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Basis & Dimension & Examples Of Basis - Linear Algebra Video Lecture | Engineering Mathematics - Civil Engineering (CE)
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FAQs on Basis & Dimension & Examples Of Basis - Linear Algebra Video Lecture - Engineering Mathematics - Civil Engineering (CE)
| 1. What is a basis in linear algebra and why is it important? |  |
Ans. In linear algebra, a basis is a set of linearly independent vectors that span a vector space. It is important because it provides a way to represent any vector in the vector space by a unique combination of the basis vectors. Additionally, the dimension of a vector space is defined as the number of vectors in its basis, so the basis also determines the dimension of the vector space.
| 2. How do you determine if a set of vectors is a basis? | |
Ans. To determine if a set of vectors is a basis, you need to check two conditions:
1. The vectors must be linearly independent, meaning that no vector in the set can be expressed as a linear combination of the others.
2. The vectors must span the vector space, meaning that every vector in the vector space can be expressed as a linear combination of the vectors in the set.
| 3. Can a vector space have more than one basis? | |
Ans. Yes, a vector space can have more than one basis. In fact, any vector space with dimension greater than zero will have infinitely many bases. However, all bases of a given vector space will have the same number of vectors, which is the dimension of the vector space.
| 4. What is the dimension of a vector space? | |
Ans. The dimension of a vector space is the number of vectors in its basis. It represents the maximum number of linearly independent vectors that can exist in the vector space. For example, the dimension of the vector space R^3 (3-dimensional Euclidean space) is 3, as any set of 3 linearly independent vectors can form a basis for this vector space.
| 5. Can a vector space have a basis with fewer vectors than its dimension? | |
Ans. No, a vector space cannot have a basis with fewer vectors than its dimension. The basis of a vector space must have the same number of vectors as the dimension of the vector space. If a vector space has dimension n, then any basis of that vector space will contain exactly n linearly independent vectors.
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Dec 23, 2024 Last updated |
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