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📚 Section I: Theoretical Foundations & Algebraic Structures

Overview:
This section introduces the essential theoretical pillars of linear algebra, equipping students with a rigorous and practical understanding of matrices, vectors, and transformations. Students will build an intuitive and formal foundation that enables the application of linear algebra across computer science and electrical engineering domains.

Chapter 1: Foundations of Linear Algebra

  • Key Concepts: Scalars, vectors, matrices, systems of linear equations.
  • Focus: Understand basic objects and how linear systems are expressed and solved mathematically.
  • Skills: Visualize matrices and vectors as fundamental building blocks of computation.

Chapter 2: Matrix Operations and Properties

  • Key Concepts: Matrix addition, subtraction, scalar multiplication, matrix multiplication, transpose, identity matrix, zero matrix, and special matrices (diagonal, symmetric, triangular, block matrices).
  • Focus: Master the algebra of matrices and recognize patterns in matrix structure.
  • Skills: Perform and simplify complex matrix operations.

Chapter 3: Vector Spaces and Subspaces

  • Key Concepts: Vector spaces, subspaces, span, basis, dimension, row space, column space, null space.
  • Focus: Understand spaces generated by vectors and their structural properties.
  • Skills: Classify subspaces and analyze dimensions of solutions to linear systems.

Chapter 4: Linear Independence and Rank

  • Key Concepts: Linear independence, dependence, rank of a matrix, row rank = column rank theorem, rank-nullity theorem.
  • Focus: Explore the relationships between vector combinations, solvability, and matrix structure.
  • Skills: Determine matrix rank and diagnose solution behaviors of linear systems.

Chapter 5: Inner Products and Orthogonality

  • Key Concepts: Inner product, norm, distance between vectors, orthogonality, orthogonal projections, orthogonal complement, orthonormal basis, Gram-Schmidt process.
  • Focus: Extend the geometric view of vectors to orthogonality and projection concepts.
  • Skills: Construct orthonormal bases and apply orthogonal projections in applications like least squares problems.

Chapter 6: Linear Transformations and Eigenanalysis

  • Key Concepts: Linear transformations, matrix representation, kernel, image, change of basis, similar matrices, eigenvalues, eigenvectors, characteristic polynomial, eigenspaces, algebraic and geometric multiplicities.
  • Focus: Translate between abstract transformations and their matrix representations; study intrinsic properties through eigenanalysis.
  • Skills: Diagonalize matrices, compute eigenvalues and eigenvectors, and understand the fundamental significance of spectral properties.

✨ Learning Outcomes:

By the end of this section, students will: - Comprehend the structures and operations central to linear algebra. - Analyze matrix behaviors and subspace relationships. - Apply transformations and eigenvalue techniques in practical settings. - Prepare for advanced topics in numerical methods, control systems, signal processing, and machine learning.