### DefinitionLet \( V \) 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:1. \( V = U_1 + U_2 + \cdots + U_k \);2. For every \( \mathbf{v} \in V \), there exist *unique* vectors \( \mathbf{u}_i \in U_i \), \( 1 \leq i \leq k \), such that \( \mathbf{v} = \mathbf{u}_1 + \cdots + \mathbf{u}_k \).### Problems9. (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: 1. \( V = U_1 + \cdots + U_k \). *Proved.* 2. 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 \( \mathbf{u}_j \) equal to zero. *Proved.*(b) Suppose that \( V \) is a finite dimensional vector space, with \( \dim(V) = n \). Prove that there exist 1-dimensional subspaces \( U_1, \ldots, U_n \) of \( V \) such that \[ V = U_1 \oplus U_2 \oplus \cdots \oplus U_n. \] *Proved.*(c) Give an example to show that condition (ii) (in the definition) cannot be replaced with \( U_i \cap U_j = \{ \mathbf{0}_V \} \), for

I have constructed subspaces of R^2 and explained the counter example.

Definition: Let V be a vector space and U₁, U₂, ,Uk be subspaces of V. Then V is said to be a direct sum of subspaces U₁, U2,...,Uk, denoted by, V = U₁ ĐU₂ ☺…ÐUk, if the following two conditions hold: V=U₁+U₂+... + Uk; (ii) For every v € V, there exist unique vectors uį € Uį, 1 ≤ i ≤ k, such that V = U₁ + ··· + Uk. 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. Proved. (ii) The only way to write Oy as a sum of u₁ + ··· + uk, where each u; € Uj, is by taking all u,'s equal to Proved. zero. (b) Suppose that V is a finite dimensional vector space, with dim(V) = n. Prove that there exist 1-dimensional subspaces U₁,..., Un of V such that V = U₁₁ U₁₂ Un. Proved. Give an example to show that condition (ii) (in definition) can not be replaced with U₂ nu; = {0v}, for i ‡ j.

Definition: Let V be a vector space and U₁, U₂, ,Uk be subspaces of V. Then V is said to be a direct sum of subspaces U₁, U2,...,Uk, denoted by, V = U₁ ĐU₂ ☺…ÐUk, if the following two conditions hold: V=U₁+U₂+... + Uk; (ii) For every v € V, there exist unique vectors uį € Uį, 1 ≤ i ≤ k, such that V = U₁ + ··· + Uk. 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. Proved. (ii) The only way to write Oy as a sum of u₁ + ··· + uk, where each u; € Uj, is by taking all u,'s equal to Proved. zero. (b) Suppose that V is a finite dimensional vector space, with dim(V) = n. Prove that there exist 1-dimensional subspaces U₁,..., Un of V such that V = U₁₁ U₁₂ Un. Proved. Give an example to show that condition (ii) (in definition) can not be replaced with U₂ nu; = {0v}, for i ‡ j.

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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Question

### 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:
1. \( V = U_1 + U_2 + \cdots + U_k \);
2. For every \( \mathbf{v} \in V \), there exist *unique* vectors \( \mathbf{u}_i \in U_i \), \( 1 \leq i \leq k \), such that \( \mathbf{v} = \mathbf{u}_1 + \cdots + \mathbf{u}_k \).

### Problems
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:
1. \( V = U_1 + \cdots + U_k \). *Proved.*
2. 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 \( \mathbf{u}_j \) equal to zero. *Proved.*

(b) Suppose that \( V \) is a finite dimensional vector space, with \( \dim(V) = n \). Prove that there exist 1-dimensional subspaces \( U_1, \ldots, U_n \) of \( V \) such that
\[
V = U_1 \oplus U_2 \oplus \cdots \oplus U_n.
\]
*Proved.*

(c) Give an example to show that condition (ii) (in the definition) cannot be replaced with \( U_i \cap U_j = \{ \mathbf{0}_V \} \), for

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