Basis and Dimension
Basis & Dimension
To understand the structure of a vector space, we need to know what creates it (Spanning) and what is redundant (Dependence).
Core Concepts
- Linear Combination: Creating new vectors by scaling and adding existing ones ().
- Linear Independence: No vector in the set can be written as a linear combination of the others.
- Spanning Set: The set of all possible linear combinations.
The Basis
A Basis is the “Goldilocks” set of a vector space. It is:
- Large enough to Span the space.
- Small enough to be Linearly Independent .
Dimension
The dimension of a vector space is simply the number of vectors in its basis. This number is unique for a given space.
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.