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O Dr. Manning (Vector spaces) Let F be the vector space of continuous functions f: R → R and let H be the subspace of F spanned by {1, sin(t), cos(t), sin(t) cos(t)}. answer. (a) Let K be the set of functions K - {a sin(2t) + cos(t) a ER}. Of the three properties of subspaces, which does K satisfy? Is K a vector subspace of H or not? Justify your (b) Let L be the set of functions L = {a cos²(t): a ER}. Determine whether L is a vector subspace of H or not. Justify your answer. (c) Consider R as a real vector space (under the usual real number addition and multiplica- tion). Let T: HR be the function given by T(f) = f(t) dt. Show that T is a linear transformation. (d) Find a nonzero vector in ker(7). Justify your answer. (e) Let D: HF be given by D(f) = f', the first derivative. Show that D is a lincar transformation. (f) Find a spanning set for ker(D), where D is the linear transformation from part (c). Justify your answer.

O Dr. Manning (Vector spaces) Let F be the vector space of continuous functions f: R → R and let H be the subspace of F spanned by {1, sin(t), cos(t), sin(t) cos(t)}. answer. (a) Let K be the set of functions K - {a sin(2t) + cos(t) a ER}. Of the three properties of subspaces, which does K satisfy? Is K a vector subspace of H or not? Justify your (b) Let L be the set of functions L = {a cos²(t): a ER}. Determine whether L is a vector subspace of H or not. Justify your answer. (c) Consider R as a real vector space (under the usual real number addition and multiplica- tion). Let T: HR be the function given by T(f) = f(t) dt. Show that T is a linear transformation. (d) Find a nonzero vector in ker(7). Justify your answer. (e) Let D: HF be given by D(f) = f', the first derivative. Show that D is a lincar transformation. (f) Find a spanning set for ker(D), where D is the linear transformation from part (c). Justify your answer.

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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Part e and f

(Vector spaces) Let F be the vector space of continuous functions f: R → R and let H be the subspace of F spanned by {1, sin(t), cos(t), sin(t) cos(t)}. Dr. Manning (a) Let K be the set of functions K {a sin(2t) + cos(t): a ER}. Of the three properties of subspaces, which does K satisfy? Is K a vector subspace of H or not? Justify your answer. DIRE (b) Let L be the set of functions L = {a cos² (t) : aE R}. Determine whether L is a vector subspace of H or not. Justify your answer. (c) Consider R as a real vector space (under the usual real number addition and multiplica- tion). Let T: HR be the function given by T(ƒ) = fő ƒ (t) dt. Show that T is a linear transformation. (d) Find a nonzero vector in ker(7). Justify your answer. (e) Let D: HF be given by D(f) = f', the first derivative. Show that D is a linear transformation. (f) Find a spänning set for ker(D), where D is the lincar transformation from part (c). Justify your answer.
Transcribed Image Text:(Vector spaces) Let F be the vector space of continuous functions f: R → R and let H be the subspace of F spanned by {1, sin(t), cos(t), sin(t) cos(t)}. Dr. Manning (a) Let K be the set of functions K {a sin(2t) + cos(t): a ER}. Of the three properties of subspaces, which does K satisfy? Is K a vector subspace of H or not? Justify your answer. DIRE (b) Let L be the set of functions L = {a cos² (t) : aE R}. Determine whether L is a vector subspace of H or not. Justify your answer. (c) Consider R as a real vector space (under the usual real number addition and multiplica- tion). Let T: HR be the function given by T(ƒ) = fő ƒ (t) dt. Show that T is a linear transformation. (d) Find a nonzero vector in ker(7). Justify your answer. (e) Let D: HF be given by D(f) = f', the first derivative. Show that D is a linear transformation. (f) Find a spänning set for ker(D), where D is the lincar transformation from part (c). Justify your answer.
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