(c) Define the vectors ₁ = f(v₁) and ₂ W = (₁, ₂) forms a basis for the image of f. = f(v₂), where w₁, 2 E Im(f). Show that the set You may assume without proof that the image of the linear map Im(f) forms a subspace of R³. d) Define a linear map g: Im(ƒ) → R³ given by: g (w₁) = V₁, g (W₂) = V₂ Show that fogof=f (e) Based on your results, determine the validity of the following statement: "The linear map go f is the identity map."
(c) Define the vectors ₁ = f(v₁) and ₂ W = (₁, ₂) forms a basis for the image of f. = f(v₂), where w₁, 2 E Im(f). Show that the set You may assume without proof that the image of the linear map Im(f) forms a subspace of R³. d) Define a linear map g: Im(ƒ) → R³ given by: g (w₁) = V₁, g (W₂) = V₂ Show that fogof=f (e) Based on your results, determine the validity of the following statement: "The linear map go f is the identity map."
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
Question
I just need help with c), d) and e)
![2. (a) Show that the set B = (v1, V2, V3) defined by:
1
0
V₁
[f]
2 =
=
form a basis of R3
(b) Consider a linear map f: R³ → R³ and suppose the matrix associated with f under the
standard basis of R3 is given by:
1
-1
-1
1
2
Find the image vectors f (v₁), f (v₂), f (√3)
1
*-()
1
2
-3 1
-3 2
-3 2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b60fa95-9b4b-4fa6-a572-04dc2e43c26e%2F6a2166b6-d0ec-4bd9-a9e4-48febaaf7c44%2Fsu4u3pkj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. (a) Show that the set B = (v1, V2, V3) defined by:
1
0
V₁
[f]
2 =
=
form a basis of R3
(b) Consider a linear map f: R³ → R³ and suppose the matrix associated with f under the
standard basis of R3 is given by:
1
-1
-1
1
2
Find the image vectors f (v₁), f (v₂), f (√3)
1
*-()
1
2
-3 1
-3 2
-3 2
Transcribed Image Text:Define the vectors w₁ = f(v₁) and w₂= f(v2), where w₁, W₂ € Im(f). Show that the set
W = (₁, ₂) forms a basis for the image of f.
You may assume without proof that the image of the linear map Im(f) forms a subspace
of R³.
(d) Define a linear map g: Im(f) → R³ given by:
g (w₁) = v₁,
g (W₂) = V₂
Show that fogof=f
(e) Based on your results, determine the validity of the following statement:
"The linear map go f is the identity map."
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