1. Let V be an n -dimensional vector space and let W CV be an m -dimensional subspace. For each v e V, define S, = U = {Sv : v € V}. Define addition in U so that for any x, y E V {v +w: w E W}, and let 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 E R and x1, ..., X E V, if v = E C;X; then Sy = 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 o(Sx) = (e) Show that o is linear. (f) For what values of dim(V') is ø injective? (g) For what values of dim(V') is o surjective? $(Sy).
1. Let V be an n -dimensional vector space and let W CV be an m -dimensional subspace. For each v e V, define S, = U = {Sv : v € V}. Define addition in U so that for any x, y E V {v +w: w E W}, and let 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 E R and x1, ..., X E V, if v = E C;X; then Sy = 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 o(Sx) = (e) Show that o is linear. (f) For what values of dim(V') is ø injective? (g) For what values of dim(V') is o surjective? $(Sy).
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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