📖 Course Systems Overview

🌰 Plant Growth System: 7 progressive tiers
🌶️ Difficulty Scale: 7 levels of complexity
⚡ Thunderstorm System: Tier advancement exams

🌱 Plant Growth System (PGS)

Progress through 7 growth stages by mastering sets of topics

🌰
🌱
🌿
🪴
🌳
🌸
🍎

Seed → Sprout → Seedling → Plant → Tree → Flowering Tree → Fruit Tree

🌶️ Difficulty System

Each topic rated on 7-level complexity scale

🌶️ 🌶️ 🌶️ 🌶️ 🌶️ 🌶️ 🌶️

Very Easy → Easy → Fair → Moderate → Hard → Very Hard → Astro-Complexity

⚡ Thunderstorm System

Cumulative exams to advance between tiers

⚡🌩️⚡

Pass Requirements:

Seed→Sprout: 70% | Sprout→Seedling: 72% | Seedling→Plant: 75%
Plant→Tree: 78% | Tree→Flowering: 82% | Flowering→Fruit: 85%

🎯 Progressive Learning Path

🌰 SEED TIER - Foundation Building

Master the fundamental concepts of linear algebra

Linear Equations
🌶️
Understanding what linear equations are and how to solve simple systems.
📚 Free Textbooks:
Hefferon - Linear Algebra Treil - Done Wrong
🎥 Video Courses:
Khan Academy 3Blue1Brown
🏫 University Courses:
MIT 18.06
System of Linear Equations
🌶️
Solving multiple equations simultaneously using various methods.
📚 Resources:
Khan Academy Hefferon Text
Gaussian Elimination
🌶️
Systematic method for solving systems of linear equations.
📚 Resources:
3Blue1Brown MIT Course
Gauss-Jordan Elimination
🌶️
Extended form of Gaussian elimination to reach reduced row echelon form.
📚 Resources:
Hefferon Chapter 1
⚡

Seed Tier Exit Exam

Test your understanding of basic linear equations and systems

Exam Resources:

🌱 SPROUT TIER - Matrix Fundamentals

Learn the building blocks of linear algebra: matrices and vectors

Matrix Introduction
🌶️
Understanding what matrices are and basic notation.
🎥 Visual Learning:
3Blue1Brown - Matrices Khan Academy
Matrix Addition
🌶️
Adding matrices element by element.
📚 Resources:
Hefferon Ch.2
Matrix Multiplication
🌶️
Understanding the dot product rule for matrix multiplication.
🎥 Visual Learning:
3Blue1Brown - Matrix Mult
Vector Spaces
🌶️
Introduction to vector spaces and their properties.
📚 Advanced Resources:
Treil - Done Wrong MIT Course
Linear Combination
🌶️
Combining vectors using scalar multiplication and addition.
🎥 Resources:
3Blue1Brown - Linear Combinations
Linear Independence
🌶️
Understanding when vectors are linearly independent.
📚 Resources:
Hefferon Ch.2
⚡

Sprout Tier Exit Exam

Cumulative test: Linear equations + Matrix basics + Vector spaces

Exam Resources:

🌿 SEEDLING TIER - Linear Transformations & Determinants

Explore how matrices transform space and measure their properties

Linear Transformations
🌶️
Understanding how matrices transform vectors geometrically.
🎥 Essential Videos:
3Blue1Brown - Transformations
Determinant
🌶️
Understanding determinants as area/volume scaling factors.
🎥 Visual Learning:
3Blue1Brown - Determinant
Matrix Inversion
🌶️
Finding inverse matrices and understanding when they exist.
📚 Resources:
Hefferon Ch.3 MIT Lectures
Cramer's Rule
🌶️
Using determinants to solve systems of linear equations.
🎥 Resources:
3Blue1Brown - Cramer's Rule
Basis and Dimension
🌶️
Understanding coordinate systems and vector space dimensions.
📚 Resources:
Treil Advanced Treatment
Column & Row Spaces
🌶️
Understanding the fundamental subspaces of a matrix.
🏫 University Resources:
MIT 18.06
⚡

Seedling Tier Exit Exam

Cumulative: All previous + Transformations + Determinants + Spaces

Exam Resources:

🪴 PLANT TIER - Eigenvalues & Advanced Matrix Theory

Master the most important concepts in linear algebra

Eigenvalues & Eigenvectors
🌶️
Understanding the most fundamental concept in linear algebra.
🎥 Essential Learning:
3Blue1Brown - Eigenthings MIT Strang Lectures
Characteristic Polynomial
🌶️
Computing eigenvalues through polynomial equations.
📚 Resources:
Hefferon Ch.5
Diagonalization
🌶️
Transforming matrices to diagonal form using eigenvectors.
📚 Advanced Study:
Treil Advanced
Orthogonal Matrices
🌶️
Special matrices that preserve lengths and angles.
📚 Resources:
MIT Course
Gram-Schmidt Process
🌶️
Creating orthogonal bases from arbitrary vectors.
📚 Resources:
Hefferon Ch.6
Inner Product Spaces
🌶️
Generalizing dot products to abstract vector spaces.
📚 Advanced Resources:
Treil Rigorous Treatment
⚡

Plant Tier Exit Exam

Cumulative: All previous + Eigenvalues + Orthogonality + Inner Products

Exam Resources:

🌳 TREE TIER - Matrix Decompositions & Applications

Advanced techniques for matrix analysis and computation

Singular Value Decomposition (SVD)
🌶️
The most important matrix factorization in applied mathematics.
🏫 University Resources:
MIT Matrix Methods Duke University Notes
LU Decomposition
🌶️
Factoring matrices into lower and upper triangular parts.
📊 Computational Resources:
Eigen Library Docs
QR Decomposition
🌶️
Orthogonal-triangular matrix factorization.
📚 Resources:
Duke Linear Algebra
Cholesky Decomposition
🌶️
Special decomposition for positive definite matrices.
🔧 Tools:
Wolfram Examples
Jordan Normal Form
🌶️
Canonical form for matrices that cannot be diagonalized.
📚 Advanced Theory:
Treil Advanced Topics
Least Squares
🌶️
Solving overdetermined systems and data fitting.
🏫 Resources:
MIT Applications
⚡

Tree Tier Exit Exam

Cumulative: All previous + Matrix Decompositions + Applications

Exam Resources:

🌸 FLOWERING TREE TIER - Multilinear Algebra & Geometry

Advanced topics bridging linear algebra and geometry

Tensor Algebra
🌶️
Generalizing vectors and matrices to higher-order tensors.
📚 Advanced Resources:
MIT Multilinear Book Texas A&M Tensor Text
Exterior Algebra
🌶️
Antisymmetric tensors and differential forms.
📚 Resources:
Multilinear Guide
Geometric Algebra
🌶️
Clifford algebras unifying vector algebra and geometry.
📚 Specialized Resources:
Hestenes-Sobczyk Modern GA Computing
Affine Spaces
🌶️
Geometric spaces without a fixed origin.
🎓 Academic Resources:
Transformations Paper
Affine Transformations
🌶️
Linear transformations that preserve parallelism.
💻 Computational:
Graphics Implementation
Bilinear Forms
🌶️
Functions linear in two variables, generalizing dot products.
📚 Theory:
Treil Advanced
⚡

Flowering Tree Tier Exit Exam

Cumulative: All previous + Multilinear algebra + Geometric structures

Exam Resources:
  • Research-level problem sets from graduate courses
  • Pass Requirement: 85% to advance to Fruit Tree Tier

🍎 FRUIT TREE TIER - Projective Geometry & Research Topics

Master-level topics for research and applications

Projective Spaces
🌶️
Geometry where parallel lines meet at infinity.
📚 Advanced Resources:
Penn Projective Geometry CS Perspective
Projective Transformations
🌶️
Most general linear transformations preserving lines.
🎨 Applications:
Computer Vision Tutorial 2D Projective Transforms
Homogeneous Coordinates
🌶️
Coordinate system for projective geometry.
💻 Implementation:
GitHub Implementation
Spectral Theorem
🌶️
Deep theory of symmetric and normal operators.
📚 Research Level:
Treil Research Topics
Perron-Frobenius Theorem
🌶️
Eigenvalue theory for non-negative matrices.
🔬 Applications:
MIT Advanced Methods
Advanced Matrix Decompositions
🌶️
Schur, polar, and specialized decompositions.
🎓 Graduate Level:
Advanced Computational LA
⚡

Master's Level Final Exam

Comprehensive: Entire linear algebra curriculum + Research applications

Congratulations! You've completed the full journey!
  • You now have research-grade expertise in linear algebra
  • Ready for graduate-level mathematics and advanced applications
  • Continue learning through research papers and specialized courses

🎉 Congratulations on Your Linear Algebra Journey! 🎉

You've grown from a 🌰 Seed to a 🍎 Fruit Tree - from basic equations to research-level expertise!

All resources are free and open. Continue growing your mathematical knowledge!