The previous exercise: Consider a matrix A Mmxn(F), and its induced linear map f

Question

The previous exercise: Consider a matrix A Mmxn(F), and its induced linear map fa: F^n -> F^n. (I) Assume that AB = Im for some matrix B Mnxm(F). Show that fa is onto. (ii) Assume that BA = In for some matrix BMnxm(F). Show that fa is one-to-one.

Please solve Execrise 8! Thank you.

Exercise 8. Recall that a matrix A E Mn(F) is said to be invertible if AB- BA -In for some matrix BEMn(F (i) Show that A is invertible if and only if the linear map fA : Fn Fn is an isomorphism (ii) Using the previous exercise, show that the following are equivalent: (1) AB = In for some matrix B E Mn(F) (2) BA = In for some matrix B E Mn(F) (3) AB = BA = In for some matrix Mn (F)

Explanation / Answer

i) let f is invertible the f must be one one and onto, also f is linear so f is an isomorphism

now conversly let f is an isomorphism then f must be one one onto linear mapping, so that f is invertible since f is one one onto.

ii)let AB=In then f is onto from previous exercise, now since f:V->V is onto it must be one one as well so that f is one one and onto, hence f is invertible,

so that BA=In

NOW conversly let BA=In, then f must be one one so onto as well, so that AB=In,

from 1 and 2 now its clear AB=In=BA

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