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.

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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 = {β12,...,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.

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