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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.

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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**Definition:** A vector space \( \mathbb{V} \) is called the **direct sum** of \( \mathbb{W}_1 \) and \( \mathbb{W}_2 \), denoted by \( \mathbb{V} = \mathbb{W}_1 \oplus \mathbb{W}_2 \), where \( \mathbb{W}_1 \) and \( \mathbb{W}_2 \) are subspaces of \( \mathbb{V} \) and: (a) \( \mathbb{W}_1 + \mathbb{W}_2 = \mathbb{V} \). (b) \( \mathbb{W}_1 \cap \mathbb{W}_2 = \{0_\nu\} \). \hspace{10pt} \( (0_\nu \text{ is the zero element of } \mathbb{V}) \) Prove that \( \mathbb{V} = \mathbb{W}_1 \oplus \mathbb{W}_2 \) *if and only if* each element in \( \mathbb{V} \) can be *uniquely* written as \( \mathbf{x}_1 + \mathbf{x}_2 \), where \( \mathbf{x}_1 \in \mathbb{W}_1 \) and \( \mathbf{x}_2 \in \mathbb{W}_2 \).
Transcribed Image Text:**Definition:** A vector space \( \mathbb{V} \) is called the **direct sum** of \( \mathbb{W}_1 \) and \( \mathbb{W}_2 \), denoted by \( \mathbb{V} = \mathbb{W}_1 \oplus \mathbb{W}_2 \), where \( \mathbb{W}_1 \) and \( \mathbb{W}_2 \) are subspaces of \( \mathbb{V} \) and: (a) \( \mathbb{W}_1 + \mathbb{W}_2 = \mathbb{V} \). (b) \( \mathbb{W}_1 \cap \mathbb{W}_2 = \{0_\nu\} \). \hspace{10pt} \( (0_\nu \text{ is the zero element of } \mathbb{V}) \) Prove that \( \mathbb{V} = \mathbb{W}_1 \oplus \mathbb{W}_2 \) *if and only if* each element in \( \mathbb{V} \) can be *uniquely* written as \( \mathbf{x}_1 + \mathbf{x}_2 \), where \( \mathbf{x}_1 \in \mathbb{W}_1 \) and \( \mathbf{x}_2 \in \mathbb{W}_2 \).
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