Need help with part (c). Thank you :) 1. Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V, define Sy = {v+w:w€ W}, and letU = {S, : v E V}. Define addition in U so that for any x, y E VSx + Sy = Sx+y||and define scalar multiplication so that for any k E RkSxSkxIt can be shown that U is vector space (you do not need to prove this).(a) Explain why the zero vector in U is a subspace of V.(b) Prove, by induction, that for any k > 1 and any choice of c1, ..., Ck ER and x1, ..., X E V, if v = E C;X; thenSy = c;Sx,i=1(c) What is dim(U)? (Hint: Consider a basis for V which contains a basis for W, and use it to construct a basis for U.)(d) Let T: V →V' be a linear transformation. Let Wker(T), let U be as defined above, and for each v E V define=$(Sv) = T(v) (*)Since it is possible for Sy = Sw with v w, it is not immediately clear that o is well defined. Prove that (*) does indeed define a function$: U → V', by showing that for any x, y E V satisfying Sx = Sy we have (Sx) = ¢(Sy).(e) Show that ø is linear.(f) For what values of dim(V') is ø injective?(g) For what values of dim(V') is ø surjective?||W||

Name: Need help with part (c). Thank you :) 1. Let V be an n -dimensional ve
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Description: Need help with part (c). Thank you :) 1. Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V, define Sy = {v+w:w€ W}, and letU = {S, : v E V}. Define addition in U so that for any x, y E VSx + Sy = Sx+y

(c) Given W⊆V with dimW=m≤n=dimV. For each v∈V, define Sv=v+w:w∈W and let U=Sv:v∈V. Let the basis…

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Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V, define Sy = {v+w:w E W}, and let = {Sv : v e V}. Define addition in U so that for any x, y E V Sx + Sy = Sx+y d define scalar multiplication so that for any k ER kSx = Skx can be shown that U is vector space (you do not need to prove this). Explain why the zero vector in U is a subspace of V. O Prove, by induction, that for any k >1 and any choice of c1,..., Ck ER and x1,..., X E V, if v = ECx; then Sy = c:Sx, i=1 What is dim(U)? (Hint: Consider a basis for V which contains a basis for W, and use it to construct a basis for U.)

Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V, define Sy = {v+w:w E W}, and let = {Sv : v e V}. Define addition in U so that for any x, y E V Sx + Sy = Sx+y d define scalar multiplication so that for any k ER kSx = Skx can be shown that U is vector space (you do not need to prove this). Explain why the zero vector in U is a subspace of V. O Prove, by induction, that for any k >1 and any choice of c1,..., Ck ER and x1,..., X E V, if v = ECx; then Sy = c:Sx, i=1 What is dim(U)? (Hint: Consider a basis for V which contains a basis for W, and use it to construct a basis for U.)

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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Need help with part (c). Thank you :)

1. Let V be an n -dimensional vector space and let W C V be an m -dimensional subspace. For each v E V, define Sy = {v+w:w€ W}, and let
U = {S, : v E V}. Define addition in U so that for any x, y E V
Sx + Sy = Sx+y
||
and define scalar multiplication so that for any k E R
kSx
Skx
It can be shown that U is vector space (you do not need to prove this).
(a) Explain why the zero vector in U is a subspace of V.
(b) Prove, by induction, that for any k > 1 and any choice of c1, ..., Ck ER and x1, ..., X E V, if v = E C;X; then
Sy = c;Sx,
i=1
(c) What is dim(U)? (Hint: Consider a basis for V which contains a basis for W, and use it to construct a basis for U.)
(d) Let T: V →V' be a linear transformation. Let W
ker(T), let U be as defined above, and for each v E V define
=
$(Sv) = T(v) (*)
Since it is possible for Sy = Sw with v w, it is not immediately clear that o is well defined. Prove that (*) does indeed define a function
$: U → V', by showing that for any x, y E V satisfying Sx = Sy we have (Sx) = ¢(Sy).
(e) Show that ø is linear.
(f) For what values of dim(V') is ø injective?
(g) For what values of dim(V') is ø surjective?
||
W
||

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