Let U,V ,W be finite-dimensional F-vector spaces and let T:U → V, S:V→WE linear transformations. Prove that nullity(S • T)- nullity(T ) = dim(im(T )N ker(S) following steps (a)–(c) below. a) LetU:=ker(S-T)andT:=T|u.Prove:im(T').=T(U')=im(T)N ker(S ). b) Prove:ker(T)=ker(T). c) Deduce:nullity(S•T)-nullity(T)=dim(im(T)Nker(S)).

Let U,V ,W be finite-dimensional F-vector spaces and let T:U → V, S:V→WE linear transformations. Prove that nullity(S • T)- nullity(T ) = dim(im(T )N ker(S) following steps (a)–(c) below. a) LetU:=ker(S-T)andT:=T|u.Prove:im(T').=T(U')=im(T)N ker(S ). b) Prove:ker(T)=ker(T). c) Deduce:nullity(S•T)-nullity(T)=dim(im(T)Nker(S)).

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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