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2) Let V be an F vector space and let V₁,..., Vn be a subspaces of V. The point of this problem is to show the "internal" and "external" definitions of direct sum are equivalent. That is: prove the following are equivalent (a) V = V₁ ++ Vn and VinΣjti Vj = {Ov} (b) Every vector ve V has a unique expression v = V₁ + + Un with v₂ € Vi (c) V = V₁ ++ Vn and the map T: V₁ V₂ → V₁ + ... + Vn T(V₁,..., Un) = V₁ + + Vn is an isomorphism Can we replace the sum condition in (a) by just asking that Vin V₁ = {Ov} for all i j?

2) Let V be an F vector space and let V₁,..., Vn be a subspaces of V. The point of this problem is to show the "internal" and "external" definitions of direct sum are equivalent. That is: prove the following are equivalent (a) V = V₁ ++ Vn and VinΣjti Vj = {Ov} (b) Every vector ve V has a unique expression v = V₁ + + Un with v₂ € Vi (c) V = V₁ ++ Vn and the map T: V₁ V₂ → V₁ + ... + Vn T(V₁,..., Un) = V₁ + + Vn is an isomorphism Can we replace the sum condition in (a) by just asking that Vin V₁ = {Ov} for all i j?

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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please answer a,b,c

2) Let V be an F vector space and let V₁,..., Vn be a subspaces of V. The point of this problem is to show the "internal" and "external" definitions of direct sum are equivalent. That is: prove the following are equivalent (a) V = V₁ ++ Vn and Vin Σjti Vj = {Ov} (b) Every vector v € V has a unique expression v = v₁ + ... + Un with v¿ € Vi (c) V = V₁ + + Vn and the map T: V₁0 Vn → V₁ + ... + Vn + Vn T(V₁,..., Un) = V₁ + · is an isomorphism Can we replace the sum condition in (a) by just asking that Vin V₁ = {0v} for all i j?
Transcribed Image Text:2) Let V be an F vector space and let V₁,..., Vn be a subspaces of V. The point of this problem is to show the "internal" and "external" definitions of direct sum are equivalent. That is: prove the following are equivalent (a) V = V₁ ++ Vn and Vin Σjti Vj = {Ov} (b) Every vector v € V has a unique expression v = v₁ + ... + Un with v¿ € Vi (c) V = V₁ + + Vn and the map T: V₁0 Vn → V₁ + ... + Vn + Vn T(V₁,..., Un) = V₁ + · is an isomorphism Can we replace the sum condition in (a) by just asking that Vin V₁ = {0v} for all i j?
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