Definition: Let V be a vector space and U₁, U₂, U be subspaces of V. Then V is said to be a direct sum of subspaces U₁, U₂, ..., Uk, denoted by, V = U₁ U₂ зÐUk, if the following two conditions hold: (i) V=U₁+U₂ + ... ·+Uzi (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¡ € U¡, 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 = U₁ U₂ ... Un.
Definition: Let V be a vector space and U₁, U₂, U be subspaces of V. Then V is said to be a direct sum of subspaces U₁, U₂, ..., Uk, denoted by, V = U₁ U₂ зÐUk, if the following two conditions hold: (i) V=U₁+U₂ + ... ·+Uzi (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¡ € U¡, 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 = U₁ U₂ ... Un.
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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Transcribed Image Text: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 ÐU₂ з · Uk, if the following two conditions hold:
(i) 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.
The only way to write Oy as a sum of u₁ +…+ Uê, where each u¡ € U¡, 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 = U₁ U₂ ... Un.
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