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Linear Algebra

About This Course

Course Description: The concepts found in linear algebra form a basis of mathematical understanding. Throughout this course, considerable effort will be spent on developing and exploring these concepts in detail. Students will explore how solutions can be developed by integrating various concepts together. Students will not only know how to perform computations, with sufficient proficiency but also understand the basic concepts and methodologies commonly found within linear algebra. We will investigate computational procedures as well as some of their theoretical implications.

Students will become more proficient with basic matrix computations, as well as being able to recognize certain characteristics of a linear system. A focus of this course is to help prepare the students to become better problem solvers, by exploring the conceptual components of different types of linear systems. Students will gain an adequate understanding of why some linear systems are inconsistent, and how to work with these types of systems. Students will not only learn how to perform basic matrix computations but also how to use various matrix structures to develop a solution.

Course Information

Course Topics

  •  Linear Systems and Solutions Set
  • Row Reduction and Echelon Form
  • Vector and Matrix Equation Analogues
  • Linear Dependence and Linear Independence 
  • Linear Transformations
  • Applications of Linear Systems
  • Matrices and Matrix Operations
  • Partitioned Matrices
  • Inverse of a Matrix
  • LU Factorization
  • Column Space and Null Space
  • The Rank Nullity Theorem
  • Determinants 
    • Algebraically and Combinatorially
  • Properties of Determinants
  • Crammer’s Rule
  • Applications
  • Eigenectors and Eigenvalues
  • The Characteristic Equation
  • Cayley-Hamilton Theorem
  • Complex Eigenvalues
  • Diagonalization
  • Differential Equations
  • Recurrence Relations
  • Power Method
  • Vector Spaces and Subspaces
  • Basis Sets 
  • Coordinate Systems
    • Change of Basis
  • Orthogonal and Orthonormal Sets
  • Orthogonal Projections
  • The Gram-Schmidt Process
  • Least Squares and Normal Equations
  • Characteristics of Symmetric Matrices 
  • Quadratic Forms
  • Constrained Optimization
  • The Singular Value Decomposition

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