9. (a) Suppose that U₁,. ,Uk are subspaces of V. Prove that V = U₁ Uk if and only if the following two conditions hold: (i) V=U₁+...+ Uk. (ii) The only way to write Oy as a sum of u₁ + zero. + uk, where each u, EU,, is by taking all u,'s equal to
9. (a) Suppose that U₁,. ,Uk are subspaces of V. Prove that V = U₁ Uk if and only if the following two conditions hold: (i) V=U₁+...+ Uk. (ii) The only way to write Oy as a sum of u₁ + zero. + uk, where each u, EU,, is by taking all u,'s equal to
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:** Let \( V \) be a vector space and \( U_1, U_2, \ldots, U_k \) be subspaces of \( V \). Then \( V \) is said to be a **direct sum** of subspaces \( U_1, U_2, \ldots, U_k \), denoted by \( V = U_1 \oplus U_2 \oplus \cdots \oplus U_k \), if the following two conditions hold:
(i) \( V = U_1 + U_2 + \cdots + U_k \);
(ii) For every \( \mathbf{v} \in V \), there exist **unique** vectors \( \mathbf{u}_i \in U_i \), \( 1 \leq i \leq k \), such that
\[
\mathbf{v} = \mathbf{u}_1 + \cdots + \mathbf{u}_k.
\]
---
9. (a) Suppose that \( U_1, \ldots, U_k \) are subspaces of \( V \). Prove that \( V = U_1 \oplus \cdots \oplus U_k \) if and only if the following two conditions hold:
(i) \( V = U_1 + \cdots + U_k \).
(ii) The only way to write \( \mathbf{0}_V \) as a sum of \( \mathbf{u}_1 + \cdots + \mathbf{u}_k \), where each \( \mathbf{u}_j \in U_j \), is by taking all \( \mathbf{u}_j \)'s equal to zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffce2c460-26f0-4070-86e1-b1876f7380a8%2F0a7c2042-271a-49b4-ad91-848c30ca4a80%2F95vt2m4_processed.png&w=3840&q=75)
Transcribed Image Text:**Definition:** Let \( V \) be a vector space and \( U_1, U_2, \ldots, U_k \) be subspaces of \( V \). Then \( V \) is said to be a **direct sum** of subspaces \( U_1, U_2, \ldots, U_k \), denoted by \( V = U_1 \oplus U_2 \oplus \cdots \oplus U_k \), if the following two conditions hold:
(i) \( V = U_1 + U_2 + \cdots + U_k \);
(ii) For every \( \mathbf{v} \in V \), there exist **unique** vectors \( \mathbf{u}_i \in U_i \), \( 1 \leq i \leq k \), such that
\[
\mathbf{v} = \mathbf{u}_1 + \cdots + \mathbf{u}_k.
\]
---
9. (a) Suppose that \( U_1, \ldots, U_k \) are subspaces of \( V \). Prove that \( V = U_1 \oplus \cdots \oplus U_k \) if and only if the following two conditions hold:
(i) \( V = U_1 + \cdots + U_k \).
(ii) The only way to write \( \mathbf{0}_V \) as a sum of \( \mathbf{u}_1 + \cdots + \mathbf{u}_k \), where each \( \mathbf{u}_j \in U_j \), is by taking all \( \mathbf{u}_j \)'s equal to zero.
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