Linear Algebra

1. Foundations of Linear Systems

• Linear equations and systems of equations

• Geometric interpretation: lines, planes, hyperplanes

• Solution classification: consistent vs inconsistent systems

2. Matrix Theory

• Matrix operations: addition, multiplication, transpose, inverse

• Determinants and their properties

• Row reduction and echelon forms

3. Vector Spaces

• Vector spaces and subspaces

• Basis, dimension, and span

• Linear independence and dependence

• Orthogonality and inner product spaces

4. Eigenvalues and Eigenvectors

• Definition and computation via characteristic polynomials

• Geometric meaning: invariant directions under transformations

• Applications in stability analysis and data compression

5. Diagonalization and Matrix Decomposition

• Diagonalization of square matrices

• Spectral theorem and orthogonal diagonalization

• Applications: principal axes of inertia, systems of differential equations

• Introduction to Singular Value Decomposition (SVD)

6. Advanced Topics

• Jordan canonical form

• Quadratic forms and optimization

• Applications in machine learning and engineering

7. Computational Methods

• Numerical techniques for solving large systems

• Matrix factorization methods (LU, QR)

• Real-world modeling in physics, computer science, and economics

 Learning Outcomes

By the end of the course, students will be able to:

• Solve and classify linear systems using matrices

• Understand and apply vector space concepts

• Compute and interpret eigenvalues/eigenvectors

• Use diagonalization and decomposition for simplification

• Apply linear algebra to real-world problems in science and engineering

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