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(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
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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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
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
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."
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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