Notes on the proof of direct sum for linear subspace

Transcription

1 Notes o the proof of drect sum for lear subspace Da u, Qa Guo, Huzhou Xag, B uo, Zhoghua Ta, Jgbo Xa* College of scece, Huazhog Agrcultural Uversty, Wuha, Hube, Cha * Correspodece should be addressed to jgbo_xa@yahooc Keywords: ear space, Drect sum, Proof Abstract ear space s a abstract cocept algebra, ad results related to drect sum are complcated oes amog the research of lear subspace I ths paper, two models for drecto sum proof are proposed, ad a typcal proposto s troduced, ad proved by usg proposed model, so as to gve a uversally applcable proof method Itroducto Hgher algebra s a fudametal ad mportat course for polytechc studets [], whch helps to cultvate studets' ablty of abstract thg ad problem solvg [] Though theorems related to lear space are comparatvely dffcult to grasp, research of lear space s worthwhle sce t s a ma part of ths course ad ts applcato plays a mportat role may aspects, cludg desg of urba grad street [3], detfcato of Volterra erels of olearly dyamcal systems [4], ad lear space campus [5] ear space s a mportat ad abstract cocept of hgher algebra, ad t possesses specal property For example, lear space has ts ow subspaces, deftos ad operatos le sum ad tersecto ad t ca be geerated by other spaces mutually, ad these provg detals sum ad tersecto of spaces are sophstcated that always mae the proof harder Fortuately, despte the abstracto of the cocepts ad the dffcultes of the proof, there are regulatos ca be used to prove the sum especally the drect sum May researchers utlzed sorts of cocrete methods (eg, drect sum ad pecular matrx, drect sum ad lear trasformato, decomposto theorem of drect sum, drect sum ad Cayley Hamlto theorem to solve problems of drect sum of lear space, but a few focused a basc model whch has great sgfcace Therefore, the purpose of ths paper s to study the basc models of drect sum for lear space By usg these models, researchers ca drectly orgaze proper argumets to prove the exstece of drect sum certa crcumstace I ths paper, we propose two basc models ad gve summares to the commo questos I model I, sce the frst three equvalet propostos have bee show (see [6], atteto are pad to put forward ew solutos to prove the fourth equvalet proposto [7, 8] I model II, ew methods ad typcal example are combed to prove the thrd combato [9] The rest of the paper s orgazed as follows: Secto gves the prelmary dscusso of the cocept of drect sum for lear subspace Secto 3 proposes ew solutos to prove the drect sum Secto 4 presets the aalyss ad proof of typcal example Secto 5 provdes the cocluso Tradtoal proof method of drect sum, 4 equvalece (Model I Defto:

2 et W V V V, where V are subspaces of V If decomposto of a vector α, α α α, α V (,, s uque, the the sum s called the drect sum, deoted as W V [6] Proof of the equvalet proposto of W V forms model I Theorem (Model I If W V V, the the followg four propostos [7, 8, 0] are equvalet to W V Decomposto le α α α, α V (, of every vector α lear space W s uque α α 0, α V (, ca oly be set up whe α 0, (, ; 3 V V ; 4 Vector wth uque decomposto ca be foud W Proof: Sce the frst three propostos have bee proved to be equvalet see [6], oly the equvalece of the 4-th proposto s eeded Two proofs are proposed to show the equvalece of the frst ad the fourth proposto Proof method I ( Codto codto 4: Suppose the decomposto s ot uque Sce decomposto of every vector lear space V V s uque, 0 ca also be dvded as uquely, where 0 V, 0 V Therefore, vector wth uque decomposto ca be foud V V ( Codto 4 : et V ( α, α,,, α K V (,,, β β βk, V V (,, K, the for ay α ( α 3,, α K, β,, βk ( V V, we have α α l α l β l β However, sce V V { 0}, 0 {,, } l ( 3 To smplfy the process, let's assume 0, the α ( lα lα lα ( l β l β l β (,,, where ( l α lα lα V ad l β l β l β V Furthermore, decomposto of α dffers whe dffers, ( aother word, decomposto of α ( V V s ot uque Therefore, codto 4 codto 3 as well as codto 4 codto are proved Thus the theorem follows Proof method II ( Codto codto 4: As the defto clams, decomposto of every vector V V s uque, therefore, vector wth uque decomposto ca be foud V V ( Codto 4 codto : Suppose the decomposto s ot uque et V ( α, α, α, K V ( β, β,,, β K the α l α lα l, so α l β l β l β maps to α ( V V ( α, α, β,, β Sce the decomposto of vector 0 s ot uque, α ( lα lα l α ( l β l β l β s ot uque, where, satsfy 0 0 0, ( 0 or 0 ad V, V Therefore, codto 4 codto s proved Thus the theorem s prove Note: Several otes are obtaed from the above dscusso Drect sum: Decomposto of every vector sum space s uque Vector wth uque decomposto ca be foud sum space Decomposto of 0 s uque Decomposto of 0 sum space s uque V V {0}

3 3 New deas for proof of drect sum (Model II The ew dea of drect sum proof V ( V are subspaces of V s gve ths secto, deoted as model II I order to clarfy that what codtos are eeded to prove V V f V are subspaces of V,the followg three codtos are straghtforward V V V dm( V dm( V dm( V 3 V V Theorem (Model II Combato of ay two of above codtos suffces to prove V V Proof: Frst we cosder combato : V V V ad dm( V dm( V dm( V Compared wth Dmeso formula, dm( V V 0 ca get from dm( V V dm( V V dm( V dm( V V Furthermore V V Therefore V For combato : V V V ad V V From the thrd equvalet codto, t's easy to get V V Fally, combato 3: dm( V dm( V dm( V ad V V [9], suppose r ( V, ad α, α, α s bass of V, β, β, β K s bass of V, ad V V, ca be lear represeted as lα l α l α mβ m β m β Sce V V {0}, t s easy lα l α β mβ to obta that l α 0 ad also we have m l β 0 Now α, α, α ad β, β, β are learly depedet, therefore m m 0, ad α, α, α, β,,, β β α, α,, α, β,,, β β V V V l l l are learly depedet, whch meas that s bass of V So V Wth cosderato of V, we obta V 4 Applcato: Put the two models to provg a typcal proposto 4 Proposto Pleas show that ( AA ( E AA m A ( A R, whch satsfes AA A A ad the colum vectors of A 4 emma If A P, B P, the we have r( A B R, where A ( A A AA A, (A r( A r( B [6] m R s the geeralzed verse of s lear space geerated from By usg theorem (model II, ths problem ca be solved by provg the three codtos 43 Proof of codto 43 Proof method I For ay X R, deoted X EX AA E AA X AA X (E A A X, whch shows ( ( that X s lear represeted by the colum vectors of A A ad E A A Therefore, we have E AA R ( AA ( 43 Proof method II

4 Suppose A A ( α, α,, α, E A A ( ε α, ε α,, ε α stadard bass of shows that stadard bass of whch meas ( AA 433 Proof method III et dm ( AA (, E A A ε, ε,, ε s a, where R So ( AA ( E AA ( α, α,, α, ε α, ε α, ε α R s lear represeted by α, α,, α, ε α, ε α,, ε E AA R (, It α, m, the m r( A A, E AA raae (, Whle combed wth A A R, E A A R ad the result r( A A, E A A, we get r ( A A E A A m Hece, learly depedet vectors ( A A ( E A A form a bass of A A E A A R suffces to show that ( ( 44 Proof of codto 44 Proof method I Note that α, A AA A, the we have the result ( A A A A 0,, R, the t E Sce A A (α, α,, α, E A A X, ad t esures that, α, α ca be cosdered as the soluto of ( 0 ( A A r( E A A r But we have ( E A A A A r A A r E A A accordg to the emma Therefore, all of the deducto show that r ( A A r( E A A, ad dm( ( E A A dm( ( A A 44 Proof method II It s easy to show that r( AA r( E AA 0 AA 0 0 r r 0 E AA 0 E dm E A A dm A A ( ( ( ( 45 Proof for codto 3 45 Proof method I Now let A A ( α, α,, α r ( ( AA 0 AA AA r r 0 E AA 0 E AA, suffces to say r ( A A r( E A A ad ( E A A ( β, β, β α ( A A ad ( E AA I, the A AX ( E A A X satsfes If there exsts a vectorα α, ad R that Tmes A A to the both sdes of A AX ( E A A X, the we have the A A A AX A X A A AA AAX AAA AX equato ( A ( E 0 However, we have ( Therefore, AX A 0 ad AA ( I ( E AA AAX 45 Proof method II Smlarly as former secto, let A A ( α, α,, α satsfes ( A A α ad ( E A A α ca be foud, ad ( E A A ( β, β,, β If vectorα that A AX E A A X α R, the (

5 Based o the defto of AAA A, we have A AA A A A ad ( A A A AX 0 E A A α Smlarly, we have A A E A A X ad A Aα 0 ⑵ E That suffces to ( 0 ( 0, whch yelds to ( E A A A A 0 ⑴ Combed ⑴ wth ⑵, we haveα 0 So ( A A ( E A A I all, ay two proofs of the three codtos are suffcet to prove the proposto ( A A ( E A A R, ad there are at least sxtee dfferetal methods 5 Coclusos As show ths paper, there are three drectos to prove the drect sum for lear space ( Macroscopc vew: To prove the drect sum s to prove sum of dmesos equals to dmeso of sum of spaces ( Mcroscopc vew: The tas s to prove the commo elemets of the two subspaces s 0 (3 Dversty vew: Whe provg from the vew of dmeso, t's feasble to use geerated subspace, matrx ad equvalet base Whe provg the tersecto of two spaces s 0, we troduced the system of lear equato, rather tha the commoly used methods, eg, reducto to absurdty, bass of lear space Furthermore, ths paper presets strateges to commo drect sum problems from the vew of cocept of drect sum, ad sce the relatoshp betwee the two subspaces ad the whole space s represetatve, more atteto are pad to the relatoshp betwee them The same goes well for problems wth more tha two spaces However, rag ad ucleus are two specal subspaces, where rag lmts the sze of the space uder trasformato, ad ucleus defes the correspodg elemet to the ull elemet Therefore, the two subspaces satsfy the requremet that sum of dmesos equals to the dmeso of the sum space, whch meas the extra tas s to prove oe of the other two codtos metoed model II whe provg they are the two subspaces of the drect sum Research of decomposto of drect sum has sgfcat mportace trasformg problems wth hgh dmeso to problems wth lower dmeso For example, to smplfy the process, two dmesoal vector problems could be coverted to problems two orthogoal complemet spaces wth oe dmeso by usg the propertes of orthogoal complemet space ad dmesos of lower dmeso subspace Besdes, t's theoretcal feasble to dvde the drect sum, ad geeratg a geerato space by combg bases of oe space radomly s a good example However, sce spaces related to problems are always mostrous, proof of drect sum s harder But, fortuately, base forms the space, ad reflects the relatoshp betwee subspaces ad sum space Therefore, we ca solve drect sum problems by provg bases of subspaces are learly depedet ad the umber of them equals to the umber of the base of the sum space I ths paper, model I was set up uder the premse of V V V, ad model II was set up uder the premse of V s the lear subspaces of V Whe compared the two models, model I s a smple proof model, ad model II s a comparatvely complcated oe But both of them have covered the commo proof models of drect sum, cludg the abstruse models 6 Acowledgemet Ths research s partly supported by Natoal Natural Scece Foudato of Cha (Grat No ad Natoal Creatve Iovato Pla of College Studets (Grat No Refereces [] Xaopg Ye Cogto ad practce of reformato of lear algebra teachg method Joural of Su Yat-Se Uversty,998, : 8-30

6 [] je Ma Notes of educato of lear algebra Reform ad Ope ( 009, Reform ad opeg, 009, :98-99 [3] Fuzh Xue, FagHu A Explorato of ear Spatal Plag ad Desg of Urba Grad Street The Case of the Three Tru Roads Shezhe Joural of Urba Plag, 00, 9 (7: [4] We S, Zhem Dua, Tao Ha Novel method based o projecto of vectors lear space to detfy Volterra erels of arbtrary orders Applcato Research of Computers, 008, 5 (: [5] Ya Tao ear Space Composto of Campus Evromet Tag souther campus of East Cha Jaotog Uversty as xample Joural of East Cha Jaotog Uversty, 0, 8 (4: [6] Departmet of mathematc of Peg uversty Hgher algebra, Hgher Educato Press, 003 [7] Zomg Su Equvalet codtos of drect sum for lear subspace Joural of Nacho Normal College, 988, 9 (: - [8] Zomg Su, Zheguo, Mechag Me Equvalet codtos of dmeso formula ad drect sum for subspace Joural of Chagsha Uversty, 998, : [9] Q Yag Proof of drect sum of subspaces Joural of Fuyag Normal College, 009, 6 (4: 7-8 [0] Yg u, Dogl u Dscusso of drect sum for lear space Joural of Scece of Teachers' College ad Uversty 0, 3 (: 40-45