14. Given that the linear map F: C→Cover C be defined by F(x, y, z) = (2r-y-z, 4x-y) relative to the following basis of C3 and C², respectively: B₁ = {u₁ = (1,1,0), u₂ = (1, 2, 3), u3 = (1,3,5)} and B₂ = {v₁ = (1, 1), v₂ = (0, 1)}. a. Find an arbitrary vector (a, b) E C² with respect to the basis vectors U₁ and v₂. b. Write F(u₁) as a linear combination of v₁ and v₂. c. Write F(u2) as a linear combination of v₁ and v2. d. Write F(uz) as a linear combination of v₁ and v₂. e. Find [F], the matrix representation of F relative to the bases B₁ and B2.
14. Given that the linear map F: C→Cover C be defined by F(x, y, z) = (2r-y-z, 4x-y) relative to the following basis of C3 and C², respectively: B₁ = {u₁ = (1,1,0), u₂ = (1, 2, 3), u3 = (1,3,5)} and B₂ = {v₁ = (1, 1), v₂ = (0, 1)}. a. Find an arbitrary vector (a, b) E C² with respect to the basis vectors U₁ and v₂. b. Write F(u₁) as a linear combination of v₁ and v₂. c. Write F(u2) as a linear combination of v₁ and v2. d. Write F(uz) as a linear combination of v₁ and v₂. e. Find [F], the matrix representation of F relative to the bases B₁ and B2.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
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![14. Given that the linear map F: C→Cover C be defined by F(x, y, z) =
(2x-y-z, 4x - y) relative to the following basis of C³ and C², respectively:
B₁ = {u₁ = (1, 1,0), u₂ = (1, 2, 3), u3 = (1,3,5)} and B₂ = {v₁ = (1, 1), v₂ =
(0, 1)}.
a. Find an arbitrary vector (a, b) E C² with respect to the basis vectors
v₁ and v₂.
b. Write F(u₁) as a linear combination of v₁ and v2.
c. Write F(u2) as a linear combination of v₁ and v2.
d. Write F(uz) as a linear combination of v₁ and v₂.
e. Find [F], the matrix representation of F relative to the bases B₁ and
B2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd911cf9a-d01b-41a6-be72-d88d8aa52e5a%2Fc359a17a-fdc2-47e3-90d4-78fc0e6ecd1f%2F272mfp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:14. Given that the linear map F: C→Cover C be defined by F(x, y, z) =
(2x-y-z, 4x - y) relative to the following basis of C³ and C², respectively:
B₁ = {u₁ = (1, 1,0), u₂ = (1, 2, 3), u3 = (1,3,5)} and B₂ = {v₁ = (1, 1), v₂ =
(0, 1)}.
a. Find an arbitrary vector (a, b) E C² with respect to the basis vectors
v₁ and v₂.
b. Write F(u₁) as a linear combination of v₁ and v2.
c. Write F(u2) as a linear combination of v₁ and v2.
d. Write F(uz) as a linear combination of v₁ and v₂.
e. Find [F], the matrix representation of F relative to the bases B₁ and
B2.
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