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4. Let W be a subspace of a vector space V. We define a relation: V₁ V₂ if V₁ V₂ € W. ~ 4a. Show that this relation '~' is an equivalence relation on V. Namely, show: (i) Reflexive: v~ v; (ii) Symmetric: v₁ ~ V2 ⇒ V2 ~ V₁; and (iii) Transitive: V₁1~ V2, V2 V3 ⇒ V₁ ~ V3. 4b. Denote by v := {x € V | x ~ v} the equivalence class contain- ing v (surely, v Ev), and we call v a representative of the class v. Show that v = v+W:= {v + w|w€ W}. = or 4c. For v₂ € V, show: either v₁V₂00 V₁ = V₂. Show: there are equivalence classes Va (a € A) such that V is a disjoint union of them (for some index set A): V = LlaEA Vai show: V₁ = V₂ ⇒ V₁ ~ V2.

4. Let W be a subspace of a vector space V. We define a relation: V₁ V₂ if V₁ V₂ € W. ~ 4a. Show that this relation '~' is an equivalence relation on V. Namely, show: (i) Reflexive: v~ v; (ii) Symmetric: v₁ ~ V2 ⇒ V2 ~ V₁; and (iii) Transitive: V₁1~ V2, V2 V3 ⇒ V₁ ~ V3. 4b. Denote by v := {x € V | x ~ v} the equivalence class contain- ing v (surely, v Ev), and we call v a representative of the class v. Show that v = v+W:= {v + w|w€ W}. = or 4c. For v₂ € V, show: either v₁V₂00 V₁ = V₂. Show: there are equivalence classes Va (a € A) such that V is a disjoint union of them (for some index set A): V = LlaEA Vai show: V₁ = V₂ ⇒ V₁ ~ V2.

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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4. Let W be a subspace of a vector space V. We define a relation: V₁ ~ V₂ if V₁ V₂ € W. V1 4a. Show that this relation '~' is an equivalence relation on V. Namely, show: (i) Reflexive: v~ v; (ii) Symmetric: v₁ ~ V2 ⇒ V2 ~ V₁; and (iii) Transitive: V₁ V2, V₂ V3 ⇒ V₁ ~ V3. 4b. Denote by V := = {x EV|x~v} the equivalence class contain- ing v (surely, v Ev), and we call v a representative of the class v. Show that v = v+W := {v + w|w€ W}. 4c. For v₂ EV, show: either V₁ V₂ = Ø or V₁ = V₂. Show: there are equivalence classes Va (a E A) such that V is a disjoint union of them (for some index set A): V = LlaEA Vaj show: V₁ = V₂ ⇒ V₁ ~ V₂. (Remark. 4c hold for any equivalence relation on a set.) 4d. Show that the cardinality Va| = |VB| for any a, ß € A. Hint. Use 4b and show the bijectivity of the map: f: Va → VB (Va+W+Vg+w).
Transcribed Image Text:4. Let W be a subspace of a vector space V. We define a relation: V₁ ~ V₂ if V₁ V₂ € W. V1 4a. Show that this relation '~' is an equivalence relation on V. Namely, show: (i) Reflexive: v~ v; (ii) Symmetric: v₁ ~ V2 ⇒ V2 ~ V₁; and (iii) Transitive: V₁ V2, V₂ V3 ⇒ V₁ ~ V3. 4b. Denote by V := = {x EV|x~v} the equivalence class contain- ing v (surely, v Ev), and we call v a representative of the class v. Show that v = v+W := {v + w|w€ W}. 4c. For v₂ EV, show: either V₁ V₂ = Ø or V₁ = V₂. Show: there are equivalence classes Va (a E A) such that V is a disjoint union of them (for some index set A): V = LlaEA Vaj show: V₁ = V₂ ⇒ V₁ ~ V₂. (Remark. 4c hold for any equivalence relation on a set.) 4d. Show that the cardinality Va| = |VB| for any a, ß € A. Hint. Use 4b and show the bijectivity of the map: f: Va → VB (Va+W+Vg+w).
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