: solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities, get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation. All definitions and theorem statements (that will be given in lectures and that are needed to explain the keywords listed above) will be formal, but just few of these theorems will be proven formally. Instead (in the tutorial and practice classes) we will try these definitions and theorems on work with routine exercises and applied problems.