In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics.
Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
Definition
Suppose that K is a field (for example, the real numbers) and V is a vector space over K. As usual, we call elements of Vvectors and call elements of Kscalars.
If v1,...,vn are vectors and a1,...,an are scalars, then the linear combination of those vectors with those scalars as coefficients is
There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, like in the assertion "the set of all linear combinations of v1,...,vn always forms a subspace". However, one could also say "two different linear combinations can have the same value" in which case the expression must have been meant. The subtle difference between these uses is the essence of the notion of linear dependence: a family F of vectors is linearly independent precisely if any linear combination of the vectors in F (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each vi; trivial modifications such as permuting the terms or adding terms with zero coefficient do not give distinct linear combinations.
is a generating set for ; that is to say, every element of is a finite sum of elements of multiplied by coefficients in ;
is linearly independent, that is, for distinct elements of implies that (where is the zero element of and is the zero element of ).
If has invariant basis number, then by definition any two bases have the same cardinality. The cardinality of any (and therefore every) basis is called the rank of the free module . The free module is said to be free of rank n, or simply free of finite rank if the cardinality is finite.
A module is a generalization of a vector space. You can think of it as a group of vectors with scalars from a ring instead of a field. In this lesson, we introduce the module, give a variety of examples, and talk about the ways in which modules and vector spaces are different from one another.
Be sure to subscribe so you don't miss new lessons from Socratica:
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We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
http://amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
http://www.jmilne.org/math/CourseNotes/index.html
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published: 21 Apr 2017
Basis of a Module and Free Module
For any query, ask in the comment box.
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published: 26 Apr 2020
Rings 8 Free modules
This lecture is part of an online course on rings and modules.
We mainly discuss the problem of whether free modules over a ring have a well defined ran, generalizing the dimension of a vector space. We show that they do over many rings, including all non-zero commutative rings, but give an example of a non-commutative ring where the rank of a free module is not defined.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj52XDLrmvrFDgwcf6XOm2TEE
published: 06 Oct 2021
Module theory - Lecture 5 - Free modules
Generating sets of modules, Linearly independent subset, Basis of a module, Free modules, Examples of free and non-free modules, Rank of a free module over a commutative ring with 1.
For all the lectures on Module Theory, follow the link:
https://www.youtube.com/playlist?list=PLY2U8CcmALO4HqPsKdghs0JMy0HUL7TLs
published: 15 Sep 2020
Torsion and rorsion free module.د.كريم صبر
published: 27 May 2021
Free modules
published: 26 May 2021
MELC BASED MODULES FREE TO DOWNLOAD IN DEPED COMMONS #DEPEDCOMMONS #MODULESDEPEDCOMMONS
Ang bidyong ito ay nagtuturo kung paano magdownload ng MELC BASED modules sa DEPED COMMONS. Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete
Hello Teachers! For you to be updated for more learning videos/tutorials, webinars, CPD accredited seminars, trainings, MELC module resou...
published: 12 Aug 2020
Start learning with ISA today! Choose your FREE module.
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In this beginner video you will learn complete basic concepts about networking, VLAN Basics, Trunk port vs Access Ports, Native VLAN . connecting multiple devices across local area network and wide area network. Functions of each networking device are explained briefly. Physical layer, cabeling copper and optical fiber are also explained. are also explained in this video.
In this Free and Complete course of CCNA Routing & Switching (200-301) you will get complete practical hands on training on Cisco Packet Tracer. You will also get free complete course material, Packet tracer Lab activity files, dump questions which would help you to pass CCNA Certification exam.
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published: 04 Nov 2024
Basis of R-module | Free R- Module | Linear independent & Dependent elements on Module
Basis of R-module | Free Module | Lenear independent & Dependent elements of module
Definition of Free module
definition of basis of module
Lenear independent and Dependent element of module
Schur's Lemma | Direct Sum of submodules | Direct Summand of Module
Maximal submodule , Minimal submodule , Simple module ,Semi-Simple module, Fundamental theorem of Homomorphism, Imp theorem
Homomorphism and Isomorphism of R- Module , epomorphism, endomorphism , Automorphism Imp theorem
what is isomorphism?
what is homomorphism in Module?
Definition of Homomorphism,
Definition of Isomorphism
Submodule | Finitely generated module | cyclic module over ring
what is submodule ? Define submodul , definition of submodule
Definition of Finitely generated module, Definition of cyclic module
sum of sub...
A module is a generalization of a vector space. You can think of it as a group of vectors with scalars from a ring instead of a field. In this lesson, we intr...
A module is a generalization of a vector space. You can think of it as a group of vectors with scalars from a ring instead of a field. In this lesson, we introduce the module, give a variety of examples, and talk about the ways in which modules and vector spaces are different from one another.
Be sure to subscribe so you don't miss new lessons from Socratica:
http://bit.ly/1ixuu9W
♦♦♦♦♦♦♦♦♦♦
We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
http://amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
http://www.jmilne.org/math/CourseNotes/index.html
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A module is a generalization of a vector space. You can think of it as a group of vectors with scalars from a ring instead of a field. In this lesson, we introduce the module, give a variety of examples, and talk about the ways in which modules and vector spaces are different from one another.
Be sure to subscribe so you don't miss new lessons from Socratica:
http://bit.ly/1ixuu9W
♦♦♦♦♦♦♦♦♦♦
We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
http://amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
http://www.jmilne.org/math/CourseNotes/index.html
♦♦♦♦♦♦♦♦♦♦
Ways to support our channel:
► Join our Patreon : https://www.patreon.com/socratica
► Make a one-time PayPal donation: https://www.paypal.me/socratica
► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
Thank you!
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Connect with us!
Facebook: https://www.facebook.com/SocraticaStudios/
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Twitter: https://twitter.com/Socratica
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Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison
♦♦♦♦♦♦♦♦♦♦
This lecture is part of an online course on rings and modules.
We mainly discuss the problem of whether free modules over a ring have a well defined ran, gener...
This lecture is part of an online course on rings and modules.
We mainly discuss the problem of whether free modules over a ring have a well defined ran, generalizing the dimension of a vector space. We show that they do over many rings, including all non-zero commutative rings, but give an example of a non-commutative ring where the rank of a free module is not defined.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj52XDLrmvrFDgwcf6XOm2TEE
This lecture is part of an online course on rings and modules.
We mainly discuss the problem of whether free modules over a ring have a well defined ran, generalizing the dimension of a vector space. We show that they do over many rings, including all non-zero commutative rings, but give an example of a non-commutative ring where the rank of a free module is not defined.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj52XDLrmvrFDgwcf6XOm2TEE
Generating sets of modules, Linearly independent subset, Basis of a module, Free modules, Examples of free and non-free modules, Rank of a free module over a co...
Generating sets of modules, Linearly independent subset, Basis of a module, Free modules, Examples of free and non-free modules, Rank of a free module over a commutative ring with 1.
For all the lectures on Module Theory, follow the link:
https://www.youtube.com/playlist?list=PLY2U8CcmALO4HqPsKdghs0JMy0HUL7TLs
Generating sets of modules, Linearly independent subset, Basis of a module, Free modules, Examples of free and non-free modules, Rank of a free module over a commutative ring with 1.
For all the lectures on Module Theory, follow the link:
https://www.youtube.com/playlist?list=PLY2U8CcmALO4HqPsKdghs0JMy0HUL7TLs
Ang bidyong ito ay nagtuturo kung paano magdownload ng MELC BASED modules sa DEPED COMMONS. ...
Ang bidyong ito ay nagtuturo kung paano magdownload ng MELC BASED modules sa DEPED COMMONS. Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete
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Ang bidyong ito ay nagtuturo kung paano magdownload ng MELC BASED modules sa DEPED COMMONS. Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete
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Subscribe and hit the notification bell of my Youtube Channel:
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✅ Download free Daily Lesson Log here ➜ https://tinyurl.com/ye9jwetf
✅ Download free Homeroom Guidance Modules ➜ https://tinyurl.com/3ybswrwd
✅ Download free Deped aligned Periodical Test here ➜ https://tinyurl.com/4x3frys6
✅ Download free Deped PowerPoint Lesson here ➜ https://tinyurl.com/kjp9xf8e
✅ CPD Webinars and Trainings ➜ https://tinyurl.com/3tmsek7h
✅ Prayers for Meetings, Seminars and Trainings ➜ http://bit.ly/38QWj9o
✅ Download Powerpoint Games and Templates here ➜ http://bit.ly/3hA7Hue
✅ Learn and utilize Online Learning Tools ➜ http://bit.ly/3n4CUqG
✅ Download Free Printable materials here ➜ https://bit.ly/3BJx355
✅ Deped Files ➜ https://learningpal.net/deped-forms/
✅ Downloadable free Worksheets here ➜
___________________________________________________________________________
Looking to bridge skill gaps without breaking the bank? ISA self-paced, modular training can help. Choose your free module and start learning with us today!
CH...
Looking to bridge skill gaps without breaking the bank? ISA self-paced, modular training can help. Choose your free module and start learning with us today!
CHOOSE YOUR FREE MODULE: https://programs.isa.org/free-training?utm_source=YouTube&utm_medium=paid_social&utm_campaign=Bridge+Skill+Gaps+M+Course+Campaign
BECOME AN ISA MEMBER: https://www.isa.org/membership/join
Looking to bridge skill gaps without breaking the bank? ISA self-paced, modular training can help. Choose your free module and start learning with us today!
CHOOSE YOUR FREE MODULE: https://programs.isa.org/free-training?utm_source=YouTube&utm_medium=paid_social&utm_campaign=Bridge+Skill+Gaps+M+Course+Campaign
BECOME AN ISA MEMBER: https://www.isa.org/membership/join
In this beginner video you will learn complete basic concepts about networking, VLAN Basics, Trunk port vs Access Ports, Native VLAN . connecting multiple devic...
In this beginner video you will learn complete basic concepts about networking, VLAN Basics, Trunk port vs Access Ports, Native VLAN . connecting multiple devices across local area network and wide area network. Functions of each networking device are explained briefly. Physical layer, cabeling copper and optical fiber are also explained. are also explained in this video.
In this Free and Complete course of CCNA Routing & Switching (200-301) you will get complete practical hands on training on Cisco Packet Tracer. You will also get free complete course material, Packet tracer Lab activity files, dump questions which would help you to pass CCNA Certification exam.
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In this beginner video you will learn complete basic concepts about networking, VLAN Basics, Trunk port vs Access Ports, Native VLAN . connecting multiple devices across local area network and wide area network. Functions of each networking device are explained briefly. Physical layer, cabeling copper and optical fiber are also explained. are also explained in this video.
In this Free and Complete course of CCNA Routing & Switching (200-301) you will get complete practical hands on training on Cisco Packet Tracer. You will also get free complete course material, Packet tracer Lab activity files, dump questions which would help you to pass CCNA Certification exam.
Subscribe my Channel
@ccna-academy
SUPPORT MY CHANNEL
The best way to support my channel is to like, comment, subscribe, and share my videos to help spread this free series of videos please.
If you can spare to leave a tip, here are some options:
UBL IBAN No: PK78UNIL0109000208845526
Account Title: Muhammad Sharif Awan
Contact me:
Email: [email protected]
Whatsapp: +92 317 5668778
#ccna #ccnp #ccie #network_troubleshooting #networking_interview #network_configuration #network_engineer #network_engineer_training #cisco_network_configuration #router_configuration #switch_configuration #firewall_configuration #interview_questions_network_engineer #real-time_network_configuration #scenario-based_network #network_troubleshooting #network_troubleshooting_network_engineer #cisco_network_configuration #basic_network #password_recovery #Forgotton_password #lost_password #Password_bhol_gya
Basis of R-module | Free Module | Lenear independent & Dependent elements of module
Definition of Free module
definition of basis of module
Lenear independent...
Basis of R-module | Free Module | Lenear independent & Dependent elements of module
Definition of Free module
definition of basis of module
Lenear independent and Dependent element of module
Schur's Lemma | Direct Sum of submodules | Direct Summand of Module
Maximal submodule , Minimal submodule , Simple module ,Semi-Simple module, Fundamental theorem of Homomorphism, Imp theorem
Homomorphism and Isomorphism of R- Module , epomorphism, endomorphism , Automorphism Imp theorem
what is isomorphism?
what is homomorphism in Module?
Definition of Homomorphism,
Definition of Isomorphism
Submodule | Finitely generated module | cyclic module over ring
what is submodule ? Define submodul , definition of submodule
Definition of Finitely generated module, Definition of cyclic module
sum of submodules,
module Lecture -1 https://youtu.be/Pr38T5994_4
Module of ring | Advanced Abstract Algebra
Module over Ring , Msc Mathematics
Abstract Algebra in hindi,
R-Module over ring , Left R- Module , Right R-Module, Ring therory , Group theory, Vector space,
R-Module with example ,Direct product of module
properties of module , Submodules,
what is module over ring, cyclic module , Finitely generated module,
isomorphisms of modules
homomorphism of modules
Semi simple module, Direct summand ,schurs Lemma , Noetherian and Artinian Modules and Ring
your Queries
Homomorphism of Module
Explain Homomorphism
Explain Isomorphism
Automorphism
what is module
definition of module
Advanced abstract algebra
Msc mathematics video lectures
Ring theory
#ring
#module
#msc
Fuzzy Relation | Fuzzy Graph full explanation in hindi
Definition of fuzzy Relation
Definition of fuzzy Graph
Fuzzy equation full explaination with examples
fuzzy equation full explaination in Hindi
fuzzy set and fuzzy equation in Hindi
what is fuzzy equation
fuzzy equation Kya hai
fuzzy set Kya hai
Fuzzy Arithmetic And Example full explanation
Fuzzy set and their application hindi lectures
Fuzzy set in Hindi
Fuzzy Arithmetic in soft Computing
Fuzzy Arithmetic operations
Fuzzy Arithmetic tutorial
Fuzzy Arithmetic mcq
T-Norm, T-Conorm or S-Norm
Fuzzy intersection and union
Fuzzy set and their Applications in hindi
Meaning of fuzzy
Crisp set
Fuzzy set
Definition of Fuzzy set
Academy of higher Mathematics
Sir Y.K.Sahu*
This video is very important for your University exams
PDF Unit -1 Fuzzy set 👇
https://drive.google.com/file/d/1AUPI363iDGhv_lov8tfPUWA442ygofAD/view?usp=drivesdk
Some other operations on fuzzy set
Product of fuzzy set
Product of fuzzy set with crisp number
Power of fuzzy set
Important theorems
Fuzzy sets and their Applications, Characteristic of fuzzy set
Fundamental Operation on fuzzy set,
Union
Intersection
Compliment
Normal fuzzy set
Support of fuzzy set
Convex fuzzy set
Decompostion of fuzzy set
Cardinality of fuzzy set
Operation on fuzzy set and important Theorems
Academy of higher Mathematics mahasamund
By Sir Y.K.Sahu,
Best maths Teacher in Mahasaund CG,
AOHM
Academy of higher Mathematics
Fuzzy set
Fuzzy set in hindi
Fuzzy set with Example
Fuzzy intersection (T-Norm )
T-Norm operators
S-Norm operators
Fuzzy logic system
Fuzzy implication
What is Fuzzy set?
Extension of Fuzzy set
Basis of R-module | Free Module | Lenear independent & Dependent elements of module
Definition of Free module
definition of basis of module
Lenear independent and Dependent element of module
Schur's Lemma | Direct Sum of submodules | Direct Summand of Module
Maximal submodule , Minimal submodule , Simple module ,Semi-Simple module, Fundamental theorem of Homomorphism, Imp theorem
Homomorphism and Isomorphism of R- Module , epomorphism, endomorphism , Automorphism Imp theorem
what is isomorphism?
what is homomorphism in Module?
Definition of Homomorphism,
Definition of Isomorphism
Submodule | Finitely generated module | cyclic module over ring
what is submodule ? Define submodul , definition of submodule
Definition of Finitely generated module, Definition of cyclic module
sum of submodules,
module Lecture -1 https://youtu.be/Pr38T5994_4
Module of ring | Advanced Abstract Algebra
Module over Ring , Msc Mathematics
Abstract Algebra in hindi,
R-Module over ring , Left R- Module , Right R-Module, Ring therory , Group theory, Vector space,
R-Module with example ,Direct product of module
properties of module , Submodules,
what is module over ring, cyclic module , Finitely generated module,
isomorphisms of modules
homomorphism of modules
Semi simple module, Direct summand ,schurs Lemma , Noetherian and Artinian Modules and Ring
your Queries
Homomorphism of Module
Explain Homomorphism
Explain Isomorphism
Automorphism
what is module
definition of module
Advanced abstract algebra
Msc mathematics video lectures
Ring theory
#ring
#module
#msc
Fuzzy Relation | Fuzzy Graph full explanation in hindi
Definition of fuzzy Relation
Definition of fuzzy Graph
Fuzzy equation full explaination with examples
fuzzy equation full explaination in Hindi
fuzzy set and fuzzy equation in Hindi
what is fuzzy equation
fuzzy equation Kya hai
fuzzy set Kya hai
Fuzzy Arithmetic And Example full explanation
Fuzzy set and their application hindi lectures
Fuzzy set in Hindi
Fuzzy Arithmetic in soft Computing
Fuzzy Arithmetic operations
Fuzzy Arithmetic tutorial
Fuzzy Arithmetic mcq
T-Norm, T-Conorm or S-Norm
Fuzzy intersection and union
Fuzzy set and their Applications in hindi
Meaning of fuzzy
Crisp set
Fuzzy set
Definition of Fuzzy set
Academy of higher Mathematics
Sir Y.K.Sahu*
This video is very important for your University exams
PDF Unit -1 Fuzzy set 👇
https://drive.google.com/file/d/1AUPI363iDGhv_lov8tfPUWA442ygofAD/view?usp=drivesdk
Some other operations on fuzzy set
Product of fuzzy set
Product of fuzzy set with crisp number
Power of fuzzy set
Important theorems
Fuzzy sets and their Applications, Characteristic of fuzzy set
Fundamental Operation on fuzzy set,
Union
Intersection
Compliment
Normal fuzzy set
Support of fuzzy set
Convex fuzzy set
Decompostion of fuzzy set
Cardinality of fuzzy set
Operation on fuzzy set and important Theorems
Academy of higher Mathematics mahasamund
By Sir Y.K.Sahu,
Best maths Teacher in Mahasaund CG,
AOHM
Academy of higher Mathematics
Fuzzy set
Fuzzy set in hindi
Fuzzy set with Example
Fuzzy intersection (T-Norm )
T-Norm operators
S-Norm operators
Fuzzy logic system
Fuzzy implication
What is Fuzzy set?
Extension of Fuzzy set
A module is a generalization of a vector space. You can think of it as a group of vectors with scalars from a ring instead of a field. In this lesson, we introduce the module, give a variety of examples, and talk about the ways in which modules and vector spaces are different from one another.
Be sure to subscribe so you don't miss new lessons from Socratica:
http://bit.ly/1ixuu9W
♦♦♦♦♦♦♦♦♦♦
We recommend the following textbooks:
Dummit & Foote, Abstract Algebra 3rd Edition
http://amzn.to/2oOBd5S
Milne, Algebra Course Notes (available free online)
http://www.jmilne.org/math/CourseNotes/index.html
♦♦♦♦♦♦♦♦♦♦
Ways to support our channel:
► Join our Patreon : https://www.patreon.com/socratica
► Make a one-time PayPal donation: https://www.paypal.me/socratica
► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
Thank you!
♦♦♦♦♦♦♦♦♦♦
Connect with us!
Facebook: https://www.facebook.com/SocraticaStudios/
Instagram: https://www.instagram.com/SocraticaStudios/
Twitter: https://twitter.com/Socratica
♦♦♦♦♦♦♦♦♦♦
Teaching Assistant: Liliana de Castro
Written & Directed by Michael Harrison
Produced by Kimberly Hatch Harrison
♦♦♦♦♦♦♦♦♦♦
This lecture is part of an online course on rings and modules.
We mainly discuss the problem of whether free modules over a ring have a well defined ran, generalizing the dimension of a vector space. We show that they do over many rings, including all non-zero commutative rings, but give an example of a non-commutative ring where the rank of a free module is not defined.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj52XDLrmvrFDgwcf6XOm2TEE
Generating sets of modules, Linearly independent subset, Basis of a module, Free modules, Examples of free and non-free modules, Rank of a free module over a commutative ring with 1.
For all the lectures on Module Theory, follow the link:
https://www.youtube.com/playlist?list=PLY2U8CcmALO4HqPsKdghs0JMy0HUL7TLs
Ang bidyong ito ay nagtuturo kung paano magdownload ng MELC BASED modules sa DEPED COMMONS. Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete Download here the Deped Modules K-12 complete ➜ https://tinyurl.com/Depedmodulescomplete
Hello Teachers! For you to be updated for more learning videos/tutorials, webinars, CPD accredited seminars, trainings, MELC module resources and new trends in education.
Subscribe and hit the notification bell of my Youtube Channel:
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In this beginner video you will learn complete basic concepts about networking, VLAN Basics, Trunk port vs Access Ports, Native VLAN . connecting multiple devices across local area network and wide area network. Functions of each networking device are explained briefly. Physical layer, cabeling copper and optical fiber are also explained. are also explained in this video.
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Basis of R-module | Free Module | Lenear independent & Dependent elements of module
Definition of Free module
definition of basis of module
Lenear independent and Dependent element of module
Schur's Lemma | Direct Sum of submodules | Direct Summand of Module
Maximal submodule , Minimal submodule , Simple module ,Semi-Simple module, Fundamental theorem of Homomorphism, Imp theorem
Homomorphism and Isomorphism of R- Module , epomorphism, endomorphism , Automorphism Imp theorem
what is isomorphism?
what is homomorphism in Module?
Definition of Homomorphism,
Definition of Isomorphism
Submodule | Finitely generated module | cyclic module over ring
what is submodule ? Define submodul , definition of submodule
Definition of Finitely generated module, Definition of cyclic module
sum of submodules,
module Lecture -1 https://youtu.be/Pr38T5994_4
Module of ring | Advanced Abstract Algebra
Module over Ring , Msc Mathematics
Abstract Algebra in hindi,
R-Module over ring , Left R- Module , Right R-Module, Ring therory , Group theory, Vector space,
R-Module with example ,Direct product of module
properties of module , Submodules,
what is module over ring, cyclic module , Finitely generated module,
isomorphisms of modules
homomorphism of modules
Semi simple module, Direct summand ,schurs Lemma , Noetherian and Artinian Modules and Ring
your Queries
Homomorphism of Module
Explain Homomorphism
Explain Isomorphism
Automorphism
what is module
definition of module
Advanced abstract algebra
Msc mathematics video lectures
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Fuzzy Relation | Fuzzy Graph full explanation in hindi
Definition of fuzzy Relation
Definition of fuzzy Graph
Fuzzy equation full explaination with examples
fuzzy equation full explaination in Hindi
fuzzy set and fuzzy equation in Hindi
what is fuzzy equation
fuzzy equation Kya hai
fuzzy set Kya hai
Fuzzy Arithmetic And Example full explanation
Fuzzy set and their application hindi lectures
Fuzzy set in Hindi
Fuzzy Arithmetic in soft Computing
Fuzzy Arithmetic operations
Fuzzy Arithmetic tutorial
Fuzzy Arithmetic mcq
T-Norm, T-Conorm or S-Norm
Fuzzy intersection and union
Fuzzy set and their Applications in hindi
Meaning of fuzzy
Crisp set
Fuzzy set
Definition of Fuzzy set
Academy of higher Mathematics
Sir Y.K.Sahu*
This video is very important for your University exams
PDF Unit -1 Fuzzy set 👇
https://drive.google.com/file/d/1AUPI363iDGhv_lov8tfPUWA442ygofAD/view?usp=drivesdk
Some other operations on fuzzy set
Product of fuzzy set
Product of fuzzy set with crisp number
Power of fuzzy set
Important theorems
Fuzzy sets and their Applications, Characteristic of fuzzy set
Fundamental Operation on fuzzy set,
Union
Intersection
Compliment
Normal fuzzy set
Support of fuzzy set
Convex fuzzy set
Decompostion of fuzzy set
Cardinality of fuzzy set
Operation on fuzzy set and important Theorems
Academy of higher Mathematics mahasamund
By Sir Y.K.Sahu,
Best maths Teacher in Mahasaund CG,
AOHM
Academy of higher Mathematics
Fuzzy set
Fuzzy set in hindi
Fuzzy set with Example
Fuzzy intersection (T-Norm )
T-Norm operators
S-Norm operators
Fuzzy logic system
Fuzzy implication
What is Fuzzy set?
Extension of Fuzzy set
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics.
Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
Definition
Suppose that K is a field (for example, the real numbers) and V is a vector space over K. As usual, we call elements of Vvectors and call elements of Kscalars.
If v1,...,vn are vectors and a1,...,an are scalars, then the linear combination of those vectors with those scalars as coefficients is
There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, like in the assertion "the set of all linear combinations of v1,...,vn always forms a subspace". However, one could also say "two different linear combinations can have the same value" in which case the expression must have been meant. The subtle difference between these uses is the essence of the notion of linear dependence: a family F of vectors is linearly independent precisely if any linear combination of the vectors in F (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each vi; trivial modifications such as permuting the terms or adding terms with zero coefficient do not give distinct linear combinations.