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1. Let TR → Rm be a linear transformation, and let {V₁, V2, V3} be a linearly dependent set in Rn. Explain why the set {T(v₁), T(v₂), T(v3)} in Rm must also be linearly dependent. Remark: You argument should not be a "proof by example," that is, the argument should hold for arbitrary linear transformation T and arbitrary linearly dependent set of vectors {V1, V2, V3} in Rn.

1. Let TR → Rm be a linear transformation, and let {V₁, V2, V3} be a linearly dependent set in Rn. Explain why the set {T(v₁), T(v₂), T(v3)} in Rm must also be linearly dependent. Remark: You argument should not be a "proof by example," that is, the argument should hold for arbitrary linear transformation T and arbitrary linearly dependent set of vectors {V1, V2, V3} in Rn.

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 Dependence **Problem 1:** Let \( T: \mathbb{R}^n \rightarrow \mathbb{R}^m \) be a linear transformation, and let \( \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \) be a linearly dependent set in \( \mathbb{R}^n \). Explain why the set \( \{ T(\mathbf{v}_1), T(\mathbf{v}_2), T(\mathbf{v}_3) \} \) in \( \mathbb{R}^m \) must also be linearly dependent. **Remark:** Your argument should not be a "proof by example," that is, the argument should hold for an arbitrary linear transformation \( T \) and an arbitrary linearly dependent set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \) in \( \mathbb{R}^n \).
Transcribed Image Text:### Linear Transformations and Linear Dependence **Problem 1:** Let \( T: \mathbb{R}^n \rightarrow \mathbb{R}^m \) be a linear transformation, and let \( \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \) be a linearly dependent set in \( \mathbb{R}^n \). Explain why the set \( \{ T(\mathbf{v}_1), T(\mathbf{v}_2), T(\mathbf{v}_3) \} \) in \( \mathbb{R}^m \) must also be linearly dependent. **Remark:** Your argument should not be a "proof by example," that is, the argument should hold for an arbitrary linear transformation \( T \) and an arbitrary linearly dependent set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \} \) in \( \mathbb{R}^n \).
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