HW8MATH110S17 - Math 110 Spring 2017 with Professor...

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Math 110 Spring 2017, with Professor Stankova: Homework 8 SolutionsSection 2.4Exercise 2.4.1:(a) False; It should be ([T]βα)-1= [T-1]αβ.(b) True; This follows easily from the definition.(c) False;LAonly maps fromFdim(V)toFdim(W). (d) False, different dimensions. (e) True by dimension argument.(f) False, this could happen whenAandBare not square matrices. (g) True by definition. (h) True bySection 2.4 Theorem 2.18 of the textbook. (i) True by definition.Exercise 2.4.2:(b), (d)Tcannot be invertible as the domain and image ofThave different dimensions.(f)Tis invertible as it is one-to-one (zero kernel) and onto.Exercise 2.4.3:Two vector spaces are isomorphic if and only if they have the same dimensions. By checking dimensions,we know that: (a) no; (b) yes; (c) yes; (d) no asVonly has dimension three.
Exercise 2.4.4:LetAandBben×ninvertible matrices. Prove thatABis invertible and(AB)-1=B-1A-1.
Exercise 2.4.5:LetAbe invertible. Prove thatAtis invertible and(At)-1= (A-1)t.
Exercise 2.4.6:Prove that ifAis invertible andAB=O, thenB=O.1
(b) SupposeAis invertible, then by Exercise 6,B=O. This is a contradiction. ThereforeAcannot beinvertible.Exercise 2.4.9:LetAandBben×nmatrices such thatABis invertible.Prove thatAandBareinvertible.Give an example to show that arbitrary matricesAandBneed not be invertible ifABisinvertible.Solution:We first prove thatAandBare invertible under the original hypotheses:Proof.We’ll prove this by converting it into a claim about linear transformations: By Corollary 2 on page102 of the text,AandBare invertible matrices if and only if the left-multiplication transformations LAand LBare invertible linear transformations. Furthermore, by the same Corollary, our assumption thatABis invertible tells us that the composition LALB= LABis an invertible linear transformation.Note first that, becauseAandBaren×nmatrices, each of LAand LBmapsRnto itself. So to showthat each of LAand LBis invertible, we may showeitherthat it is one-to-oneorthat it is onto—because,

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