The lecture covers the following topics:
1. Inner product spaces
2. Orthonormal basis
3. Gram-Schmidt orthogonalization process
4. Cauchy-Riemann inequality and triangle in equalitity
5. Dual spaces and its examples
6. Adjoint linear operatorr
7. Self adjoint linear operator
8. Isometic
9. Normal operator
Inhaltsverzeichnis (Table of Contents)
- Orthogonality
- Inner product
- Inner product spaces
- Orthogonal and ortho normal sets
- Gram-schmidt orthogonalization process
- Theorem cauchy -schwarz in equality
- The Dual spaces
- Ad joint of linear operators
- Self-ad joint linear operators
- Isometric
- Normal operators
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This lecture note introduces the concept of orthogonality in linear algebra, focusing on inner product spaces. It explores the properties of inner products, orthogonal sets, and orthonormal sets, leading to the development of key theorems and concepts, including the Cauchy-Schwarz inequality, duality, and the classification of linear operators based on their relationship with orthogonality.
- Inner product spaces and their properties
- Orthogonal and orthonormal sets
- The Cauchy-Schwarz inequality and its applications
- Duality and the concept of adjoint operators
- Classification of linear operators based on orthogonality: self-adjoint, isometric, and normal operators.
Zusammenfassung der Kapitel (Chapter Summaries)
- Inner product: Introduces the definition of inner product on real or complex vector spaces, highlighting its properties and axioms. It explores examples of inner products on various spaces, such as Euclidean spaces and spaces of matrices.
- Inner product spaces: Defines inner product spaces, providing examples and exploring their properties. It discusses the norm and distance in inner product spaces, along with the concept of unit vectors.
- Orthogonality: Discusses the concept of orthogonal vectors in inner product spaces, providing examples and illustrations. It emphasizes the importance of orthogonality in linear algebra.
- Orthogonal and ortho normal sets: Defines orthogonal and orthonormal sets in inner product spaces, providing examples and exploring their properties. It demonstrates the construction of orthonormal sets and proves the linear independence of orthogonal sets.
Schlüsselwörter (Keywords)
Key terms and concepts explored in this lecture note include inner product spaces, orthogonality, orthonormal sets, Cauchy-Schwarz inequality, duality, adjoint operators, self-adjoint operators, isometric operators, and normal operators. It further examines the application of these concepts in various vector spaces, including Euclidean spaces, spaces of matrices, and spaces of polynomials.
- Quote paper
- Nure Amin (Author), 2018, Orthogonality. Lecture Note, Munich, GRIN Verlag, https://www.grin.com/document/421155
