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