Difference between revisions of "BSc: Analytic Geometry And Linear Algebra II.s23"

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# Left and right inverses, pseudoinverse
  
# Left and right inverses, pseudoinverse
 
# Singular value decomposition (SVD)
  
# Singular value decomposition (SVD)
−  
−  
 
|}
  
|}
   
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! Grade !! Range !! Description of performance
  
! Grade !! Range !! Description of performance
 
|-
  
|-
−
| A. Excellent || 85-100 || -
 +
| A. Excellent || 90-110 || -
 
|-
  
|-
−
| B. Good || 70-84 || -
 +
| B. Good || 75-89 || -
 
|-
  
|-
−
| C. Satisfactory || 55-70 || -
 +
| C. Satisfactory || 60-74 || -
 
|-
  
|-
−
| D. Fail || 0-54 || -
 +
| D. Fail || 0-59 || -
 
|}
  
|}
   
Line 101: Line 99:
 
| Midterm || 30
  
| Midterm || 30
 
|-
  
|-
−
| Tests|| 20 (10 for each)
 +
| Two intermediate tests|| 30 (15 for each)
 
|-
  
|-
−
| Final exam || 50
 +
| Final exam || 30
 
|-
  
|-
  +
| Five programming tasks || 20
−
| In-class participation || 5 extras
  
 
|}
  
|}
   
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=== Open access resources ===
  
=== Open access resources ===
  +
* Gilbert Strang. Linear Algebra and Its Applications, 4th Edition, Brooks Cole, 2006. ISBN: 9780030105678
−
* V.V. Konev. Linear Algebra, Vector Algebra and Analytical Geometry. Textbook. Tomsk: TPU Press, 2009, 114 pp [https://portal.tpu.ru/SHARED/k/KONVAL/Textbooks/Tab1/Konev-Linear_Algebra_Vector_Algebra_and_Analytical_Geome.pdf book1]
  
  +
* Gilbert Strang. Introduction to Linear Algebra, 4th Edition, Wellesley, MA: Wellesley-Cambridge Press, 2009. ISBN: 9780980232714
−
* R.A.Sharipov. Course of Analytical Geometry Textbook, Ufa, BSU, 2013. 227pp [https://arxiv.org/pdf/1111.6521.pdf book2]
  
  +
* 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
−
* P.R. Vital. Analytical Geometru 2D and 3D Analytical Geometry 2D and 3D [https://www.amazon.com/Analytical-Geometry-2D-3D-Vittal/dp/8131773604 book3]
  
−  
−  
−
=== Software and tools used within the course ===
  
−
* No.
  
   
 
== Activities and Teaching Methods ==
  
== Activities and Teaching Methods ==
Line 128: Line 122:
 
|+ Teaching and Learning Methods within each section
  
|+ Teaching and Learning Methods within each section
 
|-
  
|-
−
! Teaching Techniques !! Section 1 !! Section 2 !! Section 3
 +
! Teaching Techniques !! Section 1 !! Section 2 !! Section 3 !! Section 4
 
|-
  
|-
−
| Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) || 1 || 1 || 1
 +
| Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Project-based learning (students work on a project) || 0 || 0 || 0
 +
| Project-based learning (students work on a project) || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Modular learning (facilitated self-study) || 0 || 0 || 0
 +
| Modular learning (facilitated self-study) || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level) || 1 || 1 || 1
 +
| Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level) || 1 || 1 || 1 || 1
 
|-
  
|-
−
| Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them) || 0 || 0 || 0
 +
| Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them) || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Business game (learn by playing a game that incorporates the principles of the material covered within the course) || 0 || 0 || 0
 +
| Business game (learn by playing a game that incorporates the principles of the material covered within the course) || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Inquiry-based learning || 0 || 0 || 0
 +
| Inquiry-based learning || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Just-in-time teaching || 0 || 0 || 0
 +
| Just-in-time teaching || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Process oriented guided inquiry learning (POGIL) || 0 || 0 || 0
 +
| Process oriented guided inquiry learning (POGIL) || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Studio-based learning || 0 || 0 || 0
 +
| Studio-based learning || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Universal design for learning || 0 || 0 || 0
 +
| Universal design for learning || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Task-based learning || 0 || 0 || 0
 +
| Task-based learning || 0 || 0 || 0 || 0
 
|}
  
|}
 
{| class="wikitable"
  
{| class="wikitable"
 
|+ Activities within each section
  
|+ Activities within each section
 
|-
  
|-
−
! Learning Activities !! Section 1 !! Section 2 !! Section 3
 +
! Learning Activities !! Section 1 !! Section 2 !! Section 3 !! Section 4
 
|-
  
|-
−
| Lectures || 1 || 1 || 1
 +
| Lectures || 1 || 1 || 1 || 1
 
|-
  
|-
−
| Interactive Lectures || 1 || 1 || 1
 +
| Interactive Lectures || 1 || 1 || 1 || 1
 
|-
  
|-
−
| Lab exercises || 1 || 1 || 1
 +
| Lab exercises || 1 || 1 || 1 || 1
 
|-
  
|-
−
| Experiments || 0 || 0 || 0
 +
| Experiments || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Modeling || 0 || 0 || 0
 +
| Modeling || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Cases studies || 0 || 0 || 0
 +
| Cases studies || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Development of individual parts of software product code || 0 || 0 || 0
 +
| Development of individual parts of software product code || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Individual Projects || 0 || 0 || 0
 +
| Individual Projects || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Group projects || 0 || 0 || 0
 +
| Group projects || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Flipped classroom || 0 || 0 || 0
 +
| Flipped classroom || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Quizzes (written or computer based) || 1 || 1 || 1
 +
| Quizzes (written or computer based) || 1 || 1 || 1 || 1
 
|-
  
|-
−
| Peer Review || 0 || 0 || 0
 +
| Peer Review || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Discussions || 1 || 1 || 1
 +
| Discussions || 1 || 1 || 1 || 1
 
|-
  
|-
−
| Presentations by students || 0 || 0 || 0
 +
| Presentations by students || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Written reports || 0 || 0 || 0
 +
| Written reports || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Simulations and role-plays || 0 || 0 || 0
 +
| Simulations and role-plays || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Essays || 0 || 0 || 0
 +
| Essays || 0 || 0 || 0 || 0
 
|-
  
|-
−
| Oral Reports || 0 || 0 || 0
 +
| Oral Reports || 0 || 0 || 0 || 0
 
|}
  
|}
   
Line 201: Line 195:
   
 
==== Section 1 ====
  
==== Section 1 ====
−
# How to perform the shift of the vector?
 +
# How to perform Gauss elimination?
  +
# How to perform matrices multiplication?
−
# What is the geometrical interpretation of the dot product?
  
  +
# How to perform LU factorization?
−
# How to determine whether the vectors are linearly dependent?
  
  +
# How to find complete solution for any linear equation system Ax=b?
−
# What is a vector basis?
  
−
# What is the difference between matrices and determinants?
  
−
# Matrices <math display="inline">A</math> and <math display="inline">C</math> have dimensions of <math display="inline">m\times n</math> and <math display="inline">p\times q</math> respectively, and it is known that the product <math display="inline">ABC</math> exists. What are possible dimensions of <math display="inline">B</math> and <math display="inline">ABC</math>?
  
−
# How to determine the rank of a matrix?
  
−
# What is the meaning of the inverse matrix?
  
−
# How to restate a system of linear equations in the matrix form?
  
   
 
==== Section 2 ====
  
==== Section 2 ====
  +
# What is linear independence of vectors?
−
# How to represent a line in the vector form?
  
  +
# Define the four fundamental subspaces of a matrix?
−
# What is the result of intersection of two planes in vector form?
  
  +
# How to define orthogonal vectors and subspaces?
−
# How to derive the formula for the distance from a point to a line?
  
−
# How to interpret geometrically the distance between lines?
 +
# How to define orthogonal complements of the space?
  +
# How to find vector projection on a subspace?
−
# List all possible inter-positions of lines in the space.
  
  +
# How to perform linear regression for the given measurements?
−
# What is the difference between general and normalized forms of equations of a plane?
  
−
# How to rewrite the equation of a plane in a vector form?
 +
# How to find an orthonormal basis for the subspace spanned by the given vectors?
−
# What is the normal to a plane?
  
−
# How to interpret the cross products of two vectors?
  
−
# What is the meaning of scalar triple product of three vectors?
  
   
 
==== Section 3 ====
  
==== Section 3 ====
   
  +
# Give the definition of Hermitian Matrix.
−
# Formulate the canonical equation of the given quadratic curve.
  
  +
# Give the definition of Unitary Matrix.
−
# Which orthogonal transformations of coordinates do you know?
  
−
# How to perform a transformation of the coordinate system?
 +
# How to find matrix for the Fourier transform?
  +
# When we can make fast Fourier transform?
−
# How to represent a curve in the space?
  
  +
# How to find eigenvalues and eigenvectors of a matrix?
−
# What is the type of a quadric surface given by a certain equation?
  
−
# How to compose the equation of a surface of revolution?
 +
# How to diagonalize a square matrix?
−
# What is the difference between a directrix and generatrix?
  
−
# How to represent a quadric surface in the vector form?
  
   
−
=== Final assessment ===
 +
==== Section 4 ====
−
'''Section 1'''
  
−
# Evaluate <math display="inline">|\textbf{a}|^2-2\sqrt3\textbf{a}\cdot\textbf{b}-7|\textbf{b}|^2</math> given that <math display="inline">|\textbf{a}|=4</math>, <math display="inline">|\textbf{b}|=1</math>, <math display="inline">\angle(\textbf{a},\,\textbf{b})=150^{\circ}</math>.
  
−
# Prove that vectors <math display="inline">\textbf{b}(\textbf{a}\cdot\textbf{c})-\textbf{c}(\textbf{a}\cdot\textbf{b})</math> and <math display="inline">\textbf{a}</math> are perpendicular to each other.
  
−
# Bases <math display="inline">AD</math> and <math display="inline">BC</math> of trapezoid <math display="inline">ABCD</math> are in the ratio of <math display="inline">4:1</math>. The diagonals of the trapezoid intersect at point <math display="inline">M</math> and the extensions of sides <math display="inline">AB</math> and <math display="inline">CD</math> intersect at point <math display="inline">P</math>. Let us consider the basis with <math display="inline">A</math> as the origin, <math display="inline">\overrightarrow{AD}</math> and <math display="inline">\overrightarrow{AB}</math> as basis vectors. Find the coordinates of points <math display="inline">M</math> and <math display="inline">P</math> in this basis.
  
−
# A line segment joining a vertex of a tetrahedron with the centroid of the opposite face (the centroid of a triangle is an intersection point of all its medians) is called a median of this tetrahedron. Using vector algebra prove that all the four medians of any tetrahedron concur in a point that divides these medians in the ratio of <math display="inline">3:1</math>, the longer segments being on the side of the vertex of the tetrahedron.
  
−
# Find <math display="inline">A+B</math> and <math display="inline">2A-3B+I</math>.
  
−
# Find the products <math display="inline">AB</math> and <math display="inline">BA</math> (and so make sure that, in general, <math display="inline">AB\neq BA</math> for matrices).
  
−
# Find the inverse matrices for the given ones.
  
−
# Find the determinants of the given matrices.
  
−
# Point <math display="inline">M</math> is the centroid of face <math display="inline">BCD</math> of tetrahedron <math display="inline">ABCD</math>. The old coordinate system is given by <math display="inline">A</math>, <math display="inline">\overrightarrow{AB}</math>, <math display="inline">\overrightarrow{AC}</math>, <math display="inline">\overrightarrow{AD}</math>, and the new coordinate system is given by <math display="inline">M</math>, <math display="inline">\overrightarrow{MB}</math>, <math display="inline">\overrightarrow{MC}</math>, <math display="inline">\overrightarrow{MA}</math>. Find the coordinates of a point in the old coordinate system given its coordinates <math display="inline">x'</math>, <math display="inline">y'</math>, <math display="inline">z'</math> in the new one.
  
   
  +
# How to solve linear differential equations?
−
'''Section 2'''
  
  +
# Make the definition of symmetric matrix?
−
# Two lines are given by the equations <math display="inline">\textbf{r}\cdot\textbf{n}=A</math> and <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math>, and at that <math display="inline">\textbf{a}\cdot\textbf{n}\neq0</math>. Find the position vector of the intersection point of these lines.
  
  +
# Make the definition of positive definite matrix?
−
# Find the distance from point <math display="inline">M_0</math> with the position vector <math display="inline">\textbf{r}_0</math> to the line defined by the equation (a) <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math>; (b) <math display="inline">\textbf{r}\cdot\textbf{n}=A</math>.
  
  +
# Make the definition of similar matrices?
−
# Diagonals of a rhombus intersect at point <math display="inline">M(1;\,2)</math>, the longest of them being parallel to a horizontal axis. The side of the rhombus equals 2 and its obtuse angle is <math display="inline">120^{\circ}</math>. Compose the equations of the sides of this rhombus.
  
  +
# How to find left and right inverses matrices, pseudoinverse matrix?
−
# Compose the equations of lines passing through point <math display="inline">A(2;-4)</math> and forming angles of <math display="inline">60^{\circ}</math> with the line <math display="inline">\frac{1-2x}3=\frac{3+2y}{-2}</math>.
  
  +
# How to make singular value decomposition of the matrix?
−
# Find the cross product of (a) vectors <math display="inline">\textbf{a}(3;-2;\,1)</math> and <math display="inline">\textbf{b}(2;-5;-3)</math>; (b) vectors <math display="inline">\textbf{a}(3;-2;\,1)</math> and <math display="inline">\textbf{c}(-18;\,12;-6)</math>.
  
−
# A triangle is constructed on vectors <math display="inline">\textbf{a}(2;4;-1)</math> and <math display="inline">\textbf{b}(-2;1;1)</math>. (a) Find the area of this triangle. (b) Find the altitudes of this triangle.
  
−
# Find the scalar triple product of <math display="inline">\textbf{a}(1;\,2;-1)</math>, <math display="inline">\textbf{b}(7;3;-5)</math>, <math display="inline">\textbf{c}(3;\,4;-3)</math>.
  
−
# It is known that basis vectors <math display="inline">\textbf{e}_1</math>, <math display="inline">\textbf{e}_2</math>, <math display="inline">\textbf{e}_3</math> have lengths of <math display="inline">1</math>, <math display="inline">2</math>, <math display="inline">2\sqrt2</math> respectively, and <math display="inline">\angle(\textbf{e}_1,\textbf{e}_2)=120^{\circ}</math>, <math display="inline">\angle(\textbf{e}_1,\textbf{e}_3)=135^{\circ}</math>, <math display="inline">\angle(\textbf{e}_2,\textbf{e}_3)=45^{\circ}</math>. Find the volume of a parallelepiped constructed on vectors with coordinates <math display="inline">(-1;\,0;\,2)</math>, <math display="inline">(1;\,1\,4)</math> and <math display="inline">(-2;\,1;\,1)</math> in this basis.
  
   
  +
=== Final assessment ===
−
'''Section 3'''
  
  +
==== Section 1 ====
−
# Prove that a curve given by <math display="inline">34x^2+24xy+41y^2-44x+58y+1=0</math> is an ellipse. Find the major and minor axes of this ellipse, its eccentricity, coordinates of its center and foci. Find the equations of axes and directrices of this ellipse.
  
−
# Determine types of curves given by the following equations. For each of the curves, find its canonical coordinate system (i.e. indicate the coordinates of origin and new basis vectors in the initial coordinate system) and its canonical equation. (a) <math display="inline">9x^2-16y^2-6x+8y-144=0</math>; (b) <math display="inline">9x^2+4y^2+6x-4y-2=0</math>; (c) <math display="inline">12x^2-12x-32y-29=0</math>; (d) <math display="inline">xy+2x+y=0</math>;
 +
# Find linear independent vectors (exclude dependent): <math display="inline">\overrightarrow{a}=[4,0,3,2]^T</math>, <math display="inline">\overrightarrow{b}=[1,-7,4,5]^T</math>, <math display="inline">\overrightarrow{c}=[7,1,5,3]^T</math>, <math display="inline">\overrightarrow{d}=[-5,-3,-3,-1]^T</math>, <math display="inline">\overrightarrow{e}=[1,-5,2,3]^T</math>. Find <math display="inline">rank(A)</math> if <math display="inline">A</math> is a composition of this vectors. Find <math display="inline">rank(A^T)</math>.
−
# Find the equations of lines tangent to curve <math display="inline">6xy+8y^2-12x-26y+11=0</math> that are (a) parallel to line <math display="inline">6x+17y-4=0</math>; (b) perpendicular to line <math display="inline">41x-24y+3=0</math>; (c) parallel to line <math display="inline">y=2</math>.
 +
# Find <math display="inline">E</math>: <math display="inline">EA=U</math> (<math display="inline">U</math> – upper-triangular matrix). Find <math display="inline">L=E^-1</math>, if <math display="inline">A=\left(
  +
\begin{array}{ccc}
−
# For each value of parameter <math display="inline">a</math> determine types of surfaces given by the equations: (a) <math display="inline">x^2+y^2-z^2=a</math>; (b) <math display="inline">x^2+a\left(y^2+z^2\right)=1</math>; (c) <math display="inline">x^2+ay^2=az</math>; (d) <math display="inline">x^2+ay^2=az+1</math>.
  
  +
2 & 5 & 7 \\
−
# Find a vector equation of a right circular cone with apex <math display="inline">M_0\left(\textbf{r}_0\right)</math> and axis <math display="inline">\textbf{r}=\textbf{r}_0+\textbf{a}t</math> if it is known that generatrices of this cone form the angle of <math display="inline">\alpha</math> with its axis.
  
  +
6 & 4 & 9 \\
−
# Find the equation of a cylinder with radius <math display="inline">\sqrt2</math> that has an axis <math display="inline">x=1+t</math>, <math display="inline">y=2+t</math>, <math display="inline">z=3+t</math>.
  
  +
4 & 1 & 8 \\
−
# An ellipsoid is symmetric with respect to coordinate planes, passes through point <math display="inline">M(3;\,1;\,1)</math> and circle <math display="inline">x^2+y^2+z^2=9</math>, <math display="inline">x-z=0</math>. Find the equation of this ellipsoid.
  
  +
\end{array}
  +
\right)</math>.
  +
# Find complete solution for the system <math display="inline">Ax=b</math>, if <math display="inline">b=[7,18,5]^T</math> and <math display="inline">A=\left(
  +
\begin{array}{cccc}
  +
6 & -2 & 1 & -4 \\
  +
4 & 2 & 14 & -31 \\
  +
2 & -1 & 3 & -7 \\
  +
\end{array}
  +
\right)</math>. Provide an example of vector b that makes this system unsolvable.
  +
==== Section 2 ====
  +
# Find the dimensions of the four fundamental subspaces associated with <math display="inline">A</math>, depending on the parameters <math display="inline">a</math> and <math display="inline">b</math>: <math display="inline">A=\left(
  +
\begin{array}{cccc}
  +
7 & 8 & 5 & 3 \\
  +
4 & a & 3 & 2 \\
  +
6 & 8 & 4 & b \\
  +
3 & 4 & 2 & 1 \\
  +
\end{array}
  +
\right)</math>.
  +
# Find a vector <math display="inline">x</math> orthogonal to the Row space of matrix <math display="inline">A</math>, and a vector <math display="inline">y</math> orthogonal to the <math display="inline">C(A)</math>, and a vector <math display="inline">z</math> orthogonal to the <math display="inline">N(A)</math>: <math display="inline">A=\left(
  +
\begin{array}{ccc}
  +
1 & 2 & 2 \\
  +
3 & 4 & 2 \\
  +
4 & 6 & 4 \\
  +
\end{array}
  +
\right)</math>.
  +
# Find the best straight-line <math display="inline">y(x)</math> fit to the measurements: <math display="inline">y(-2)=4</math>, <math display="inline">y(-1)=3</math>, <math display="inline">y(0)=2</math>, <math display="inline">y(1)-0</math>.
  +
# Find the projection matrix <math display="inline">P</math> of vector <math display="inline">[4,3,2,0]^T</math> onto the <math display="inline">C(A)</math>: <math display="inline">A=\left(
  +
\begin{array}{cc}
  +
1 & -2 \\
  +
1 & -1 \\
  +
1 & 0 \\
  +
1 & 1 \\
  +
\end{array}
  +
\right)</math>.
  +
# Find an orthonormal basis for the subspace spanned by the vectors: <math display="inline">\overrightarrow{a}=[-2,2,0,0]^T</math>, <math display="inline">\overrightarrow{b}=[0,1,-1,0]^T</math>, <math display="inline">\overrightarrow{c}=[0,1,0,-1]^T</math>. Then express <math display="inline">A=[a,b,c]</math> in the form of <math display="inline">A=QR</math>
  +
==== Section 3 ====
  +
# Find eigenvector of the circulant matrix <math display="inline">C</math> for the eigenvalue = <math display="inline">{c}_1</math>+<math display="inline">{c}_2</math>+<math display="inline">{c}_3</math>+<math display="inline">{c}_4</math>: <math display="inline">C=\left(
  +
\begin{array}{cccc}
  +
{c}_1 & {c}_2 & {c}_3 & {c}_4 \\
  +
{c}_4 & {c}_1 & {c}_2 & {c}_3 \\
  +
{c}_3 & {c}_4 & {c}_1 & {c}_2 \\
  +
{c}_2 & {c}_3 & {c}_4 & {c}_1 \\
  +
\end{array}
  +
\right)</math>.
  +
# Diagonalize this matrix: <math display="inline">A=\left(
  +
\begin{array}{cc}
  +
2 & 1-i \\
  +
1+i & 3 \\
  +
\end{array}
  +
\right)</math>.
  +
# <math display="inline">A</math> is the matrix with full set of orthonormal eigenvectors. Prove that <math display="inline">AA=A^HA^H</math>.
  +
# Find all eigenvalues and eigenvectors of the cyclic permutation matrix <math display="inline">P=\left(
  +
\begin{array}{cccc}
  +
0 & 1 & 0 & 0 \\
  +
0 & 0 & 1 & 0 \\
  +
0 & 0 & 0 & 1 \\
  +
1 & 0 & 0 & 0 \\
  +
\end{array}
  +
\right)</math>.
  +
==== Section 4 ====
  +
<ol>
  +
<li><p>Find <math display="inline">det(e^A)</math> for <math display="inline">A=\left(
  +
\begin{array}{cc}
  +
2 & 1 \\
  +
2 & 3 \\
  +
\end{array}
  +
\right)</math>.</p></li>
  +
<li><p>Write down the first order equation system for the following differential equation and solve it:</p>
  +
<p><math display="inline">d^3y/dx+d^2y/dx-2dy/dx=0</math></p>
  +
<p><math display="inline">y''(0)=6</math>, <math display="inline">y'(0)=0</math>, <math display="inline">y(0)=3</math>.</p>
  +
<p>Is the solution of this system will be stable?</p></li>
  +
<li><p>For which <math display="inline">a</math> and <math display="inline">b</math> quadratic form <math display="inline">Q(x,y,z)</math> is positive definite:</p>
  +
<p><math display="inline">Q(x,y,z)=ax^2+y^2+2z^2+2bxy+4xz</math></p></li>
  +
<li><p>Find the SVD and the pseudoinverse of the matrix <math display="inline">A=\left(
  +
\begin{array}{ccc}
  +
1 & 0 & 0 \\
  +
0 & 1 & 1 \\
  +
\end{array}
  +
\right)</math>.</p></li></ol>
   
 
=== The retake exam ===
  
=== The retake exam ===
  +
Exams will be paper-based and will be conducted in a form of problem solving, where the problems will be similar to those mentioned above. Students will be given 1-2 hours to complete the exam.
−
Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.
  
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First retake will be conducted in the same form as the midterm and final exams. The weight of the retake exam will be the same as the all course.
  +
Second retake will be conducted in the same form as the midterm and final exams. The weight of the retake exam will be the same as the all course.

Latest revision as of 12:57, 27 January 2023

Analytical Geometry & Linear Algebra – II

  • Course name: Analytical Geometry & Linear Algebra – II
  • Code discipline: CSE204
  • Subject area: Math

Short Description

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 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
  3. LU and LDU factorization
  4. Transposes and permutations
  5. Vector spaces and subspaces
  6. The null space: Solving Ax=0 and Ax=b
  7. Row reduced echelon form. Matrix rank
  8. Numerical methods for solving systems of linear algebraic equations
Linear regression analysis, QR-decomposition
  1. Independence, basis and dimension
  2. The four fundamental subspaces
  3. Orthogonal vectors and subspaces
  4. Projections onto subspaces Projection matrices
  5. Least squares approximations
  6. Gram-Schmidt orthogonalization and A = QR
Matrix Diagonalization
  1. Complex Numbers
  2. Hermitian and Unitary Matrices
  3. Eigenvalues and eigenvectors
  4. Matrix diagonalization
Symmetric, positive definite and similar matrices. Singular value decomposition
  1. Linear differential equations.
  2. Symmetric matrices.
  3. Positive definite matrices
  4. Similar matrices.
  5. Left and right inverses, pseudoinverse
  6. Singular value decomposition (SVD)

Intended Learning Outcomes (ILOs)

ILOs defined at three levels

We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

  • explain the geometrical interpretation of the basic operations of vector algebra,
  • restate equations of lines and planes in different forms,
  • interpret the geometrical meaning of the conic sections in the mathematical expression,
  • give the examples of the surfaces of revolution,
  • understand the value of geometry in various fields of science and techniques.

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 ...

  • perform the basic operations of vector algebra,
  • use different types of equations of lines and planes to solve the plane and space problems,
  • represent the conic section in canonical form,
  • compose the equation of quadric surface.

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 ...

  • list basic notions of vector algebra,
  • recite the base form of the equations of transformations in planes and spaces,
  • recall equations of lines and planes,
  • identify the type of conic section,
  • recognize the kind of quadric surfaces.

Grading

Course grading range

Grade Range Description of performance
A. Excellent 90-110 -
B. Good 75-89 -
C. Satisfactory 60-74 -
D. Fail 0-59 -

Course activities and grading breakdown

Activity Type Percentage of the overall course grade
Midterm 30
Two intermediate tests 30 (15 for each)
Final exam 30
Five programming tasks 20

Recommendations for students on how to succeed in the course

  • Participation is important. Attending lectures is the key to success in this course.
  • Review lecture materials before classes to do well.
  • Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.

Resources, literature and reference materials

Open access resources

  • 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
  • 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

Activities and Teaching Methods

Teaching and Learning Methods within each section
Teaching Techniques Section 1 Section 2 Section 3 Section 4
Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) 0 0 0 0
Project-based learning (students work on a project) 0 0 0 0
Modular learning (facilitated self-study) 0 0 0 0
Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level) 1 1 1 1
Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them) 0 0 0 0
Business game (learn by playing a game that incorporates the principles of the material covered within the course) 0 0 0 0
Inquiry-based learning 0 0 0 0
Just-in-time teaching 0 0 0 0
Process oriented guided inquiry learning (POGIL) 0 0 0 0
Studio-based learning 0 0 0 0
Universal design for learning 0 0 0 0
Task-based learning 0 0 0 0
Activities within each section
Learning Activities Section 1 Section 2 Section 3 Section 4
Lectures 1 1 1 1
Interactive Lectures 1 1 1 1
Lab exercises 1 1 1 1
Experiments 0 0 0 0
Modeling 0 0 0 0
Cases studies 0 0 0 0
Development of individual parts of software product code 0 0 0 0
Individual Projects 0 0 0 0
Group projects 0 0 0 0
Flipped classroom 0 0 0 0
Quizzes (written or computer based) 1 1 1 1
Peer Review 0 0 0 0
Discussions 1 1 1 1
Presentations by students 0 0 0 0
Written reports 0 0 0 0
Simulations and role-plays 0 0 0 0
Essays 0 0 0 0
Oral Reports 0 0 0 0

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

  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?

Section 2

  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?

Section 3

  1. Give the definition of Hermitian Matrix.
  2. Give the definition of Unitary Matrix.
  3. How to find matrix for the Fourier transform?
  4. When we can make fast Fourier transform?
  5. How to find eigenvalues and eigenvectors of a matrix?
  6. How to diagonalize a square matrix?

Section 4

  1. How to solve linear differential equations?
  2. Make the definition of symmetric matrix?
  3. Make the definition of positive definite matrix?
  4. Make the definition of similar matrices?
  5. How to find left and right inverses matrices, pseudoinverse matrix?
  6. How to make singular value decomposition of the matrix?

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?

  3. For which and quadratic form is positive definite:

  4. Find the SVD and the pseudoinverse of the matrix .

The retake exam

Exams will be paper-based and will be conducted in a form of problem solving, where the problems will be similar to those mentioned above. Students will be given 1-2 hours to complete the exam. First retake will be conducted in the same form as the midterm and final exams. The weight of the retake exam will be the same as the all course. Second retake will be conducted in the same form as the midterm and final exams. The weight of the retake exam will be the same as the all course.

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