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Prove that any linear transformation T : R" → R" maps a line passing through the origin to either the zero vector or a line passing through the origin. Generalize this for planes and hyperplanes. What are the images of these under linear transformations?

Prove that any linear transformation T : R" → R" maps a line passing through the origin to either the zero vector or a line passing through the origin. Generalize this for planes and hyperplanes. What are the images of these under linear transformations?

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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(1) Prove that any linear transformation T : R" → R" maps a line passing through the origin to either the zero vector or a line passing through the origin. Generalize this for planes and hyperplanes. What are the images of these under linear transformations?
Transcribed Image Text:(1) Prove that any linear transformation T : R" → R" maps a line passing through the origin to either the zero vector or a line passing through the origin. Generalize this for planes and hyperplanes. What are the images of these under linear transformations?
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