Need help with part (g). 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 (g). Thank you :) 1. Let V be an n -dimensional ve
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Description: Need help with part (g). 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

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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 S, = {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 ER 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 € V, if v = E1 Cix; then 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 $(S,) = T(v) (*) Since it is possible for Sy = Sw with v + w, it is not immediately clear that ø 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 o(Sx) = ¢(Sy). (e) Show that o is linear. (f) For what values of dim(V') is o injective? (g) For what values of dim(V') is ø surjective?

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 S, = {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 ER 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 € V, if v = E1 Cix; then 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 $(S,) = T(v) () Since it is possible for Sy = Sw with v + w, it is not immediately clear that ø 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 o(Sx) = ¢(Sy). (e) Show that o is linear. (f) For what values of dim(V') is o injective? (g) For what values of dim(V') is ø surjective?

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

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