5. (*) Let V be a vector space over F and let U be a subspace of V. Define a relation ~on V by v~wv-w€U (a) Prove that is an equivalence relation on V. Write [v] for the equivalence class of containing v € V. (b) Show that [v] = {v+u: u € U}. For u, w € V we define [v] + [w] = [v+w]~ and for λ E F we define X[v] = [v]. (c) With this definition of addition and scalar multiplication, prove that {[v]: v € V}, the set of equivalence classes of~, is a vector space over F.
5. (*) Let V be a vector space over F and let U be a subspace of V. Define a relation ~on V by v~wv-w€U (a) Prove that is an equivalence relation on V. Write [v] for the equivalence class of containing v € V. (b) Show that [v] = {v+u: u € U}. For u, w € V we define [v] + [w] = [v+w]~ and for λ E F we define X[v] = [v]. (c) With this definition of addition and scalar multiplication, prove that {[v]: v € V}, the set of equivalence classes of~, is a vector space over F.
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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![5. (*) Let V be a vector space over F and let U be a subspace of V. Define a relation ~ on V by
v~wv-w€U
(a) Prove that is an equivalence relation on V.
Write [v] for the equivalence class of containing v € V.
(b) Show that [v] = {v+u: u € U}.
For u, w € V we define
[v] + [w] = [v+w]~
and for λ E F we define X[v] = [v].
(c) With this definition of addition and scalar multiplication, prove that
{[v]~: v € V},
the set of equivalence classes of ~, is a vector space over F.
(d) Take V = R³, U =
-{(+₁)=zex}.
: ER and define as above. Find a set of representatives for the
equivalence classes of~.
(e) With V and U as in part (d), (c) tells us the set of equivalence classes of ~ is a vector space over
R. What is the dimension of this vector space?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a4af2c9-a3ce-409e-892c-ae7106d06bb0%2Ff458f516-b6e1-495b-9577-1dcbdd70018d%2Ffoh8ixj_processed.png&w=3840&q=75)
Transcribed Image Text:5. (*) Let V be a vector space over F and let U be a subspace of V. Define a relation ~ on V by
v~wv-w€U
(a) Prove that is an equivalence relation on V.
Write [v] for the equivalence class of containing v € V.
(b) Show that [v] = {v+u: u € U}.
For u, w € V we define
[v] + [w] = [v+w]~
and for λ E F we define X[v] = [v].
(c) With this definition of addition and scalar multiplication, prove that
{[v]~: v € V},
the set of equivalence classes of ~, is a vector space over F.
(d) Take V = R³, U =
-{(+₁)=zex}.
: ER and define as above. Find a set of representatives for the
equivalence classes of~.
(e) With V and U as in part (d), (c) tells us the set of equivalence classes of ~ is a vector space over
R. What is the dimension of this vector space?
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