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Suppose that U₁,...,Uk are subspaces of V. Prove that V = U₁ ☺ · · · ÐUk if and only if the following two conditions hold: (i) V = U₁ + ... + Uk. Proved.

Suppose that U₁,...,Uk are subspaces of V. Prove that V = U₁ ☺ · · · ÐUk if and only if the following two conditions hold: (i) V = U₁ + ... + Uk. Proved.

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: 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 u¿ € Uį, 1 ≤ i ≤k, such that v = U₁ + ··· + Uk. (a) Suppose that U₁,...,Uk are subspaces of V. Prove that V = conditions hold: (i) V = U₁ + ... + Uk. Proved. (ii) The only way to write Oy as a sum of u₁ + zero. Proved. U₁ ... Uk if and only if the following two + uk, where each u; € U₁, is by taking all u,'s equal to (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. Proved. (c) Give an example to show that condition (ii) (in definition) can not be replaced with UįU; = {0v}, for i ‡ j. Proved Let V₁, V2 V be subspaces of V and let V3, V4 ≤ V₂ be subspaces of V2. Prove or disprove: If V₁ V₂ = V and V3 V4 = V2, then V₁³ © V₁ = V.
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 u¿ € Uį, 1 ≤ i ≤k, such that v = U₁ + ··· + Uk. (a) Suppose that U₁,...,Uk are subspaces of V. Prove that V = conditions hold: (i) V = U₁ + ... + Uk. Proved. (ii) The only way to write Oy as a sum of u₁ + zero. Proved. U₁ ... Uk if and only if the following two + uk, where each u; € U₁, is by taking all u,'s equal to (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. Proved. (c) Give an example to show that condition (ii) (in definition) can not be replaced with UįU; = {0v}, for i ‡ j. Proved Let V₁, V2 V be subspaces of V and let V3, V4 ≤ V₂ be subspaces of V2. Prove or disprove: If V₁ V₂ = V and V3 V4 = V2, then V₁³ © V₁ = V.
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