Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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
See similar textbooks
icon
Related questions
Question
### 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 \( \mathbb{R}^n \).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS