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If T is a linear transformation and T(u), T(v), T(w) are linearly independent, then u, v, w are linearly independent. True False

If T is a linear transformation and T(u), T(v), T(w) are linearly independent, then u, v, w are linearly independent. True False

Advanced Engineering Mathematics
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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### Linear Transformations and Independence **Question:** If \( T \) is a linear transformation and \( T(u) \), \( T(v) \), \( T(w) \) are linearly independent, then \( u \), \( v \), \( w \) are linearly independent. - True - False **Explanation:** This question involves understanding the concept of linear transformations and linear independence within the context of linear algebra. **Key Concepts:** 1. **Linear Transformation:** - A function \( T \) between two vector spaces that preserves the operations of vector addition and scalar multiplication. - Formally, \( T: V \rightarrow W \) is a linear transformation if for any vectors \( u, v \) in \( V \) and any scalar \( c \), the following holds: * \( T(u + v) = T(u) + T(v) \) * \( T(cu) = cT(u) \) 2. **Linear Independence:** - A set of vectors \( \{v_1, v_2, ..., v_n\} \) in a vector space is said to be linearly independent if the vector equation \( c_1v_1 + c_2v_2 + ... + c_nv_n = 0 \) has only the trivial solution where all scalars \( c_i = 0 \). - If any one of the scalars \( c_i \) can be nonzero and the equation still holds, the vectors are deemed linearly dependent. Given the problem, you are asked to determine whether the linear independence of \( T(u) \), \( T(v) \), \( T(w) \) implies the linear independence of \( u \), \( v \), \( w \). **Answer Options:** - **True** - **False** To answer this question, you must consider whether the transformation \( T \) preserves the linear independence of vectors in the domain (original space). This involves probing deeper into the properties of linear transformations and how they interact with the structure of vector spaces.
Transcribed Image Text:### Linear Transformations and Independence **Question:** If \( T \) is a linear transformation and \( T(u) \), \( T(v) \), \( T(w) \) are linearly independent, then \( u \), \( v \), \( w \) are linearly independent. - True - False **Explanation:** This question involves understanding the concept of linear transformations and linear independence within the context of linear algebra. **Key Concepts:** 1. **Linear Transformation:** - A function \( T \) between two vector spaces that preserves the operations of vector addition and scalar multiplication. - Formally, \( T: V \rightarrow W \) is a linear transformation if for any vectors \( u, v \) in \( V \) and any scalar \( c \), the following holds: * \( T(u + v) = T(u) + T(v) \) * \( T(cu) = cT(u) \) 2. **Linear Independence:** - A set of vectors \( \{v_1, v_2, ..., v_n\} \) in a vector space is said to be linearly independent if the vector equation \( c_1v_1 + c_2v_2 + ... + c_nv_n = 0 \) has only the trivial solution where all scalars \( c_i = 0 \). - If any one of the scalars \( c_i \) can be nonzero and the equation still holds, the vectors are deemed linearly dependent. Given the problem, you are asked to determine whether the linear independence of \( T(u) \), \( T(v) \), \( T(w) \) implies the linear independence of \( u \), \( v \), \( w \). **Answer Options:** - **True** - **False** To answer this question, you must consider whether the transformation \( T \) preserves the linear independence of vectors in the domain (original space). This involves probing deeper into the properties of linear transformations and how they interact with the structure of vector spaces.
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