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10.1 Lemma Let X be a linear space over K. Consider subsets U and V of X, and k € K such that U CV + kU. Then for every z EU, there is a sequence (v₁) in V such that x − (v₁ + kv₂ + + k”−¹vn) € k"U, n=1,2,.... Proof: Let z EU. Since UCV + kU, there is some v₁ € V such that z- ບ is in kU. Let n ≥ 1 and assume that we have found ₁,...,U₁, in V as stated in the lemma. Then z = v₁ + kv₂+...+k-vn + ku for some u EU. Since u = Un+1 + kuo for some Un+1 EV and uo € U, we see that Request Explain

10.1 Lemma Let X be a linear space over K. Consider subsets U and V of X, and k € K such that U CV + kU. Then for every z EU, there is a sequence (v₁) in V such that x − (v₁ + kv₂ + + k”−¹vn) € k"U, n=1,2,.... Proof: Let z EU. Since UCV + kU, there is some v₁ € V such that z- ບ is in kU. Let n ≥ 1 and assume that we have found ₁,...,U₁, in V as stated in the lemma. Then z = v₁ + kv₂+...+k-vn + ku for some u EU. Since u = Un+1 + kuo for some Un+1 EV and uo € U, we see that Request Explain

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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10.1 Lemma Let X be a linear space over K. Consider subsets U and V of X, and k € K such that UCV + kU. Then for every z EU, there is a sequence (v₁) in V such that x = (v₁ + kv₂ + +k²-¹vn) € k¹U, n=1,2,.... Proof: Let z EU. Since UCV+KU, there is some v₁ € V such that x- V1 is in kU. Let n ≥ 1 and assume that we have found ₁,...,Un in V as stated in the lemma. Then x = v₁ + kv₂ + ... + k²²-¹vn + ku for some u EU. Since u = Un+1 + kuo for some Un+1 EV and uo € U, we see that Thus we inductively obtain a sequence (vn) in V with the stated prop- erty. Request Explain
Transcribed Image Text:10.1 Lemma Let X be a linear space over K. Consider subsets U and V of X, and k € K such that UCV + kU. Then for every z EU, there is a sequence (v₁) in V such that x = (v₁ + kv₂ + +k²-¹vn) € k¹U, n=1,2,.... Proof: Let z EU. Since UCV+KU, there is some v₁ € V such that x- V1 is in kU. Let n ≥ 1 and assume that we have found ₁,...,Un in V as stated in the lemma. Then x = v₁ + kv₂ + ... + k²²-¹vn + ku for some u EU. Since u = Un+1 + kuo for some Un+1 EV and uo € U, we see that Thus we inductively obtain a sequence (vn) in V with the stated prop- erty. Request Explain
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