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Vector Subspace: Subspace Theorems & Examples Video Lecture | Mathematics for Competitive Exams
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FAQs on Vector Subspace: Subspace Theorems & Examples Video Lecture - Mathematics for Competitive Exams
| 1. What are vector subspaces? |  |
Ans. Vector subspaces are subsets of a vector space that satisfy three conditions: closure under addition (if u and v are in the subspace, then u+v is also in the subspace), closure under scalar multiplication (if u is in the subspace and c is a scalar, then cu is also in the subspace), and containing the zero vector (the subspace must contain the zero vector).
| 2. What are some examples of vector subspaces? | |
Ans. Examples of vector subspaces include the set of all solutions to a homogeneous linear system (also known as the null space or kernel), the span of a set of vectors, and the column space and row space of a matrix.
| 3. What are the subspace theorems related to vector subspaces? | |
Ans. The subspace theorems state that if a subset of a vector space satisfies the closure properties (closure under addition and scalar multiplication) and contains the zero vector, it is a subspace. Additionally, any intersection of subspaces is also a subspace, and the union of subspaces may or may not be a subspace.
| 4. How can vector subspaces be visualized? | |
Ans. Vector subspaces can be visualized as planes, lines, or even just the origin in three-dimensional space. In two-dimensional space, subspaces can be visualized as lines or the origin. The dimension of a subspace corresponds to the number of linearly independent vectors needed to span it.
| 5. What is the significance of vector subspaces in mathematics? | |
Ans. Vector subspaces play a fundamental role in linear algebra and various areas of mathematics. They allow for the study of linear transformations, eigenvectors, and eigenvalues, as well as providing a framework for solving systems of linear equations and understanding the properties of matrices.
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Dec 23, 2024 Last updated |
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