Material Detail

This repository offers a comprehensive, interactive curriculum for Linear Algebra, comprising 110+ Jupyter notebooks designed for advanced undergraduates and engineering graduates. Moving beyond standard introductory texts, this resource bridges theoretical rigor with computational fluency, implementing concepts in both Python (NumPy/SciPy/SymPy) and Julia. A distinguishing feature is the use of code-generated computational layouts: algorithms such as Gaussian elimination, QR decomposition, and eigendecomposition are displayed step by step with intermediate results, structured tables, and interactive sliders, inviting hands-on experimentation rather than passive reading.

The content follows a logical progression from foundational concepts to specialized graduate-level topics:

I. FOUNDATIONS AND SYSTEMS

• Vector/Matrix Algebra: Arithmetic, dot products, and algorithmic matrix multiplication.
• Linear Systems: Gaussian elimination, row echelon forms, and the geometry of solution spaces.
• Subspaces and Transformations: Basis, dimension, linear independence, and the Four Fundamental Subspaces.
• Determinants: Introduced via wedge products and exterior algebra, yielding the standard cofactor expansions and the usual theorems about determinants as consequences.

II. CORE DECOMPOSITIONS AND SPECTRAL THEORY

• Orthogonality: Inner products, Gram-Schmidt, QR decomposition, and least squares.
• Eigen-analysis: Diagonalization, the Spectral Theorem, and symmetric matrix properties.
• SVD: Singular Value Decomposition, pseudoinverses, low-rank approximations, and randomized SVD

III. ADVANCED EXTENSIONS (BEYOND STANDARD CURRICULA)

• Generalized Problems: Generalized Eigenvalue Problems (GEP) (Ax=lambda Bx) with Rayleigh quotients and Generalized SVD (GSVD).
• Geometric Structures: Exploration of the Grassmannian manifold and coordinate transformations.
• Matrix Functions: Analytic functions of matrices (matrix exponential, sine), functions of degenerate matrices, and the Cayley-Hamilton theorem.
• Numerical and Iterative Methods: Condition numbers, Conjugate Gradients, and Arnoldi/Ritz iterations for large-scale systems.
• Specialized Applications: Eigenfaces (PCA), difference equations, backpropagation as linear algebra, and quadric surface visualization.

PEDAGOGICAL FEATURES:

• Code-Generated Layouts: Step-by-step displays of algorithms with intermediate results, decomposition tables, and interactive sliders.
• Bilingual Code: Dual implementations in Python and Julia for comparative learning.
• Interactive Visualization: Embedded graphics for manipulating vectors, subspaces, and eigenvalue dynamics.
• Zero-Setup Access: Instant launch via Binder or local deployment via Docker.
• Supplements: Obsidian-compatible notes, randomized problem generators, and a companion YouTube playlist.

This collection serves as a rigorous supplement or standalone study resource for students seeking deep mastery of modern computational linear algebra.

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