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Let 1: R* → R³, 1: x Ax, where k - 1 A : 3 4 -k 3 1 1 1 a.) Establish for which value of k, the linear function I is an isomorphism. b.) Establish for which value of k, dim Kerl = 2. c.) Establish for which value of k, it is dim Iml = 2 and in this case, find a basis for Iml.

Let 1: R* → R³, 1: x Ax, where k - 1 A : 3 4 -k 3 1 1 1 a.) Establish for which value of k, the linear function I is an isomorphism. b.) Establish for which value of k, dim Kerl = 2. c.) Establish for which value of k, it is dim Iml = 2 and in this case, find a basis for Iml.

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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Let l: R3 → R³, l : xH Ax, where k – 1 A = 3 4 -k 3 | 1 1 1 a.) Establish for which value of k, the linear function l is an isomorphism. b.) Establish for which value of k, dim Kerl = 2. c.) Establish for which value of k, it is dim Iml = 2 and in this case, find a basis for Iml.
Transcribed Image Text:Let l: R3 → R³, l : xH Ax, where k – 1 A = 3 4 -k 3 | 1 1 1 a.) Establish for which value of k, the linear function l is an isomorphism. b.) Establish for which value of k, dim Kerl = 2. c.) Establish for which value of k, it is dim Iml = 2 and in this case, find a basis for Iml.
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