Note Nov 6, 2021

SOLUTIONS TO CARNEN QUIZ 3 #| SUPPOSE THAT Si= {(7,, ; --,'tim} AND Sz-:{v,,-...,v,,} Age SUBSETS OF A vecTor space | GIVEN THAT V= Sp ( Sy) AMD THAT S, 18 LINEARLY IMOSPEVDEMT, C HOOSE THE CORRECT STATEMENT RELOW. @ dim (V)>K M) — @) ARE FALSE, AUD WE ALl STAEME pPTS USIA G THE SAME CAN EXPLAIN THAT BY COUNTEREXANPLE LeT V= R, S2= {é'ié% E""]} cV AD Q= {);'J) f?]}g V. THew, m=2, K=3, ouom (V)=2. \/:: §P (Sz) | RECAUZE FOR AMVY ALSO, — _ Y _ _ ( VECTOR X (A V) X = [x'l&];.yle)_'_xz@z,—]-@.[l].

AND Sy IS £/NEALLY [NIDEPENDENT, ST — )(I:O > XZ=O N OUL EXAMPLE CLAIH m=2, k=3, dm(V)=2. '@ dm (V) > K 92 >2 FALSE @ k<M 329 FALSE @ m=K 9 =% FALSE B @ S S 2 [T s b 5, BUT T (s MOT [N Sz. | | THeneFore, FALSE. ® dm(V)£K — coprpecT \\ ANY SFANNIKE SET CONTAINS A BASIS For V ThHeeeFoee, gy (v €K !

#2 led L AND V BE Two WECTORS [N THeE vecron seace V) aup 18T W BE A aygspace OF I F THe wecTe TW+V 1S W W) THEX BOTH VECTORS oL AND V aze w W, T ® COUNTERE XAMPLE < gp {—'}' FHEL g+ U= [4] 48, s w0 W. gUT ME(THER A NVO2 v s/ #3 &7 [ BE A VETOE appce awp LET THE ST BV, 7,%] BE A BASIS TOR V eT WBeA QURSPACE OF V. CHOOSE A CORRECT STATEMENT. pace! @ dumo (W)=3 y T cace! @) THERE EXSTS 2 pser oF B, cae 7B, S THA B! 1S A BASIS FOP W. o UNTERER AP s SW H{OLi) THAT STATEHEMTS ' LeT V/@ B< {e_, 82,85},\/:"0 l,(]rgp{' f?,]} THER dm(w) | anD ANY BASIS OF () cWTAINS oy ove veeror. MO VecTor K B seans W.