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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 38EQ
Related questions
Question
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning