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Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download
Vector Spaces
Definition: A 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.
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FAQs on Vector Spaces and Subspaces - Vector Algebra, CSIR-NET Mathematical Sciences - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is a vector space and what are its properties? |  |
| 2. How do you determine if a set of vectors forms a subspace? | |
Ans. To determine if a set of vectors forms a subspace, we need to check three conditions: (1) the set is not empty, (2) it is closed under vector addition, and (3) it is closed under scalar multiplication. If all three conditions are satisfied, then the set is a subspace.
| 3. Can a subspace contain the zero vector? | |
Ans. Yes, a subspace must always contain the zero vector. This is because a subspace must be closed under scalar multiplication, and multiplying any vector in the subspace by the scalar zero would result in the zero vector. Additionally, the zero vector is also required to satisfy the condition of additive identity.
| 4. What is the dimension of a vector space? | |
Ans. The dimension of a vector space is the number of vectors in any basis for that vector space. It represents the maximum number of linearly independent vectors that can be chosen from the vector space. The dimension is a fundamental property of a vector space and helps in understanding its structure and properties.
| 5. How are vector spaces used in real-life applications? | |
Ans. Vector spaces have numerous applications in various fields. For example, in physics, vector spaces are used to represent physical quantities such as force, velocity, and displacement. In computer graphics, vector spaces are used to model and manipulate images and three-dimensional objects. They are also used in engineering, economics, and statistics for data analysis, optimization problems, and modeling real-world systems.
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