Introduction to Vector Space
The Vector Space
We are now moving from concrete lists of numbers to the abstract definition of a vector space defined over (Real numbers).
The 10 Fundamental Axioms
For a set to be a Vector Space, it must satisfy 10 axioms under two operations: Vector Addition and Scalar Multiplication.
- Closure (Addition & Multiplication)
- Associativity
- Commutativity
- Identity (Existence of Zero Vector )
- Inverse (Additive Inverse)
- …and distributive properties.
Subspaces
A Subspace is a non-empty subset of a vector space that is itself a vector space under the same operations.
The Subspace Test:
- Is ?
- Is closed under addition?
- Is closed under scalar multiplication?
Key Properties
- The zero vector is unique.
- The additive inverse of any vector is unique.
All Chapters in this Book
Vector and Matrices
Introduction to vectors, matrices, and their fundamental operations in linear algebra.
Solving Systems of Linear Equations
Mastering techniques to solve linear systems: Cramer's Rule, Inverse Matrix, and Gauss Elimination.
Introduction to Vector Space
Formal definition of vector spaces, axioms, and subspaces.
Basis and Dimension
Understanding the building blocks of vector spaces: Linear Independence, Spanning Sets, Basis, and Dimension.
Rank and Nullity
Exploring the fundamental subspaces of a matrix and the Rank-Nullity Theorem.
Linear Transformation
Mapping vector spaces: Homomorphisms, Isomorphisms, and Matrix Representations.
Equivalence and Similarity
Comparing matrices: When are two matrices really the same thing in disguise?
Affine Subspaces
Moving beyond the origin: Affine subspaces and mappings.
Inner Product Space
Geometry in vector spaces: Angles, Lengths, and Orthogonality.