**Linear Independence and Dependency in Vector Spaces** In the study of vector spaces, the concept of linear independence is fundamental. Here’s a concise overview relevant for understanding these properties: 1. **Definition of Linear Independence and Spanning a Vector Space:** - If a set of vectors in a vector space is linearly independent, then it spans that vector space. This means that no vector in the set can be written as a linear combination of the others, highlighting their unique contribution to the space. 2. **Linear Map and Dependency:** - Consider a linear map \( T: V \to W \) where \( V \) and \( W \) are vector spaces. - If a set \( \{ \mathbf{v}_1, \ldots, \mathbf{v}_n \} \subset V \) is linearly dependent, then the image \( \{ T(\mathbf{v}_1), \ldots, T(\mathbf{v}_n) \} \) under the map \( T \) is also linearly dependent. Understanding these concepts is crucial for exploring more advanced topics in linear algebra, such as transformations and applications in various fields.
**Linear Independence and Dependency in Vector Spaces** In the study of vector spaces, the concept of linear independence is fundamental. Here’s a concise overview relevant for understanding these properties: 1. **Definition of Linear Independence and Spanning a Vector Space:** - If a set of vectors in a vector space is linearly independent, then it spans that vector space. This means that no vector in the set can be written as a linear combination of the others, highlighting their unique contribution to the space. 2. **Linear Map and Dependency:** - Consider a linear map \( T: V \to W \) where \( V \) and \( W \) are vector spaces. - If a set \( \{ \mathbf{v}_1, \ldots, \mathbf{v}_n \} \subset V \) is linearly dependent, then the image \( \{ T(\mathbf{v}_1), \ldots, T(\mathbf{v}_n) \} \) under the map \( T \) is also linearly dependent. Understanding these concepts is crucial for exploring more advanced topics in linear algebra, such as transformations and applications in various fields.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Answer TRUE or FALSE. Along with your answer, provide a
proof if true and a speci c counterexample if false.
Transcribed Image Text:**Linear Independence and Dependency in Vector Spaces**
In the study of vector spaces, the concept of linear independence is fundamental. Here’s a concise overview relevant for understanding these properties:
1. **Definition of Linear Independence and Spanning a Vector Space:**
- If a set of vectors in a vector space is linearly independent, then it spans that vector space. This means that no vector in the set can be written as a linear combination of the others, highlighting their unique contribution to the space.
2. **Linear Map and Dependency:**
- Consider a linear map \( T: V \to W \) where \( V \) and \( W \) are vector spaces.
- If a set \( \{ \mathbf{v}_1, \ldots, \mathbf{v}_n \} \subset V \) is linearly dependent, then the image \( \{ T(\mathbf{v}_1), \ldots, T(\mathbf{v}_n) \} \) under the map \( T \) is also linearly dependent.
Understanding these concepts is crucial for exploring more advanced topics in linear algebra, such as transformations and applications in various fields.
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