Grid of Variables
Matrices, Systems, Transformations & Linear Algebra
The Human Problem
We have studied single equations and single vectors. But what happens when we have multiple unknowns and multiple relationships simultaneously? Instead of solving them one at a time, mathematics organizes the entire system into a rectangular structure. A matrix is a language for storing and transforming many quantities at once.
The 72 Matrix Cells
From Tables of Numbers to Spaces of Information
Matrix Foundations
- Matrix Idea & Entries (aᵢⱼ)
- Rows, Columns & Order (m×n)
- Row, Column, Square, Zero Matrices
- Identity, Diagonal, Scalar Matrices
- Upper & Lower Triangular
- Equality of Matrices
Operations
- Addition & Subtraction
- Scalar Multiplication
- Transpose & its Laws
- Symmetric Matrices
- Matrix Multiplication (Row-by-Column)
- Non-Commutativity (AB ≠ BA)
Transformations
- Matrix as a Function
- Transforming Space
- Rotation Matrices
- Scaling, Reflection & Shear
- Composition of Transformations
- Inverse as Reverse Transformation
Determinants
- 2×2 & 3×3 Determinants
- Area Scale Meaning
- Determinant Sign (Orientation)
- Minors & Cofactors
- Determinant Properties
- Singular vs Non-Singular Matrices
Inverses
- Matrix Inverse Definition
- 2×2 Inverse Formula
- Adjugate Method
- Determinant Test for Invertibility
- Undoing a Transformation
Systems (AX=B)
- Elementary Row Operations
- Augmented Matrix
- Inverse Method for Systems
- Cramer's Rule
- Consistent / Inconsistent Systems
- Geometric Interpretation
Linear Algebra Bridge
- Rank — Preview
- Linear Independence — Preview
- Basis Vectors — Preview
- Linear Transformations
- Matrix as Many-Variable Language
Applications
- Computer Graphics & Robotics
- Data Science & Datasets
- Machine Learning (Y=WX+b)
- Economics (Input-Output)
- Physics & Systems Theory
The Matrix Universe
- Matrix Dimensions Logic
- Determinant/Inverse Relationship
- Matrix → System → Geometry
- Matrix → Transformation → Space
Exit Portal
We have learned to handle entire systems of relationships simultaneously. But what happens when many outcomes are possible, and we do not know which one will occur? After learning to count possibilities (World 15) and encode systems (World 23), the next question is: How do we measure uncertainty itself?