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Vector Subspace: Linearly Independent & Dependents Vectors Video Lecture | Mathematics for Competitive Exams
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FAQs on Vector Subspace: Linearly Independent & Dependents Vectors Video Lecture - Mathematics for Competitive Exams
| 1. What is a vector subspace? |  |
A vector subspace is a subset of a vector space that is closed under addition and scalar multiplication. This means that if we take any two vectors from the subspace, their sum will still be in the subspace, and if we multiply a vector from the subspace by a scalar, the result will also be in the subspace.
| 2. How do we determine if a set of vectors is linearly independent? | |
To determine if a set of vectors is linearly independent, we need to check if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 (where c1, c2, ..., cn are scalars and v1, v2, ..., vn are the vectors) is the trivial solution, where all the scalars are zero. If the only solution is the trivial solution, then the vectors are linearly independent. Otherwise, they are linearly dependent.
| 3. Can a set of linearly dependent vectors form a vector subspace? | |
No, a set of linearly dependent vectors cannot form a vector subspace. A vector subspace requires that the set of vectors is closed under addition and scalar multiplication. However, if the vectors are linearly dependent, it means that at least one vector in the set can be written as a linear combination of the other vectors. Therefore, adding or multiplying these vectors will result in vectors that are not in the set, violating the closure property.
| 4. How many linearly independent vectors can a vector subspace have? | |
The number of linearly independent vectors that a vector subspace can have is called its dimension. The dimension of a vector subspace is a measure of its size. The maximum number of linearly independent vectors a vector subspace can have is equal to the number of dimensions of the vector space it is a subspace of. For example, in three-dimensional space, a vector subspace can have at most three linearly independent vectors.
| 5. Can a vector subspace contain the zero vector? | |
Yes, a vector subspace must contain the zero vector. The zero vector is defined as the vector that when added to any other vector, gives the same vector as the result. Since a vector subspace requires closure under addition, it must contain the zero vector. In other words, it means that the subspace must have a vector that can be obtained by multiplying the zero scalar with any vector in the subspace.
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Dec 23, 2024 Last updated |
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