Rank and Nullity
Rank and Nullity
Every matrix hides important geometric information in its columns and solutions.
Rank
The Rank of a matrix is the number of linearly independent columns (or rows). It tells us the “true” dimension of the output space.
Null Space & Nullity
The Null Space is the set of all vectors such that . The Nullity is the dimension of this Null Space.
The Rank-Nullity Theorem
For an matrix :
This fundamental theorem creates a balance: The information preserved (Rank) plus the information lost (Nullity) equals the original input dimension ().
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.