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4. Let L: R5 R¹ be a linear transformation, and let S R5. Suppose that {L(v₁), L(v₂), L(v3)} is linearly independent. independent. (v₁, U2, U3) be an indexed subset of Show that S = (v₁, U2, U3} is linearly

4. Let L: R5 R¹ be a linear transformation, and let S R5. Suppose that {L(v₁), L(v₂), L(v3)} is linearly independent. independent. (v₁, U2, U3) be an indexed subset of Show that S = (v₁, U2, U3} is linearly

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 Linear Independence in Vector Spaces** ### Problem Statement: Let \( T: \mathbb{R}^5 \to \mathbb{R} \) be a linear transformation, and let \( S = \{v_1, v_2, v_3\} \) be an indexed subset of \( \mathbb{R}^5 \). Suppose that \( \{T(v_1), T(v_2), T(v_3)\} \) is linearly independent. Show that \( S = \{v_1, v_2, v_3\} \) is linearly independent. ### Explanation: This problem involves proving the linear independence of a set of vectors in a higher-dimensional space, given the linear independence of their images under a linear transformation. It is a fundamental concept in linear algebra, connecting the properties of a linear transformation with the linear independence of vectors in vector spaces. To solve the problem, you need to use the properties of linear transformations and the definition of linear independence. The proof typically leverages the fact that a linear transformation preserves linear relations between vectors. ### Detailed Steps to Consider: 1. **Assume a Linear Combination**: Start by assuming that \( a_1v_1 + a_2v_2 + a_3v_3 = 0 \) for some scalars \( a_1, a_2, a_3 \in \mathbb{R} \). 2. **Apply the Linear Transformation**: Apply the linear transformation \( T \) to both sides of the equation. 3. **Use Linear Independence of \( \{T(v_1), T(v_2), T(v_3)\} \)**: Utilize the given that \( \{T(v_1), T(v_2), T(v_3)\} \) is linearly independent to conclude that the scalars \( a_1, a_2, a_3 \) must all be zero. 4. **Conclude Linear Independence**: Conclude that the original set \( \{v_1, v_2, v_3\} \) is linearly independent. This process solidifies your understanding of how linear transformations interact with vector spaces and the concept of independence, a cornerstone of linear algebra. Feel free to explore more resources and examples to deepen your understanding of this topic. Happy learning!
Transcribed Image Text:**Linear Transformations and Linear Independence in Vector Spaces** ### Problem Statement: Let \( T: \mathbb{R}^5 \to \mathbb{R} \) be a linear transformation, and let \( S = \{v_1, v_2, v_3\} \) be an indexed subset of \( \mathbb{R}^5 \). Suppose that \( \{T(v_1), T(v_2), T(v_3)\} \) is linearly independent. Show that \( S = \{v_1, v_2, v_3\} \) is linearly independent. ### Explanation: This problem involves proving the linear independence of a set of vectors in a higher-dimensional space, given the linear independence of their images under a linear transformation. It is a fundamental concept in linear algebra, connecting the properties of a linear transformation with the linear independence of vectors in vector spaces. To solve the problem, you need to use the properties of linear transformations and the definition of linear independence. The proof typically leverages the fact that a linear transformation preserves linear relations between vectors. ### Detailed Steps to Consider: 1. **Assume a Linear Combination**: Start by assuming that \( a_1v_1 + a_2v_2 + a_3v_3 = 0 \) for some scalars \( a_1, a_2, a_3 \in \mathbb{R} \). 2. **Apply the Linear Transformation**: Apply the linear transformation \( T \) to both sides of the equation. 3. **Use Linear Independence of \( \{T(v_1), T(v_2), T(v_3)\} \)**: Utilize the given that \( \{T(v_1), T(v_2), T(v_3)\} \) is linearly independent to conclude that the scalars \( a_1, a_2, a_3 \) must all be zero. 4. **Conclude Linear Independence**: Conclude that the original set \( \{v_1, v_2, v_3\} \) is linearly independent. This process solidifies your understanding of how linear transformations interact with vector spaces and the concept of independence, a cornerstone of linear algebra. Feel free to explore more resources and examples to deepen your understanding of this topic. Happy learning!
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