This is from advanced linear algebra. ( Reference is Linear Algebra and Matrix Theory By Nering) Please provide a complete proof. This topic is on the direct sum and uniqueness of complementary subspaces Let A = {β1,β2,...,βk} be a linearly independent set of vectors in an n-dimensional vector space V over F where n > k. (i) Let W = span(A). Show that there exist subspaces W ′ ⊆ V and W ′′ ⊆ V such that V =W⊕W′ V =W⊕W′′. but W' is not equal to W'' (ii) Show that every vector γ ∈ V can be expressed as γ = γ1 + γ2. where γ1 ∈ W and γ2 not an element of W . Show that γ1 is unique but γ2 isn’t.
This is from advanced linear algebra. ( Reference is Linear Algebra and Matrix Theory By Nering) Please provide a complete proof. This topic is on the direct sum and uniqueness of complementary subspaces Let A = {β1,β2,...,βk} be a linearly independent set of vectors in an n-dimensional vector space V over F where n > k. (i) Let W = span(A). Show that there exist subspaces W ′ ⊆ V and W ′′ ⊆ V such that V =W⊕W′ V =W⊕W′′. but W' is not equal to W'' (ii) Show that every vector γ ∈ V can be expressed as γ = γ1 + γ2. where γ1 ∈ W and γ2 not an element of W . Show that γ1 is unique but γ2 isn’t.
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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This is from advanced
Please provide a complete proof.
This topic is on the direct sum and uniqueness of complementary subspaces
Let A = {β1,β2,...,βk} be a linearly independent set of
(i) Let W = span(A). Show that there exist subspaces W ′ ⊆ V and W ′′ ⊆ V such that V =W⊕W′
V =W⊕W′′. but W' is not equal to W''
(ii) Show that every vector γ ∈ V can be expressed as γ = γ1 + γ2. where γ1 ∈ W and γ2 not an element of W . Show that γ1 is unique but γ2 isn’t.
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