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Vector Spaces over R and C Video Lecture | Mathematics Optional Notes for UPSC
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FAQs on Vector Spaces over R and C Video Lecture - Mathematics Optional Notes for UPSC
| 1. What is a vector space over R and C? |  |
Ans. A vector space over R (the real numbers) and C (the complex numbers) is a set of elements equipped with two operations, addition and scalar multiplication, satisfying certain properties such as closure under addition and scalar multiplication.
| 2. What are the key properties of vector spaces over R and C? | |
Ans. Some key properties of vector spaces over R and C include the existence of a zero vector, the existence of additive inverses, closure under addition and scalar multiplication, and the distributive property of scalar multiplication over addition.
| 3. How are vector spaces over R and C different from each other? | |
Ans. Vector spaces over R are restricted to elements that involve only real numbers, while vector spaces over C allow elements to involve complex numbers. This means that vector spaces over C have a broader range of elements compared to vector spaces over R.
| 4. Can a vector space over R be considered a vector space over C? | |
Ans. Yes, a vector space over R can be considered a vector space over C by extending the scalar multiplication operation to include complex numbers. This means that every vector space over R can also be viewed as a vector space over C.
| 5. What are some examples of vector spaces over R and C? | |
Ans. Examples of vector spaces over R include Euclidean spaces and spaces of polynomials with real coefficients. Examples of vector spaces over C include spaces of polynomials with complex coefficients and spaces of complex-valued functions.
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