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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?

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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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 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 ||
Transcribed Image Text: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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