Let L : R³ → R³ be a linear map with associated matrix [L]B= = -1 0 0 -2 2 -1 4 0 1 under the standard basis B = ((1,0,0), (0, 1, 0)¹, (0, 0, 1)ª). (a) Evaluate L((2,−1, 1)T) under the standard basis (b) Define C = ((0, 1, 1), (0, 1, 0), (-1,0, 2)). Show that C forms a basis of R³. (c) Find the change of basis matrix [C] and hence find [B]c. (d) Find [L] and hence generalise your result to ([L]c) for a positive integer N (e) Let N a positive integer. Use the result of the previous part to find ([L]ß)N.
Let L : R³ → R³ be a linear map with associated matrix [L]B= = -1 0 0 -2 2 -1 4 0 1 under the standard basis B = ((1,0,0), (0, 1, 0)¹, (0, 0, 1)ª). (a) Evaluate L((2,−1, 1)T) under the standard basis (b) Define C = ((0, 1, 1), (0, 1, 0), (-1,0, 2)). Show that C forms a basis of R³. (c) Find the change of basis matrix [C] and hence find [B]c. (d) Find [L] and hence generalise your result to ([L]c) for a positive integer N (e) Let N a positive integer. Use the result of the previous part to find ([L]ß)N.
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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Please help me solve question (d) and (e), thank you:)
![Let L: R³ → R³ be a linear map with associated matrix
-1
[L]B =
-2 2 -1
4
1
under the standard basis B = ((1,0,0)", (0, 1, 0)" , (0,0, 1)").
(a) Evaluate L((2, –1,1)T) under the standard basis
(b) Define C = ((0, 1, 1)", (0, 1, 0)T, (-1,0, 2)7). Show that C forms a basis of R3.
(c) Find the change of basis matrix [C]g and hence find [B]c.
(d) Find [L]c and hence generalise your result to ([L]c)™ for a positive integer N
(e) Let N a positive integer. Use the result of the previous part to find ([L]g)N.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b8f22fc-b305-4f4f-aaf0-4d806b773299%2Fdbb26529-e5ad-4d3c-810c-3f8a04b243a4%2Fw8mavyq_processed.png&w=3840&q=75)
Transcribed Image Text:Let L: R³ → R³ be a linear map with associated matrix
-1
[L]B =
-2 2 -1
4
1
under the standard basis B = ((1,0,0)", (0, 1, 0)" , (0,0, 1)").
(a) Evaluate L((2, –1,1)T) under the standard basis
(b) Define C = ((0, 1, 1)", (0, 1, 0)T, (-1,0, 2)7). Show that C forms a basis of R3.
(c) Find the change of basis matrix [C]g and hence find [B]c.
(d) Find [L]c and hence generalise your result to ([L]c)™ for a positive integer N
(e) Let N a positive integer. Use the result of the previous part to find ([L]g)N.
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