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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,

   \item concepts of linear algebra objects and their representation in vector-matrix form

\end{itemize}

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

   \item Understand key principles involved in solution of linear equation systems and the properties of matrices
   \item Linear regression analysis
   \item Fast Fourier Transform 
   \item 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

   \item Become familiar with the four fundamental subspaces
   \item Linear regression analysis
   \item Fast Fourier Transform
   \item 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

   \item Make linear regression analysis
   \item Fast Fourier Transform
   \item 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

   Applications, 4th Edition, Brooks Cole, 2006. ISBN: 9780030105678
   \item Gilbert Strang. Introduction to Linear Algebra, 4th Edition, Wellesley, MA: Wellesley-Cambridge Press, 2009. ISBN: 9780980232714

\end{itemize}

\paragraph{Reference material:} \begin{itemize}

   \item 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

\end{itemize}

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.

   \item Matrix operations, including inverses. $LU$ and $LDU$ factorization.
   \item Transposes and permutations. Vector spaces and subspaces.
   \item The null space: Solving $Ax=0$ and $Ax=b$. Row reduced echelon form. Matrix rank.

\end{itemize}

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?

   \item How to perform matrices multiplication?
   \item How to perform LU factorization?
   \item 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.

   \item Perform $A=LU$ factorization for the given matrix $A$.
   
   \item Factor the given symmetric matrix  $A$ into  $A=LDL^{T}$ with the diagonal pivot matrix $D$.
   
   \item Find inverse matrix $A^{-}1$ for the given matrix $A$.

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 $rank(A^{T})$.

   \item Find $E$: $EA=U$ ($U$ – upper-triangular matrix). Find $L=E^{-}1$, if
    $A=\left({\begin{array}{ccc}2&5&7\\6&4&9\\4&1&8\\\end{array}}\right)$.
   
   \item Find complete solution for the system $Ax=b$, if $b=[7,18,5]^{T}$ and 
   $A=\left({\begin{array}{cccc}6&-2&1&-4\\4&2&14&-31\\2&-1&3&-7\\\end{array}}\right)$.
      Provide an example of vector b that makes this system unsolvable.

Section 2

Section title

Linear regression analysis and decomposition $A=QR$.

Topics covered in this section

• Independence, basis and dimension. The four fundamental subspaces.

   \item Orthogonal vectors and subspaces. Projections onto subspaces
   \item Projection matrices. Least squares approximations. Gram-Schmidt and A = QR.

\end{itemize}

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?

   \item Define the four fundamental subspaces of a matrix?
   \item How to define orthogonal vectors and subspaces?
   \item How to define orthogonal complements of the space?
   \item How to find vector projection on a subspace?
   \item How to perform linear regression for the given measurements?
   \item 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

   \item Find four fundamental subspaces of the given matrix.
   \item Check out orthogonality of the given subspaces.
   \item Find orthogonal complement for the given subspace.
   \item Find vector projection on the given subspace.
   \item Perform linear regression for the given measurements.
   \item Find an orthonormal basis for the subspace spanned by the given vectors.

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 $b$:

   $A=\left({\begin{array}{cccc}7&8&5&3\\4&a&3&2\\6&8&4&b\\3&4&2&1\\\end{array}}\right)$.
   \item Find a vector $x$ orthogonal to the Row space of matrix $A$, and a vector $y$ orthogonal to the $C(A)$, and a vector $z$ orthogonal to the $N(A)$:
   $A=\left({\begin{array}{ccc}1&2&2\\3&4&2\\4&6&4\\\end{array}}\right)$.
   \item Find the best straight-line $y(x)$  fit to the measurements: $y(-2)=4$, $y(-1)=3$, $y(0)=2$, $y(1)-0$.
   \item Find the projection matrix $P$ of vector $[4,3,2,0]^{T}$ onto the $C(A)$: 
   $A=\left({\begin{array}{cc}1&-2\\1&-1\\1&0\\1&1\\\end{array}}\right)$.
      \item Find an orthonormal basis for the subspace spanned by the vectors: ${\overrightarrow {a}}=[-2,2,0,0]^{T}$, ${\overrightarrow {b}}=[0,1,-1,0]^{T}$, ${\overrightarrow {c}}=[0,1,0,-1]^{T}$. Then express $A=[a,b,c]$ in the form of $A=QR$

Section 3

Section title

Fast Fourier Transform. Matrix Diagonalization.

Topics covered in this section

• Complex Numbers. Hermitian and Unitary Matrices.

   \item Fourier Series. The Fast Fourier Transform
   \item Eigenvalues and eigenvectors. Matrix diagonalization. 

\end{itemize}

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. Make the definition of Hermitian Matrix.

   \item Make the definition of Unitary Matrix.
   \item How to find matrix for the Fourier transform?
   \item When we can make fast Fourier transform?
   \item How to find eigenvalues and eigenvectors of a matrix?
   \item How to diagonalize a square matrix?

Typical questions for seminar classes (labs) within this section

1. Check out is the given matrix Hermitian.

   \item Check out is the given matrix Unitary.
   \item Find the matrix for the given Fourier transform.
   \item Find eigenvalues and eigenvectors for the given matrix.
   \item Find diagonalize form for the given matrix.

Test questions for final assessment in this section

1. Find eigenvector of the circulant matrix $C$ for the eigenvalue = ${c}_{1}$+${c}_{2}$+${c}_{3}$+${c}_{4}$:

   $C=\left({\begin{array}{cccc}{c}_{1}&{c}_{2}&{c}_{3}&{c}_{4}\\{c}_{4}&{c}_{1}&{c}_{2}&{c}_{3}\\{c}_{3}&{c}_{4}&{c}_{1}&{c}_{2}\\{c}_{2}&{c}_{3}&{c}_{4}&{c}_{1}\\\end{array}}\right)$.
   \item Diagonalize this matrix:
    $A=\left({\begin{array}{cc}2&1-i\\1+i&3\\\end{array}}\right)$.
   \item $A$ is the matrix with full set of orthonormal eigenvectors. Prove that $AA=A^{H}A^{H}$.
   \item Find all eigenvalues and eigenvectors of the cyclic permutation matrix  
   $P=\left({\begin{array}{cccc}0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0\\\end{array}}\right)$.

Section 4

Section title

Symmetric, positive definite and similar matrices. Singular value decomposition.

Topics covered in this section

• Linear differential equations.

   \item Symmetric matrices. Positive definite matrices.
   \item Similar matrices. Left and right inverses, pseudoinverse. Singular value decomposition (SVD).

\end{itemize}

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 solve linear differential equations?

   \item Make the definition of symmetric matrix?
   \item Make the definition of positive definite matrix?
   \item Make the definition of similar matrices?
   \item How to find left and right inverses matrices, pseudoinverse matrix?
   \item How to make singular value decomposition of the matrix?

Typical questions for seminar classes (labs) within this section

1. Find solution of the linear differential equation.

   \item Make the definition of symmetric matrix.
   \item Check out the given matrix on positive definess
   \item Check out the given matrices on similarity.
   \item For the given matrix find left and right inverse matrices, pseudoinverse matrix.
   \item Make the singular value decomposition of the given matrix.

Test questions for final assessment in this section

1. Find $det(e^{A})$ for

    $A=\left({\begin{array}{cc}2&1\\2&3\\\end{array}}\right)$.
   \item Write down the first order equation system for the following differential equation and solve it:
   
   $d^{3}y/dx+d^{2}y/dx-2dy/dx=0$
   
   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y"(0)=6}
, $y'(0)=0$, $y(0)=3$.
   
   Is the solution of this system will be stable?
   
   \item For which $a$ and $b$ quadratic form $Q(x,y,z)$  is positive definite:
   
   $Q(x,y,z)=ax^{2}+y^{2}+2z^{2}+2bxy+4xz$
   
   \item Find the SVD and the pseudoinverse of the matrix
   $A=\left({\begin{array}{ccc}1&0&0\\0&1&1\\\end{array}}\right)$.