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Binary linear codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
It is a basic fact from linear algebra that each linear code (abstractly: each linear space, each vector space) has a basis. Linear algebra applies to arbitrary fields. Most people are familiar with fields like the rational numbers, the real numbers and the complex numbers, but we can apply the basics of linear algebra to finite fields like F2 as well. This leads to the following basic fact:
Applied Analysis
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
A vector space is sometimes called a linear vector space or simply a linear space. The reader should be aware that occasionally it is convenient to specify (sometimes) unambiguously the vector space V and the field F by the symbols V and F respectively.
Preliminaries
Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018
Ronald B. Guenther, John W. Lee
A linear space (or vector space) Linear spaceVector spaceis a set M together with a set of scalars S and two operations, addition and scalar multiplication such that f + g and αf belong to M for all f and g in M and all scalars α. The only scalar fields used in this book are R and C. M is called a real, respectively complex, linear space according as the scalar field is R or C. Addition and scalar multiplication in a linear space satisfy the following familiar rules. For all f, g, h in M and all scalars α and β, f+g=g+f,(f+g)+h=f+(g+h),α(f+g)=αf+αg,(α+β)f=αf+βf,1f=f.
Set-valued equilibrium problems based on the concept of null set with applications
Published in Optimization, 2022
It is easy to see that if , then , which implies that is the zero element of . However, . Then is not a linear space under the addition and scalar multiplication. From [16], the set is said to be the null set of , which can be regarded as a type of ‘zero element’ in . Clearly, .
Zero duality gap conditions via abstract convexity
Published in Optimization, 2022
Hoa T. Bui, Regina S. Burachik, Alexander Y. Kruger, David T. Yost
In accordance with Definition 2.2, the space of abstract linear functions possesses the standard addition operation. We are now going to assume additionally that is closed with respect to the ordinary scalar multiplication: given an and , the function defined by for all , belongs to . This makes a real vector space. As a consequence, is also a vector space, and they both are subspaces of the ambient vector space of all real-valued functions on X. We give a simple example on the abstract linear space that satisfies conditions in Definition 2.2 but is not closed under the ordinary scalar multiplication. Consider the set , where denotes the set of all integer numbers and for all , conditions (i) and (ii) in Definition 2.2 are satisfied since ; and , therefore is not closed under the ordinary multiplication.
Exact and approximate vector Ekeland variational principles
Published in Optimization, 2022
T. Q. Bao, C. Gutiérrez, V. Novo, J. L. Ródenas-Pedregosa
Let A be a nonempty set, Y be a real linear space and be a vector-valued function. Consider two nonempty sets and such that , E is free disposal with respect to , and . Let be satisfying the triangle inequality property.