Question

Q2 15 Points Let A € Mnxn(R). Define trace(A) = %=1 4,1 (1. e. the sum of the diagonal entries) and tr: Mnxn(R) +R, A H trace(A).

Answer 1

(with entry is zero I and all the crother tr: MAR) IR At tr (A) This isa linear map xR) is a repor space dimension n. Define Eij & Minx (R) such that the of Eij enty daj otherwix. forms a basis for 3M man lce, jew If it, then tr (E13)=0 R, I LFI 2.3 So Eij Eij & kent) of itj do fij) = 1 to CE,,- Eii) = f(f) - h Eli) = 11120 Then

2,3.. claimio-{{43 lit; so (fi, fii) e ker(t) forms a bons for Kertr). to isa lineer map which nonzero and dim By rank nullity theorem dim ker(to) rank (fr) & Tim MnR) Lim Ker (tr) range (fr) = 1 Now 1B12 hr-h non din diagonal elements in Any Need to show generates kerto). ht A E kw (fr) then to (A) = 12 w A = aij fijs. I aij fij. it j زر I aij tij ir ant Žaiti itj 3,06 € + 5 + I ari hii (2 & ai (Eri -E 1 17 3. aj tij itj t LEL So generates kw (hr). Her Bisabomina boms of Kerffor) as 1B12 dim ker(t).

24 A E MAR h then A=&aij fij n Ecij Eij of itj ai fii & Cup Ell-Lail hi + (2 ci) Ei -P 1-2 12 ta e n W ht I aii Eu i tj 12 position baij bij + I alifii a 2012an er which has lat (.th and all other entry to B) & aij to (Ei;) s ci folfinga itj - Edito (E11) i zero. 1-2 1-2 2 + I all Į aii it A B ta e where B et ker(tr) 2.5 in port 4 i2