Definition: A vector space is called the direct sun of W₁ and W2, denoted by V =W₁ W2, where W₁ and W2 are subspaces of Vand:(a) W₁ + W₂ = V.(b) W₁ W₂{0}. (Oy is the zero element of V)Prove that V =W₁ W₂ if and only if each element in V can be uniquely written as x₁ + x2where x₁ EW₁ and x2 E W₂.

Assume V = W1⊕W2 Let v be an element in V. Since V = W1 + W2 there exists x1 ∈ W1 and x2 ∈ W2…

Definition: A vector space is called the direct sun of W₁ and W2, denoted by V W₁ W2, where W₁ and W₂ are subspaces of Vand: (a) W₁ + W₂ (b) W₁N W₂ = : V. = {0}. (Oy is the zero element of V) Prove that V = W₁ W₂ if and only if each element in V can be uniquely written as x₁ + x2 where x₁ W₁ and x2 E W₂.

Definition: A vector space is called the direct sun of W₁ and W2, denoted by V W₁ W2, where W₁ and W₂ are subspaces of Vand: (a) W₁ + W₂ (b) W₁N W₂ = : V. = {0}. (Oy is the zero element of V) Prove that V = W₁ W₂ if and only if each element in V can be uniquely written as x₁ + x2 where x₁ W₁ and x2 E W₂.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.CR: Review Exercises

Problem 78CR: Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set...

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