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 v eV has a unique expression v = v₁ + ... + Un with v₂ € Vi (c) V = V₁ +. + V₁ and the map T: V₁ V₂ → V₁ + ··· + Vn π(V₁,..., Un) = V₁ + ... + Vn is an isomorphism Can we replace the sum condition in (a) by just asking that Vin V₂ = {0v} 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 v eV has a unique expression v = v₁ + ... + Un with v₂ € Vi (c) V = V₁ +. + V₁ and the map T: V₁ V₂ → V₁ + ··· + Vn π(V₁,..., Un) = V₁ + ... + Vn is an isomorphism Can we replace the sum condition in (a) by just asking that Vin V₂ = {0v} for all i + j?
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
![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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2Fb43cf025-8d61-44d8-8988-87a31806174d%2Fh8kbx5m_processed.png&w=3840&q=75)
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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