Inner Product Space
Inner Product Space
To talk about “length” or “angle” in abstract spaces, we define the Inner Product .
Core Concepts
- Inner Product: Generalization of the dot product .
- Norm (Length): Induced by the inner product .
- Orthogonality: Two vectors are orthogonal if their inner product is zero.
Gram-Schmidt Orthonormalization
An algorithm to convert any basis into an Orthonormal Basis (orthogonal unit vectors). It works by sequentially subtracting the “projection” of the new vector onto the existing ones.
Orthogonal Transformations
Transformations that preserve the inner product (and thus lengths and angles).
- Rotation Matrices are prime examples.
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.