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Vector Spaces Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
FAQs on Vector Spaces Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is a vector space? |  |
Ans. A vector space is a mathematical structure that consists of a set of vectors and two operations: vector addition and scalar multiplication. It satisfies certain properties, such as closure under addition and scalar multiplication, and the existence of an additive identity and additive inverses.
| 2. What are the properties of a vector space? | |
Ans. A vector space must satisfy several properties, including closure under vector addition and scalar multiplication, associativity and commutativity of vector addition, existence of an additive identity and additive inverses, distributivity of scalar multiplication over vector addition, and compatibility of scalar multiplication with field multiplication.
| 3. Can a set of matrices form a vector space? | |
Ans. Yes, a set of matrices can form a vector space if it satisfies the properties of a vector space. Matrices can be added together and multiplied by scalars, and these operations satisfy the required properties. For example, the set of all 2x2 matrices with real number entries is a vector space.
| 4. What is a subspace in a vector space? | |
Ans. A subspace is a subset of a vector space that also forms a vector space itself. It must satisfy the properties of closure under vector addition and scalar multiplication, and contain the additive identity and additive inverses. For example, the set of all polynomials of degree 2 or less is a subspace of the vector space of all polynomials.
| 5. What is the dimension of a vector space? | |
Ans. The dimension of a vector space is the number of vectors in a basis for that vector space. A basis is a set of linearly independent vectors that spans the entire vector space. The dimension of a vector space can be thought of as the "size" or "degree of freedom" of the space. For example, the dimension of the vector space of all 3-dimensional vectors is 3.
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