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Vector Subspace: Basis & Dimension Video Lecture | Mathematics for Competitive Exams
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FAQs on Vector Subspace: Basis & Dimension Video Lecture - Mathematics for Competitive Exams
| 1. What is a vector subspace? |  |
Ans. A vector subspace is a subset of a vector space that is closed under vector addition and scalar multiplication. It contains the zero vector and satisfies the axioms of a vector space.
| 2. What is a basis of a vector subspace? | |
Ans. A basis of a vector subspace is a set of linearly independent vectors that span the entire subspace. It provides a minimal set of vectors that can be used to represent any vector in the subspace.
| 3. How do you determine the dimension of a vector subspace? | |
Ans. The dimension of a vector subspace is equal to the number of vectors in its basis. It represents the maximum number of linearly independent vectors that can exist in the subspace.
| 4. Can a vector subspace have more than one basis? | |
Ans. Yes, a vector subspace can have multiple bases. As long as the vectors in the basis are linearly independent and span the subspace, any set of vectors that satisfies these conditions can be considered a basis.
| 5. How can the dimension of a vector subspace be used to determine if two subspaces are equal? | |
Ans. If two vector subspaces have the same dimension, it suggests that they have the same number of linearly independent vectors. However, it does not necessarily mean that the two subspaces are equal. To prove equality, one must show that every vector in one subspace can be expressed as a linear combination of vectors in the other subspace, and vice versa.
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Dec 23, 2024 Last updated |
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