Mathematics  >  Mathematics for IIT JAM, CSIR NET, UGC NET  >  Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences

Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics

Document Description: Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences for Mathematics 2022 is part of Mathematics for IIT JAM, CSIR NET, UGC NET preparation. The notes and questions for Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences have been prepared according to the Mathematics exam syllabus. Information about Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences covers topics like and Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences Example, for Mathematics 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences.

Introduction of Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences in English is available as part of our Mathematics for IIT JAM, CSIR NET, UGC NET for Mathematics & Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences in Hindi for Mathematics for IIT JAM, CSIR NET, UGC NET course. Download more important topics related with notes, lectures and mock test series for Mathematics Exam by signing up for free. Mathematics: Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics
1 Crore+ students have signed up on EduRev. Have you?

Vector Spaces

Definition:vector space is a set V on which two operations + and · are defined, called vector addition and scalar multiplication.

The operation + (vector addition) must satisfy the following conditions:

Closure: If u and v are any vectors in V, then the sum u + v  belongs to V.
(1) Commutative law: For all vectors u and v in V, u + v = v + u
(2) Associative law: For all vectors u, v, w in V, u + (v + w) = (u + v) + w
(3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V, 0 + v = v  and v + 0 = v.
(4) Additive inverses: For each vector v in V, the equations v + x = 0  and x + v = 0  have a solution x in V, called an additive inverse of v, and denoted by - v.

The operation · (scalar multiplication) is defined between real numbers (or scalars) and vectors, and must satisfy the following conditions:
Closure: If v in any vector in V, and c is any real number, then the product c · v  belongs to V.
(5) Distributive law: For all real numbers c and all vectors u, v in V, c · (u + v) = c · u + c · v
(6) Distributive law: For all real numbers c, d and all vectors v in V, (c+d) · v = c · v + d · v
(7) Associative law: For all real numbers c,d and all vectors v in V, c · (d · v) = (cd) · v
(8) Unitary law: For all vectors v in V, 1 · v = v

Subspaces

Definition: Let V be a vector space, and let W be a subset of V. If W is a vector space with respect to the operations in V, then W is called a subspace of V.

Theorem: Let V be a vector space, with operations + and  ·, and let W be a subset of V. Then W is a subspace of V if and only if the following conditions hold.
Sub0 W is nonempty: The zero vector belongs to W.
Sub1 Closure under +: If u and v are any vectors in W, then u + v   is in W.
Sub2 Closure under ·: If v is any vector in W, and c is any real number, then c · v  is in W.

The document Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences Notes | Study Mathematics for IIT JAM, CSIR NET, UGC NET - Mathematics is a part of the Mathematics Course Mathematics for IIT JAM, CSIR NET, UGC NET.
All you need of Mathematics at this link: Mathematics
Download as PDF

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

video lectures

,

UGC NET - Mathematics

,

Vector Spaces and Subspaces - Vector Algebra

,

Exam

,

MCQs

,

CSIR-NET Mathematical Sciences Notes | Study Mathematics for IIT JAM

,

Previous Year Questions with Solutions

,

study material

,

Objective type Questions

,

CSIR NET

,

mock tests for examination

,

Vector Spaces and Subspaces - Vector Algebra

,

pdf

,

practice quizzes

,

Summary

,

Vector Spaces and Subspaces - Vector Algebra

,

CSIR NET

,

Free

,

Semester Notes

,

CSIR-NET Mathematical Sciences Notes | Study Mathematics for IIT JAM

,

Important questions

,

CSIR-NET Mathematical Sciences Notes | Study Mathematics for IIT JAM

,

CSIR NET

,

ppt

,

Extra Questions

,

Sample Paper

,

UGC NET - Mathematics

,

UGC NET - Mathematics

,

shortcuts and tricks

,

Viva Questions

,

past year papers

;