Mth 204: Linear Algebra I

Course Description

MTH 204: Linear Algebra I Unijos Note, Pdf is an introductory course focused on the fundamental concepts and techniques of linear algebra. This course covers topics such as vector spaces, linear transformations, matrices, determinants, eigenvalues, and eigenvectors, providing students with the theoretical foundation and practical skills needed to solve linear algebra problems.

Course Structure

1. Introduction to Linear Algebra
   • Basic Definitions: Scalars, vectors, matrices
   • Systems of Linear Equations: Solutions using Gaussian elimination
2. Vector Spaces
   • Definitions and Examples: Vector spaces, subspaces
   • Linear Combinations: Span, linear independence
   • Bases and Dimension: Basis, dimension of vector spaces
3. Matrices
   • Matrix Operations: Addition, multiplication, scalar multiplication
   • Special Matrices: Identity, diagonal, symmetric, and invertible matrices
   • Matrix Inverses: Methods to find the inverse of a matrix
4. Determinants
   • Definition and Properties: Calculation of determinants
   • Applications: Determinants in solving linear systems, Cramer’s Rule
5. Linear Transformations
   • Definition and Examples: Linear transformations between vector spaces
   • Kernel and Image: Understanding null space and range
   • Matrix Representation: Representation of linear transformations as matrices
6. Eigenvalues and Eigenvectors
   • Definitions: Eigenvalues, eigenvectors
   • Characteristic Equation: Finding eigenvalues and eigenvectors
   • Diagonalization: Diagonalizing matrices and applications
7. Inner Product Spaces
   • Definition and Examples: Inner product, norm, orthogonality
   • Gram-Schmidt Process: Orthogonalization of vectors
   • Orthogonal Matrices: Properties and applications
8. Applications of Linear Algebra
   • Linear Systems: Solutions and applications in various fields
   • Least Squares: Approximation solutions to overdetermined systems
   • Markov Chains: Application of linear algebra in probability

Learning Outcomes

By the end of this course, students should be able to:

• Understand and apply the fundamental concepts of vector spaces and linear transformations.
• Perform operations with matrices and solve systems of linear equations.
• Compute determinants and understand their properties and applications.
• Find eigenvalues and eigenvectors, and understand their significance in diagonalization.
• Utilize inner product spaces and orthogonalization processes.
• Apply linear algebra techniques to solve real-world problems.

Assessment Methods

• Exams and Quizzes: To evaluate theoretical understanding and problem-solving skills.
• Homework and Assignments: To practice and reinforce concepts learned in lectures.
• Projects: To apply linear algebra concepts to more complex problems and real-world scenarios.
• Class Participation: To encourage engagement and active learning.