Syllabus - Yousef's Notes
Syllabus

Syllabus

#MODULE 1: VECTORS AND MATRICES

#SESSION 1 (LIVE IN-PERSON)

  • Introduction to Vectors
  • Vectors and Linear Combination
  • Length, Dot Product, and Unit Vectors
  • Minkowski Distances
  • Cosine Similarity
  • Vectors Applications

#SESSION 2 (LIVE IN-PERSON)

  • Matrix Definition
  • Types and Properties of Matrices
  • Transpose and Inverse of a Matrix
  • Matrix Multiplication
  • Strassen Algorithm
  • Matrices Exercises - Simplifications based on transpose and inverse properties.

#SESSION 3 (LIVE IN-PERSON)

  • Practice: Review Key Ideas, Vectors, and Matrices Applications

#MODULE 2: SOLVING LINEAR EQUATIONS

#SESSION 4 (LIVE IN-PERSON)

  • Linear Equations (n Equations, n Unknowns)
  • Row Picture and Column Picture of Equations in 2D and 3D
  • Matrix Picture
  • The Idea of Gauss Elimination

#SESSION 5 (LIVE IN-PERSON)

  • Gauss Elimination Using Matrices
  • Elimination Matrix
  • Permutation Matrix
  • Row Echelon Form (REF)
  • Gauss Jordan Elimination
  • Row Reduced Echelon Form (RREF)

#SESSION 6 (LIVE IN-PERSON)

  • Factorization A=LU
  • Factorization A=LDU
  • Factorizations PA=LU
  • Computational Cost A=LU

#SESSION 7 (LIVE IN-PERSON)

  • Practice
  • Application: Balance Chemistry
  • Application: Leontief
  • Application: Electric Circuit

#SESSION 8 (LIVE IN-PERSON)

  • Gauss Jordan Inverse of A
  • Round Off Error
  • Partial Pivoting
  • Network Flow Application

#SESSION 9 (LIVE IN-PERSON)

  • Jacobi Numerical Method
  • Gauss-Seidel Numerical Method
  • Finite Linear Games
  • GPS Calculation

#MODULE 3: VECTOR SPACES

#SESSION 10 (LIVE IN-PERSON)

  • Vector Spaces and Subspaces
  • Span, Base, and Rank
  • Echelon Form and Pivots
  • Reduced Row Echelon Form
  • Column Space of A: Solving Ax=b
  • Nullspace of A: Solving Ax=0

#SESSION 11 (LIVE IN-PERSON)

  • Subspace Dimension
  • Complete Solution of Ax=b
  • Row Space of A
  • Nullspace of A Transpose
  • The 4 Fundamental Subspaces Picture

#SESSION 12 (LIVE IN-PERSON)

  • Independence Vectors
  • Basis of Subspaces
  • Complete Solution of Ax=b

#SESSION 13 (LIVE IN-PERSON)

  • Subspaces
  • 4 Fundamental Subspaces
  • Complete Solution of Ax=b
  • Short Questions

#SESSION 14 (LIVE IN-PERSON)

  • Midterm Review

#SESSION 15 (LIVE IN-PERSON)

  • Midterm Exam

#MODULE 4: ORTHOGONALITY

#SESSION 16 (LIVE IN-PERSON)

  • Orthogonal Vectors
  • Orthogonal Subspaces
  • Fundamental 4 Subspaces Orthogonality
  • Projections
  • Least Squares

#SESSION 17 (LIVE IN-PERSON)

  • Orthonormal Bases
  • Orthogonal Matrix Q
  • Gram-Schmidt
  • QR Decomposition

#SESSION 18 (LIVE IN-PERSON)

  • Practice:
    • Least Squares for Different Equations
    • Gram-Schmidt
    • QR Decomposition

#MODULE 5: DETERMINANTS

#SESSION 19 (LIVE IN-PERSON)

  • Determinant Properties
  • Formula of Determinant
  • Cofactors
  • Cross Product Application
  • Condensed Method for Determinants

#SESSION 20 (LIVE IN-PERSON)

  • Formula for Inverse of A
  • Cramer’s Rule
  • Areas
  • Volumes
  • Application: Hill Cipher

#MODULE 6: EIGENVALUES AND EIGENVECTORS

#SESSION 21 (LIVE IN-PERSON)

  • Introduction to Eigenvalues and Eigenvectors
  • Diagonalizing a Matrix
  • Matrix Powers via Diagonalization

#SESSION 22 (LIVE IN-PERSON)

  • Fibonacci Sequence
  • Markov Matrices
  • Differential Equations

#SESSION 23 (LIVE IN-PERSON)

  • Math Challenge: Team Competition

#SESSION 24 (LIVE IN-PERSON)

  • Symmetric Matrices
  • Singular Value Decomposition (SVD)

#SESSION 25 (LIVE IN-PERSON)

  • Practice:
    • Singular Value Decomposition (SVD)
    • Principal Component Analysis

#MODULE 7: LINEAR TRANSFORMATIONS

#SESSION 26 (LIVE IN-PERSON)

  • Linear Transformations
  • Matrix of Linear Transformations
  • Inverse of Transformation
  • Composition of Linear Transformations

#SESSION 27 (LIVE IN-PERSON)

  • Practice: Review Key Ideas, Linear Transformations

#SESSION 28 (LIVE IN-PERSON)

  • Presentation of Linear Algebra Applications by Students

#SESSION 29 (LIVE IN-PERSON)

  • Final Review Session: Vectors and Matrices, Solving Linear Equations, Vector Spaces, Orthogonality, Determinants, Eigenvalues, and Linear Transformations

#SESSION 30 (LIVE IN-PERSON)

  • Final Exam

#Evaluation Method

#A. CLASS PARTICIPATION

  • Worth 10% of the overall grade.
  • Assesses students’ active engagement during lectures.
  • Students are expected to contribute meaningfully to discussions, ask questions, provide relevant remarks, participate in class exercises, and engage in work group challenges.
  • Active participation is measured by the quality and relevance of contributions.

#B. GROUP WORK

  • Worth 30% of the final grade.
  • Students will work in groups on a particular application in Linear Algebra selected from the modules.
  • Graded after the deadline.

#C. INTERMEDIATE TESTS: EXERCISES + MIDTERM

  • Worth 30% of the final grade.
  • Includes exercises assignments based on class exercises.

#D. FINAL-EXAM

  • Worth 30% of the overall grade.
  • Minimum score of 3.5 required to pass the course.
  • Detailed characteristics of the final exam will be provided at the beginning of the semester.

#Prep Roadmap

To prepare for each session of your linear algebra course, you should study and practice specific sections from the book “Elementary Linear Algebra” based on the syllabus provided. Here’s a breakdown of which parts of the book to focus on before each session:

#MODULE 1: VECTORS AND MATRICES

#SESSION 1 (LIVE IN-PERSON)

  • Introduction to Vectors: Chapter 3.1 (Vectors in 2-Space, 3-Space, and n-Space)
  • Vectors and Linear Combination: Chapter 3.1
  • Length, Dot Product, and Unit Vectors: Chapter 3.2 (Norm, Dot Product, and Distance in ( R^n ))
  • Minkowski Distances: Not explicitly covered in the table of contents, but related to norms in Chapter 3.2.
  • Cosine Similarity: Related to dot product in Chapter 3.2.
  • Vectors Applications: Chapter 3.5 (Cross Product) and Chapter 4.9 (Basic Matrix Transformations in ( R^2 ) and ( R^3 ))

#SESSION 2 (LIVE IN-PERSON)

  • Matrix Definition: Chapter 1.3 (Matrices and Matrix Operations)
  • Types and Properties of Matrices: Chapter 1.3
  • Transpose and Inverse of a Matrix: Chapter 1.4 (Inverses; Algebraic Properties of Matrices)
  • Matrix Multiplication: Chapter 1.3
  • Strassen Algorithm: Not explicitly covered in the table of contents.
  • Matrices Exercises: Chapter 1.3 and 1.4

#SESSION 3 (LIVE IN-PERSON)

  • Practice: Review Key Ideas, Vectors, and Matrices Applications: Review Chapters 3.1, 3.2, 1.3, and 1.4

#MODULE 2: SOLVING LINEAR EQUATIONS

#SESSION 4 (LIVE IN-PERSON)

  • Linear Equations (n Equations, n Unknowns): Chapter 1.1 (Introduction to Systems of Linear Equations)
  • Row Picture and Column Picture of Equations in 2D and 3D: Chapter 1.1
  • Matrix Picture: Chapter 1.1
  • The Idea of Gauss Elimination: Chapter 1.2 (Gaussian Elimination)

#SESSION 5 (LIVE IN-PERSON)

  • Gauss Elimination Using Matrices: Chapter 1.2
  • Elimination Matrix: Chapter 1.5 (Elementary Matrices and a Method for Finding ( A^{-1} ))
  • Permutation Matrix: Chapter 1.5
  • Row Echelon Form (REF): Chapter 1.2
  • Gauss Jordan Elimination: Chapter 1.2
  • Row Reduced Echelon Form (RREF): Chapter 1.2

#SESSION 6 (LIVE IN-PERSON)

  • Factorization A=LU: Chapter 9.1 (LU-Decompositions)
  • Factorization A=LDU: Chapter 9.1
  • Factorizations PA=LU: Chapter 9.1
  • Computational Cost A=LU: Chapter 9.1

#SESSION 7 (LIVE IN-PERSON)

  • Practice: Review Chapters 1.1, 1.2, and 9.1
  • Application: Balance Chemistry: Chapter 1.9 (Balancing Chemical Equations)
  • Application: Leontief: Chapter 1.10 (Application: Leontief Input-Output Models)
  • Application: Electric Circuit: Chapter 1.9 (Electrical Circuits)

#SESSION 8 (LIVE IN-PERSON)

  • Gauss Jordan Inverse of A: Chapter 1.5
  • Round Off Error: Not explicitly covered in the table of contents.
  • Partial Pivoting: Chapter 1.2
  • Network Flow Application: Chapter 1.9 (Network Analysis (Traffic Flow))

#SESSION 9 (LIVE IN-PERSON)

  • Jacobi Numerical Method: Not explicitly covered in the table of contents.
  • Gauss-Seidel Numerical Method: Not explicitly covered in the table of contents.
  • Finite Linear Games: Not explicitly covered in the table of contents.
  • GPS Calculation: Not explicitly covered in the table of contents.

#MODULE 3: VECTOR SPACES

#SESSION 10 (LIVE IN-PERSON)

  • Vector Spaces and Subspaces: Chapter 4.1 (Real Vector Spaces) and 4.2 (Subspaces)
  • Span, Base, and Rank: Chapter 4.4 (Coordinates and Basis) and 4.8 (Rank, Nullity, and the Fundamental Matrix Spaces)
  • Echelon Form and Pivots: Chapter 1.2
  • Reduced Row Echelon Form: Chapter 1.2
  • Column Space of A: Solving Ax=b: Chapter 4.7 (Row Space, Column Space, and Null Space)
  • Nullspace of A: Solving Ax=0: Chapter 4.7

#SESSION 11 (LIVE IN-PERSON)

  • Subspace Dimension: Chapter 4.5 (Dimension)
  • Complete Solution of Ax=b: Chapter 4.7
  • Row Space of A: Chapter 4.7
  • Nullspace of A Transpose: Chapter 4.7
  • The 4 Fundamental Subspaces Picture: Chapter 4.7

#SESSION 12 (LIVE IN-PERSON)

  • Independence Vectors: Chapter 4.3 (Linear Independence)
  • Basis of Subspaces: Chapter 4.4
  • Complete Solution of Ax=b: Chapter 4.7

#SESSION 13 (LIVE IN-PERSON)

  • Subspaces: Chapter 4.2
  • 4 Fundamental Subspaces: Chapter 4.7
  • Complete Solution of Ax=b: Chapter 4.7
  • Short Questions: Review Chapters 4.1-4.8

#SESSION 14 (LIVE IN-PERSON)

  • Midterm Review: Review all previous chapters covered in Modules 1-3

#SESSION 15 (LIVE IN-PERSON)

  • Midterm Exam: Covers Modules 1-3

#MODULE 4: ORTHOGONALITY

#SESSION 16 (LIVE IN-PERSON)

  • Orthogonal Vectors: Chapter 3.3 (Orthogonality)
  • Orthogonal Subspaces: Chapter 3.3
  • Fundamental 4 Subspaces Orthogonality: Chapter 4.7
  • Projections: Chapter 6.2 (Angle and Orthogonality in Inner Product Spaces)
  • Least Squares: Chapter 6.4 (Best Approximation; Least Squares)

#SESSION 17 (LIVE IN-PERSON)

  • Orthonormal Bases: Chapter 6.3 (Gram-Schmidt Process; QR-Decomposition)
  • Orthogonal Matrix Q: Chapter 7.1 (Orthogonal Matrices)
  • Gram-Schmidt: Chapter 6.3
  • QR Decomposition: Chapter 6.3

#SESSION 18 (LIVE IN-PERSON)

  • Practice:
    • Least Squares for Different Equations: Chapter 6.4
    • Gram-Schmidt: Chapter 6.3
    • QR Decomposition: Chapter 6.3

#MODULE 5: DETERMINANTS

#SESSION 19 (LIVE IN-PERSON)

  • Determinant Properties: Chapter 2.1 (Determinants by Cofactor Expansion) and 2.3 (Properties of Determinants; Cramer’s Rule)
  • Formula of Determinant: Chapter 2.1
  • Cofactors: Chapter 2.1
  • Cross Product Application: Chapter 3.5 (Cross Product)
  • Condensed Method for Determinants: Chapter 2.2 (Evaluating Determinants by Row Reduction)

#SESSION 20 (LIVE IN-PERSON)

  • Formula for Inverse of A: Chapter 1.4
  • Cramer’s Rule: Chapter 2.3
  • Areas: Chapter 2.3
  • Volumes: Chapter 2.3
  • Application: Hill Cipher: Not explicitly covered in the table of contents.

#MODULE 6: EIGENVALUES AND EIGENVECTORS

#SESSION 21 (LIVE IN-PERSON)

  • Introduction to Eigenvalues and Eigenvectors: Chapter 5.1 (Eigenvalues and Eigenvectors)
  • Diagonalizing a Matrix: Chapter 5.2 (Diagonalization)
  • Matrix Powers via Diagonalization: Chapter 5.2

#SESSION 22 (LIVE IN-PERSON)

  • Fibonacci Sequence: Not explicitly covered in the table of contents.
  • Markov Matrices: Chapter 5.5 (Application: Dynamical Systems and Markov Chains)
  • Differential Equations: Chapter 5.4 (Application: Differential Equations)

#SESSION 23 (LIVE IN-PERSON)

  • Math Challenge: Team Competition: Review relevant chapters based on competition topics.

#SESSION 24 (LIVE IN-PERSON)

  • Symmetric Matrices: Chapter 7.5 (Hermitian, Unitary, and Normal Matrices)
  • Singular Value Decomposition (SVD): Chapter 9.4 (Singular Value Decomposition)

#SESSION 25 (LIVE IN-PERSON)

  • Practice:
    • Singular Value Decomposition (SVD): Chapter 9.4
    • Principal Component Analysis: Chapter 9.5 (Application: Data Compression Using Singular Value Decomposition)

#MODULE 7: LINEAR TRANSFORMATIONS

#SESSION 26 (LIVE IN-PERSON)

  • Linear Transformations: Chapter 8.1 (General Linear Transformations)
  • Matrix of Linear Transformations: Chapter 8.4 (Matrices for General Linear Transformations)
  • Inverse of Transformation: Chapter 8.2 (Compositions and Inverse Transformations)
  • Composition of Linear Transformations: Chapter 8.2

#SESSION 27 (LIVE IN-PERSON)

  • Practice: Review Key Ideas, Linear Transformations: Review Chapters 8.1, 8.2, and 8.4

#SESSION 28 (LIVE IN-PERSON)

  • Presentation of Linear Algebra Applications by Students: Review relevant chapters based on presentation topics.

#SESSION 29 (LIVE IN-PERSON)

  • Final Review Session: Vectors and Matrices, Solving Linear Equations, Vector Spaces, Orthogonality, Determinants, Eigenvalues, and Linear Transformations: Review all relevant chapters from Modules 1-7.

#SESSION 30 (LIVE IN-PERSON)

  • Final Exam: Covers all modules.

This plan should help you stay super prepared for each session by focusing on the relevant sections of the book. Good luck with your studies!