ST259 Lecture : 2.4 - Invertibility and Isomorphisms
Document Summary
We assumethat vand w are finitedimensional vectorspaces over f. Defilet t e l v w a function u w v issaidtobe an inverseof t if tu iw and ut iv. If t has an inversethen t is said tobeinvertible if t is invertible thentheinverseof t is uniqueandis denotedbyt t thefollowingfactsholdforinvertiblefunctionstand u. Let t v w be a lineartransformation where vand w are finite dimensional spacesofequal dimension then tis invertibleiff rankg dim v. Thaket ti v w be linearandinvertible then t l w v is linear. Lemmalet t be an invertiblelinear transformationfromv tow then v is finite dimensionaliff wis finite dimensional. Thm_let vand w befinite dimensionalvectorspaces withordered basesbandv respectively let. 1el v w then t is invertibleiff itis isinvertible futhermore it ii. Corollarvet v be a finite dimensionalvectorspacewith an ordered basis13 and let t v v be linearthen t isinvertible iff its isinvertible furthermore it is itis. Corollarketabe an mxn matrix then a isinvertibleiff lais invertible furthermore la.
