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What is a Vector Space? Video Lecture | Mathematics Optional Notes for UPSC
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FAQs on What is a Vector Space? Video Lecture - Mathematics Optional Notes for UPSC
| 1. What is a vector space in mathematics? |  |
Ans. A vector space in mathematics is a set of objects called vectors, along with operations such as addition and scalar multiplication, which satisfy certain requirements like closure under addition and scalar multiplication.
| 2. What are the basic properties of a vector space? | |
Ans. The basic properties of a vector space include the existence of an additive identity, closure under addition and scalar multiplication, associativity of addition, distributivity of scalar multiplication over addition, and the existence of additive inverses.
| 3. How is a vector space different from a field? | |
Ans. A field is a set with two operations, addition and multiplication, that satisfy certain properties, while a vector space is a set of objects with operations defined on them that satisfy specific requirements related to linear algebra.
| 4. Can a set of vectors form a vector space? | |
Ans. A set of vectors can form a vector space if it satisfies the properties of closure under addition and scalar multiplication, associativity of addition, distributivity of scalar multiplication over addition, and the existence of an additive identity and inverses.
| 5. How are vector spaces used in practical applications? | |
Ans. Vector spaces are used in various fields such as physics, engineering, computer science, and economics to represent quantities with both magnitude and direction, model systems with multiple variables, and solve systems of linear equations.
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