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Basis and Dimension Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
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FAQs on Basis and Dimension Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is the definition of basis in linear algebra? |  |
Ans. In linear algebra, the basis of a vector space is a set of vectors that are linearly independent and span the entire vector space. It means that any vector in the vector space can be expressed as a unique linear combination of the basis vectors.
| 2. How is the dimension of a vector space related to its basis? | |
Ans. The dimension of a vector space is equal to the number of vectors in any basis for that vector space. In other words, all bases of the same vector space will have the same number of vectors, and this number is called the dimension of the vector space.
| 3. Can a vector space have multiple bases? | |
Ans. Yes, a vector space can have multiple bases. In fact, any vector space with dimension greater than zero will have infinitely many different bases. However, all these bases will have the same number of vectors, which is equal to the dimension of the vector space.
| 4. What is the significance of basis and dimension in solving systems of linear equations? | |
Ans. The basis and dimension of a vector space play a crucial role in solving systems of linear equations. The dimension determines the number of variables in the system, while the basis provides a set of vectors that can be used to express any solution to the system. By finding the basis and dimension, we can identify the number of independent equations needed to find a unique solution to the system.
| 5. Can the basis of a vector space be empty? | |
Ans. No, the basis of a vector space cannot be empty. It must contain at least one vector. An empty set of vectors cannot span any vector space, and therefore cannot form a basis. The dimension of a vector space is always a positive integer, indicating the minimum number of vectors required to form a basis.
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Dec 23, 2024 Last updated |
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