Let W₁, W2, ..., We are subspaces of a finite-dimensional vector space V such that V=W₁ +W₂ + ... + Wk. Let 3; denote a basis for W, and let 3 = and only if ß is a basis for V. UB. Prove that V = W₁ W₂2 ... Wk if O i=1
Let W₁, W2, ..., We are subspaces of a finite-dimensional vector space V such that V=W₁ +W₂ + ... + Wk. Let 3; denote a basis for W, and let 3 = and only if ß is a basis for V. UB. Prove that V = W₁ W₂2 ... Wk if O i=1
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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![**Problem Statement:**
12. Let \( W_1, W_2, \ldots, W_k \) be subspaces of a finite-dimensional vector space \( V \) such that
\[
V = W_1 + W_2 + \cdots + W_k.
\]
Let \( \beta_i \) denote a basis for \( W_i \) and let \( \beta = \bigcup_{i=1}^k \beta_i \). Prove that \( V = W_1 \oplus W_2 \oplus \cdots \oplus W_k \) if and only if \( \beta \) is a basis for \( V \).
**Explanation:**
This problem involves understanding direct sums and basis concepts in linear algebra. You are required to demonstrate the equivalence between \( V \) being the direct sum of subspaces \( W_1 \) through \( W_k \) and the union of individual bases \( \beta_i \) forming a basis for \( V \). This task essentially tests your ability to apply knowledge about vector spaces, subspaces, bases, and linear independence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F97a1bd76-bab9-4190-9395-a3bd033a29ce%2Fyic2ss_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
12. Let \( W_1, W_2, \ldots, W_k \) be subspaces of a finite-dimensional vector space \( V \) such that
\[
V = W_1 + W_2 + \cdots + W_k.
\]
Let \( \beta_i \) denote a basis for \( W_i \) and let \( \beta = \bigcup_{i=1}^k \beta_i \). Prove that \( V = W_1 \oplus W_2 \oplus \cdots \oplus W_k \) if and only if \( \beta \) is a basis for \( V \).
**Explanation:**
This problem involves understanding direct sums and basis concepts in linear algebra. You are required to demonstrate the equivalence between \( V \) being the direct sum of subspaces \( W_1 \) through \( W_k \) and the union of individual bases \( \beta_i \) forming a basis for \( V \). This task essentially tests your ability to apply knowledge about vector spaces, subspaces, bases, and linear independence.
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