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R.sirgalina (talk | contribs) (Replaced content with " = Analytical Geometry \& Linear Algebra -- II = * Course name: Analytical Geometry \& Linear Algebra -- II * Course number: XYZ == Course Characteristics == === Key con...") Tags: Manual revert Replaced |
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=== What is the purpose of this course? === |
=== What is the purpose of this course? === |
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| − | === Course objectives based on Bloom’s taxonomy === |
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| − | |||
| − | ==== - What should a student remember at the end of the course? ==== |
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| − | By the end of the course, the students should be able to |
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| − | * List basic notions of linear algebra |
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| − | * Understand key principles involved in solution of linear equation systems and the properties of matrices |
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| − | * Linear regression analysis |
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| − | * Fast Fourier Transform |
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| − | * How to find eigenvalues and eigenvectors for matrix diagonalization and single value decomposition |
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| − | |||
| − | ==== - What should a student be able to understand at the end of the course? ==== |
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| − | By the end of the course, the students should be able to |
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| − | * Key principles involved in solution of linear equation systems and the properties of matrices |
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| − | * Become familiar with the four fundamental subspaces |
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| − | * Linear regression analysis |
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| − | * Fast Fourier Transform |
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| − | * How to find eigenvalues and eigenvectors for matrix diagonalization and single value decomposition |
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| − | |||
| − | ==== - What should a student be able to apply at the end of the course? ==== |
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| − | By the end of the course, the students should be able to |
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| − | * Linear equation system solving by using the vector-matrix approach |
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| − | * Make linear regression analysis |
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| − | * Fast Fourier Transform |
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| − | * To find eigenvalues and eigenvectors for matrix diagonalization and single value decomposition |
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| − | === Course evaluation === |
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| − | {| class="wikitable" |
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| − | |+ Course grade breakdown |
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| − | |- |
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| − | ! Type !! Points |
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| − | |- |
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| − | | Labs/seminar classes || 20 |
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| − | |- |
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| − | | Interim performance assessment || 30 |
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| − | |- |
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| − | | Exams || 50 |
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| − | |} |
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| − | |||
| − | === Grades range === |
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| − | {| class="wikitable" |
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| − | |+ Course grading range |
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| − | |- |
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| − | ! Grade !! Points |
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| − | |- |
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| − | | A || [85, 100] |
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| − | |- |
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| − | | B || [65, 84] |
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| − | |- |
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| − | | C || [50, 64] |
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| − | |- |
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| − | | D || [0, 49] |
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| − | |} |
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| − | === Resources and reference material === |
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| − | * Gilbert Strang. Linear Algebra and Its |
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| − | Applications, 4th Edition, Brooks Cole, 2006. ISBN: 9780030105678 |
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| − | * Gilbert Strang. Introduction to Linear Algebra, 4th Edition, Wellesley, MA: Wellesley-Cambridge Press, 2009. ISBN: 9780980232714 |
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| − | == Course Sections == |
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| − | The main sections of the course and approximate hour distribution between them is as follows: |
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| − | === Section 1 === |
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| − | |||
| − | ==== Section title ==== |
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| − | Linear equation system solving by using the vector-matrix approach |
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| − | |||
| − | ==== Topics covered in this section ==== |
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| − | * The geometry of linear equations. Elimination with matrices. |
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| − | * Matrix operations, including inverses. <math>LU</math> and <math>LDU</math> factorization. |
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| − | * Transposes and permutations. Vector spaces and subspaces. |
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| − | * The null space: Solving <math>Ax = 0</math> and <math>Ax = b</math>. Row reduced echelon form. Matrix rank. |
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| − | |||
| − | ==== What forms of evaluation were used to test students’ performance in this section? ==== |
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| − | {| class="wikitable" |
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| − | |+ |
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| − | |- |
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| − | ! Form !! Yes/No |
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| − | |- |
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| − | | Development of individual parts of software product code || 1 |
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| − | |- |
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| − | | Homework and group projects || 1 |
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| − | |- |
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| − | | Midterm evaluation || 1 |
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| − | |- |
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| − | | Testing (written or computer based) || 1 |
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| − | |- |
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| − | | Reports || 0 |
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| − | |- |
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| − | | Essays || 0 |
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| − | |- |
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| − | | Oral polls || 0 |
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| − | |- |
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| − | | Discussions || 1 |
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| − | |} |
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| − | |||
| − | ==== Typical questions for ongoing performance evaluation within this section ==== |
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| − | # How to perform Gauss elimination? |
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| − | # How to perform matrices multiplication? |
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| − | # How to perform LU factorization? |
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| − | # How to find complete solution for any linear equation system Ax=b? |
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| − | |||
| − | ==== Typical questions for seminar classes (labs) within this section ==== |
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| − | # Find the solution for the given linear equation system <math>Ax=b</math> by using Gauss elimination. |
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| − | # Perform <math>A=LU</math> factorization for the given matrix <math>A</math>. |
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| − | # Factor the given symmetric matrix <math>A</math> into <math>A=LDL^T</math> with the diagonal pivot matrix <math>D</math>. |
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| − | # Find inverse matrix <math>A^-1</math> for the given matrix <math>A</math>. |
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| − | |||
| − | ==== Tasks for midterm assessment within this section ==== |
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| − | |||
| − | |||
| − | ==== Test questions for final assessment in this section ==== |
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| − | # Find linear independent vectors (exclude dependent): <math>\overrightarrow{a}=[4,0,3,2]^T</math>, <math>\overrightarrow{b}=[1,-7,4,5]^T</math>, <math>\overrightarrow{c}=[7,1,5,3]^T</math>, <math>\overrightarrow{d}=[-5,-3,-3,-1]^T</math>, <math>\overrightarrow{e}=[1,-5,2,3]^T</math>. Find <math>rank(A)</math> if <math>A</math> is a composition of this vectors. Find <math>rank(A^T)</math>. |
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| − | # Find <math>E</math>: <math>EA=U</math> (<math>U</math> – upper-triangular matrix). Find <math>L=E^-1</math>, if |
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| − | <math>A=\left( |
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| − | \begin{array}{ccc} |
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| − | 2 & 5 & 7 \\ |
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| − | 6 & 4 & 9 \\ |
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| − | 4 & 1 & 8 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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| − | # Find complete solution for the system <math>Ax=b</math>, if <math>b=[7,18,5]^T</math> and |
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| − | <math>A=\left( |
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| − | \begin{array}{cccc} |
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| − | 6 & -2 & 1 & -4 \\ |
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| − | 4 & 2 & 14 & -31 \\ |
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| − | 2 & -1 & 3 & -7 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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| − | Provide an example of vector b that makes this system unsolvable. |
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| − | === Section 2 === |
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| − | |||
| − | ==== Section title ==== |
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| − | Linear regression analysis and decomposition <math>A=QR</math>. |
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| − | |||
| − | ==== Topics covered in this section ==== |
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| − | * Independence, basis and dimension. The four fundamental subspaces. |
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| − | * Orthogonal vectors and subspaces. Projections onto subspaces |
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| − | * Projection matrices. Least squares approximations. Gram-Schmidt and A = QR. |
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| − | |||
| − | ==== What forms of evaluation were used to test students’ performance in this section? ==== |
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| − | {| class="wikitable" |
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| − | |+ |
||
| − | |- |
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| − | ! Form !! Yes/No |
||
| − | |- |
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| − | | Development of individual parts of software product code || 1 |
||
| − | |- |
||
| − | | Homework and group projects || 1 |
||
| − | |- |
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| − | | Midterm evaluation || 1 |
||
| − | |- |
||
| − | | Testing (written or computer based) || 1 |
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| − | |- |
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| − | | Reports || 0 |
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| − | |- |
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| − | | Essays || 0 |
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| − | |- |
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| − | | Oral polls || 0 |
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| − | |- |
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| − | | Discussions || 1 |
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| − | |} |
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| − | |||
| − | ==== Typical questions for ongoing performance evaluation within this section ==== |
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| − | # What is linear independence of vectors? |
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| − | # Define the four fundamental subspaces of a matrix? |
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| − | # How to define orthogonal vectors and subspaces? |
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| − | # How to define orthogonal complements of the space? |
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| − | # How to find vector projection on a subspace? |
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| − | # How to perform linear regression for the given measurements? |
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| − | # How to find an orthonormal basis for the subspace spanned by the given vectors? |
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| − | |||
| − | ==== Typical questions for seminar classes (labs) within this section ==== |
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| − | # Check out linear independence of the given vectors |
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| − | # Find four fundamental subspaces of the given matrix. |
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| − | # Check out orthogonality of the given subspaces. |
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| − | # Find orthogonal complement for the given subspace. |
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| − | # Find vector projection on the given subspace. |
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| − | # Perform linear regression for the given measurements. |
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| − | # Find an orthonormal basis for the subspace spanned by the given vectors. |
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| − | |||
| − | ==== Tasks for midterm assessment within this section ==== |
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| − | |||
| − | |||
| − | ==== Test questions for final assessment in this section ==== |
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| − | # Find the dimensions of the four fundamental subspaces associated with <math>A</math>, depending on the parameters <math>a</math> and <math>b</math>: |
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| − | <math>A=\left( |
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| − | \begin{array}{cccc} |
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| − | 7 & 8 & 5 & 3 \\ |
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| − | 4 & a & 3 & 2 \\ |
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| − | 6 & 8 & 4 & b \\ |
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| − | 3 & 4 & 2 & 1 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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| − | # Find a vector <math>x</math> orthogonal to the Row space of matrix <math>A</math>, and a vector <math>y</math> orthogonal to the <math>C(A)</math>, and a vector <math>z</math> orthogonal to the <math>N(A)</math>: |
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| − | <math>A=\left( |
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| − | \begin{array}{ccc} |
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| − | 1 & 2 & 2 \\ |
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| − | 3 & 4 & 2 \\ |
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| − | 4 & 6 & 4 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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| − | # Find the best straight-line <math>y(x)</math> fit to the measurements: <math>y(-2)=4</math>, <math>y(-1)=3</math>, <math>y(0)=2</math>, <math>y(1)-0</math>. |
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| − | # Find the projection matrix <math>P</math> of vector <math>[4,3,2,0]^T</math> onto the <math>C(A)</math>: |
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| − | <math>A=\left( |
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| − | \begin{array}{cc} |
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| − | 1 & -2 \\ |
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| − | 1 & -1 \\ |
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| − | 1 & 0 \\ |
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| − | 1 & 1 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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| − | # Find an orthonormal basis for the subspace spanned by the vectors: <math>\overrightarrow{a}=[-2,2,0,0]^T</math>, <math>\overrightarrow{b}=[0,1,-1,0]^T</math>, <math>\overrightarrow{c}=[0,1,0,-1]^T</math>. Then express <math>A=[a,b,c]</math> in the form of <math>A=QR</math> |
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| − | === Section 3 === |
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| − | |||
| − | ==== Section title ==== |
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| − | Fast Fourier Transform. Matrix Diagonalization. |
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| − | |||
| − | ==== Topics covered in this section ==== |
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| − | * Complex Numbers. Hermitian and Unitary Matrices. |
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| − | * Fourier Series. The Fast Fourier Transform |
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| − | * Eigenvalues and eigenvectors. Matrix diagonalization. |
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| − | |||
| − | ==== 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 |
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| − | |- |
||
| − | | Discussions || 1 |
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| − | |} |
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| − | |||
| − | ==== Typical questions for ongoing performance evaluation within this section ==== |
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| − | # Make the definition of Hermitian Matrix. |
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| − | # Make the definition of Unitary Matrix. |
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| − | # How to find matrix for the Fourier transform? |
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| − | # When we can make fast Fourier transform? |
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| − | # How to find eigenvalues and eigenvectors of a matrix? |
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| − | # How to diagonalize a square matrix? |
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| − | |||
| − | ==== Typical questions for seminar classes (labs) within this section ==== |
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| − | # Check out is the given matrix Hermitian. |
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| − | # Check out is the given matrix Unitary. |
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| − | # Find the matrix for the given Fourier transform. |
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| − | # Find eigenvalues and eigenvectors for the given matrix. |
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| − | # Find diagonalize form for the given matrix. |
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| − | |||
| − | ==== Tasks for midterm assessment within this section ==== |
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| − | |||
| − | |||
| − | ==== Test questions for final assessment in this section ==== |
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| − | # Find eigenvector of the circulant matrix <math>C</math> for the eigenvalue = <math>{c}_1</math>+<math>{c}_2</math>+<math>{c}_3</math>+<math>{c}_4</math>: |
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| − | <math>C=\left( |
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| − | \begin{array}{cccc} |
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| − | {c}_1 & {c}_2 & {c}_3 & {c}_4 \\ |
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| − | {c}_4 & {c}_1 & {c}_2 & {c}_3 \\ |
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| − | {c}_3 & {c}_4 & {c}_1 & {c}_2 \\ |
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| − | {c}_2 & {c}_3 & {c}_4 & {c}_1 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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| − | # Diagonalize this matrix: |
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| − | <math>A=\left( |
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| − | \begin{array}{cc} |
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| − | 2 & 1-i \\ |
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| − | 1+i & 3 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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| − | # <math>A</math> is the matrix with full set of orthonormal eigenvectors. Prove that <math>AA=A^HA^H</math>. |
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| − | # Find all eigenvalues and eigenvectors of the cyclic permutation matrix |
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| − | <math>P=\left( |
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| − | \begin{array}{cccc} |
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| − | 0 & 1 & 0 & 0 \\ |
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| − | 0 & 0 & 1 & 0 \\ |
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| − | 0 & 0 & 0 & 1 \\ |
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| − | 1 & 0 & 0 & 0 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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| − | === Section 4 === |
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| − | |||
| − | ==== Section title ==== |
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| − | Symmetric, positive definite and similar matrices. Singular value decomposition. |
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| − | |||
| − | ==== Topics covered in this section ==== |
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| − | * Linear differential equations. |
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| − | * Symmetric matrices. Positive definite matrices. |
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| − | * Similar matrices. Left and right inverses, pseudoinverse. Singular value decomposition (SVD). |
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| − | |||
| − | ==== 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 |
||
| − | |} |
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| − | |||
| − | ==== Typical questions for ongoing performance evaluation within this section ==== |
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| − | # How to solve linear differential equations? |
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| − | # Make the definition of symmetric matrix? |
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| − | # Make the definition of positive definite matrix? |
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| − | # Make the definition of similar matrices? |
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| − | # How to find left and right inverses matrices, pseudoinverse matrix? |
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| − | # How to make singular value decomposition of the matrix? |
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| − | |||
| − | ==== Typical questions for seminar classes (labs) within this section ==== |
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| − | # Find solution of the linear differential equation. |
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| − | # Make the definition of symmetric matrix. |
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| − | # Check out the given matrix on positive definess |
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| − | # Check out the given matrices on similarity. |
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| − | # For the given matrix find left and right inverse matrices, pseudoinverse matrix. |
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| − | # Make the singular value decomposition of the given matrix. |
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| − | |||
| − | ==== Tasks for midterm assessment within this section ==== |
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| − | |||
| − | |||
| − | ==== Test questions for final assessment in this section ==== |
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| − | # Find <math>det(e^A)</math> for |
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| − | <math>A=\left( |
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| − | \begin{array}{cc} |
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| − | 2 & 1 \\ |
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| − | 2 & 3 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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| − | # Write down the first order equation system for the following differential equation and solve it: |
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| − | |||
| − | <math>d^3y/dx+d^2y/dx-2dy/dx=0</math> |
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| − | |||
| − | <math>y"(0)=6</math>, <math>y'(0)=0</math>, <math>y(0)=3</math>. |
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| − | |||
| − | Is the solution of this system will be stable? |
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| − | # For which <math>a</math> and <math>b</math> quadratic form <math>Q(x,y,z)</math> is positive definite: |
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| − | |||
| − | <math>Q(x,y,z)=ax^2+y^2+2z^2+2bxy+4xz</math> |
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| − | # Find the SVD and the pseudoinverse of the matrix |
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| − | <math>A=\left( |
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| − | \begin{array}{ccc} |
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| − | 1 & 0 & 0 \\ |
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| − | 0 & 1 & 1 \\ |
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| − | \end{array} |
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| − | \right)</math>. |
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Revision as of 16:19, 6 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