Math
These are some personal notes on mathematics collected over time while studying different topics in math and physics. They include definitions, theorems, examples, and short explanations written mainly for learning, reference, and personal understanding.
Set
- Set Theory
- Empty Set
- Subsets
- Improper Subsets
- Proper Subsets
- Empty Set
- Universe Set
- Complement Set
- Union, Intersection, and Difference
- Union
- Intersection
- Difference Set
- Equal Sets
- Disjoint Sets
- Power Set
- Set Partitions
- Upper and Lower Bounds (majorants and minorants)
- Maximum and Supremum (Least Upper Bound)
- Minimum and Infimum (Greatest Lower Bound)
- Extremes
- Cartesian Product
- Relations
- Set Collections
- Open and Closed Sets
Geometry
- Geometry
- Lines
- Plane
- Triangle
- Circumnference
- Conic sections
- Similarity
- Criteria for Triangle Similarity
- Similarity Criteria for Polygons with the Same Number of Sides
- Golden Ratio
- The Pythagorean Theorem
- The Extended Pythagorean Theorem
- Radius of an Inscribed Circle in a Triangle
Algebra
Functions
Linear Algebra
Calculus
Statistics
Matrices
- Matrices
- Square matrix
- Matrix addition
- Matrix multiplication
- Multiplying matrices by scalars
- Identity matrix
- Zero matrix
- Matrix power
- Determinant of square matrix
- Matrix rank
- Invertible and inverse matrices
- Row equivalent matrices
- Pivot operations in a matrix
- Complementary minor in a matrix
- Characteristic polynomial
- The Bordered Theorem ( Kronecker's theorem )
- Gauss' Method of elimination
Vectors
- Vectors
- Adding vectors
- Vector subtraction
- Scalar multiplication of a vector
- Dot product
- Cross product
- Norm
- Angle between two vectors
- Linear Combination of Vectors
- Zero vector
- Opposite vector
- Parallel vectors
- Equivalent vectors
- Decomposing a vector
- Linear dependence and independence of Vectors
- Vector representation of a line
- Vector representation of a plane
Complex numbers
Vector space
- Vector spaces
- Basis of a vector space
- Coordinates (or Weights)
- Uniqueness of the Linear Combination of a Basis
- Dimension of the Basis
- Codimension
- Grassmann's Formula
- Orthogonal vectors
- Orthogonal Basis
- Orthonormal Basis
- Converting Any Basis to Orthogonal
- Converting an Orthogonal Basis to Orthonormal
- Orthogonal Projection of a Vector onto a Subspace
- Orthogonal Complement
- Linear span
- Generators of a vector space
- Vector Subspace
- Subspaces Generated by a Set of Vectors
- Union of Subspaces
- Intersection of Vector Subspaces
- Sum of Vector Subspaces
- Direct Sum
- Complementary Subspaces
- Decomposition of Vector Space
Topology
- Topology
- Topological Spaces
- Topology of Open Sets
- Open Sets
- Trivial Topology (or Minimal Topology)
- Discrete Topology
- Finite Complement Topology
- Example of Topology
- Particolar Point Topology
- Excluded Point Topology
- Basis of Topology
Abstract Algebra
Modular Arithmetic
Group Theory
- Groups
- Group Theory
- Identity Element in a Group
- Inverse Elements
- Power of an Element
- Centre
- Compositions in Symmetric Groups
Logic
Numerical Computation
Category Theory
Physics
