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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 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
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, ‚U be subspaces of V. Then V is said to be a direct sum of subspaces U₁, U2, ..., Uk, denoted by, V = U₁ ÐU₂ з··ÐUk, 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. 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, i's 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 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 U₂ nu; = = {Ov}, for i ‡ j.
Transcribed Image Text: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 ÐU₂ з··ÐUk, 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. 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, i's 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 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 U₂ nu; = = {Ov}, for i ‡ j.
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