Linear Algebra Introduction


 Milton Wilson
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1 Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x + = 0, ut lter on it diversified due its vivid pplition in vrious fields The si ide ehind finding the solution of system of liner equtions is y elimintion of vriles The min reson ehind the populrity of lgeri mthemtis is following generi solution whih most of the prolems follow. E.g. strt from prolem in geometry, trnslte the prolem in the lnguge of lger, solve the resulting lger prolem using the lgeri tools nd finlly trnsport the solution k to geometry. History Mtrix, set of numers rrnged in rows nd olumns so s to form retngulr rry. The numers re lled the elements, or entries, of the mtrix. Mtries hve wide pplitions in engineering, physis, eonomis, nd sttistis s well s in vrious rnhes of mthemtis. Historilly, it ws not the mtrix ut ertin numer ssoited with squre rry of numers lled the determinnt tht ws first reognized. Only grdully did the ide of the mtrix s n lgeri entity emerge. Tht suh n rrngement ould e tken s n utonomous mthemtil ojet, sujet to speil rules tht llow for mnipultion like ordinry numers, ws first oneived in the 180s y Cyley nd his good friend the ttorney nd mthemtiin Jmes Joseph Sylvester.
2 Wht do you lern here? Key Terms: 1. Mtrix. Vetor. Sumtrix 4. Squre mtrix. Equl mtries. Zero mtrix 7. Identity mtrix 8. Digonl mtrix 9. Upper tringulr mtrix 10. Lower tringulr mtrix 11. Tridigonl mtrix 1. Digonlly dominnt mtrix 1. Addition of mtries 14. Sutrtion of mtries 1. Multiplition of mtries 1. Slr Produt of mtries 17. Liner Comintion 18. Rules of Binry Mtrix Opertion 19. Trnspose 0. Symmetri Mtrix 1. SkewSymmetri Mtrix. Tre of Mtrix. Determinnt 4. Consistent system. Inonsistent. Infinite solutions 7. Unique solution 8. Rnk 9. Inverse 0. Eigenvlue 1. Eigenvetors. Power method
3 Wht does mtrix look like? Mtries re everywhere. Look t the mtrix elow out the sle of jens in Deprtmentl Store given y qurter nd mke of jens. Levis Newport Pepe Q1 Q Q Q If one wnts to know how mny Pepe jens were sold in Qurter 4, we go long the row Pepe nd olumn Q4 nd find tht it is 7. So wht is mtrix? A mtrix is retngulr rry of elements. The elements n e symoli expressions or numers. Mtrix is denoted y 11 1 [ A] m1 1 m n n mn Row i of hs n elements nd is... i1 i in nd olumn j of hs m elements nd is 1 j j mj Eh mtrix hs rows nd olumns nd this defines the size of the mtrix. If mtrix hs m rows nd n olumns, the size of the mtrix is denoted y m n. The mtrix my lso e denoted y [ A] m n to show tht is mtrix with m rows nd n olumns. Eh entry in the mtrix is lled the entry or element of the mtrix nd is denoted y numer nd j is the olumn numer of the element. where i is the row The mtrix for the jens sles exmple ould e denoted y the mtrix s
4 There re rows nd 4 olumns, so the size of the mtrix is 4. In the ove mtrix, 7 Wht re the speil types of mtries? Vetor: A vetor is mtrix tht hs only one row or one olumn. There re two types of vetors row vetors nd olumn vetors. Row Vetor: If mtrix [B] hs one row, it is lled row vetor B] [ ] nd n is the dimension of the row vetor. Exmple Give n exmple of row vetor. [ B ] [ 0 0] is n exmple of row vetor of dimension. Column vetor: If mtrix [C] hs one olumn, it is lled olumn vetor 1 [ C] m nd m is the dimension of the vetor. Exmple Give n exmple of olumn vetor. [C] is n exmple of olumn vetor of dimension. Sumtrix: [ 1 n If some row(s) or/nd olumn(s) of mtrix re deleted (no rows or olumns my e deleted), the remining mtrix is lled sumtrix of. Exmple Find some of the sumtries of the mtrix 4
5 ,, 1 1 4, 4, re some of the sumtries of. Cn you find other sumtries of? Squre mtrix: If the numer of rows m of mtrix is equl to the numer of olumns n of mtrix, ( m n ), then is lled squre mtrix. The entries 11,,..., nn re lled the digonl elements of squre mtrix. Sometimes the digonl of the mtrix is lso lled the prinipl or min of the mtrix. Exmple Give n exmple of squre mtrix is squre mtrix s it hs the sme numer of rows nd olumns, tht is,. The digonl elements of re, 10, Upper tringulr mtrix: A m n mtrix for whih 0, i j is lled n upper tringulr mtrix. Tht is, ll the elements elow the digonl entries re zero. Exmple Give n exmple of n upper tringulr mtrix is n upper tringulr mtrix. Lower tringulr mtrix: A m n mtrix for whih 0, j i is lled lower tringulr mtrix. Tht is, ll the elements ove the digonl entries re zero.
6 Exmple Give n exmple of lower tringulr mtrix is lower tringulr mtrix. Digonl mtrix: A squre mtrix with ll nondigonl elements equl to zero is lled digonl mtrix, tht is, only the digonl entries of the squre mtrix n e nonzero, ( 0, i j ). Exmple Give exmples of digonl mtrix is digonl mtrix. Any or ll the digonl entries of digonl mtrix n e zero. For exmple is lso digonl mtrix. Identity mtrix: A digonl mtrix with ll digonl elements equl to one is lled n identity mtrix, ( 0, i j nd 1 for ll i ). ii Exmple Give n exmple of n identity mtrix is n identity mtrix.
7 Zero mtrix: A mtrix whose ll entries re zero is lled zero mtrix, ( 0 for ll i nd j ). Exmple Give exmples of zero mtrix [B] [C] [ D ] re ll exmples of zero mtrix. Tridigonl mtries: A tridigonl mtrix is squre mtrix in whih ll elements not on the following re zero  the mjor digonl, the digonl ove the mjor digonl, nd the digonl elow the mjor digonl. Exmple Give n exmple of tridigonl mtrix is tridigonl mtrix. Do non squre mtries hve digonl entries? Yes, for Exmple m n mtrix, the digonl entries re 11,..., k 1, k 1, kk where k min{ m, n}. Wht re the digonl entries of
8 The digonl elements of re. nd Digonlly Dominnt Mtrix: A n n squre mtrix is digonlly dominnt mtrix if n ii for ll i 1,,..., n nd j1 i j n ii for t lest one i, i j 1 j tht is, for eh row, the solute vlue of the digonl element is greter thn or equl to the sum of the solute vlues of the rest of the elements of tht row, nd tht the inequlity is stritly greter thn for t lest one row. Digonlly dominnt mtries re importnt in ensuring onvergene in itertive shemes of solving simultneous liner equtions. Exmple Give exmples of digonlly dominnt mtries nd not digonlly dominnt mtries is digonlly dominnt mtrix s nd for t lest one row, tht is Rows 1 nd in this se, the inequlity is stritly greter thn inequlity.
9 [B] is digonlly dominnt mtrix s The inequlities re stisfied for ll rows nd it is stisfied stritly greter thn for t lest one row (in this se it is Row ). C is not digonlly dominnt s When re two mtries onsidered to e equl? Two mtries nd [B] re equl if the size of nd [B] is the sme (numer of rows nd olumns re sme for nd [B]) nd = for ll i nd j. Exmple Wht would mke 7 to e equl to 11 [ B ] The two mtries nd [B] ould e equl if nd How do you dd two mtries? How do you dd Two mtries nd [B] n e dded only if they re the sme size. The ddition is then shown s [ C] [ A] [ B] where
10 Exmple Add the following two mtries. 7 [B] [ C] [ A] [ B] Exmple ABC Deprtmentl Store hs two store lotions A nd B, nd their sles of jens re given y mke (in rows) nd qurters (in olumns) s shown elow. 0 [B] 4 where the rows represent the sle of Levis, Newport nd Pepe jens respetively nd the olumns represent the qurter numer: 1,, nd 4. Wht re the totl jens sles for the two lotions y mke nd qurter? 0 = [ C] [ A] [ B] 0 0 =
11 So if one wnts to know the totl numer of Pepe jens sold in qurter 4 t the two lotions, we would look t Row Column 4 to give How do you sutrt two mtries? Two mtries nd [B] n e sutrted only if they re the sme size. The sutrtion is then given y [ D] [ A] [ B] Where d Exmple Sutrt mtrix [B] from mtrix. 1 [B] 7 7 [ D] [ A] [ B] 1 ( ) (1 ) ( 7) ( ) ( ( )) (7 19) Exmple ABC Deprtmentl Store hs two store lotions A nd B nd their sles of jens re given y mke (in rows) nd qurters (in olumns) s shown elow
12 0 [B] 4 = where the rows represent the sle of Levis, Newport nd Pepe jens respetively nd the olumns represent the qurter numer: 1,,, nd 4. How mny more jens did store A sell thn store B of eh rnd in eh qurter? [ D] [ A] [ B] So if you wnt to know how mny more Pepe jens were sold in qurter 4 in store A thn store B, d Note tht d implies tht store A sold 1 less Newport jens thn store B in qurter. How do I multiply two mtries? Two mtries nd [B] n e multiplied only if the numer of olumns of is equl to the numer of rows of [B] to give [ C] A B mn [ ] m p[ ] pn If is m p mtrix nd [B] is p n mtrix, the resulting mtrix [C] is m n mtrix. So how does one lulte the elements of [C] mtrix? p k 1 i1 ik 1 j kj i j for eh i 1,,, m nd j 1,,, n. ip pj
13 th j ] th To put it in simpler terms, the i row nd olumn of the [C mtrix in [ C] [ A][ B] is lulted th j ] th y multiplying the i row of y the olumn of [B, tht is, i1 i1 i 1j i ip j 1 j j pj... ip pj. p k 1 ik kj Exmple Given 1 [B] 9 Find C AB n e found y multiplying the first row of y the seond olumn of [B], ( )( ) ()( 8) ()( 10) Similrly, one n find the other elements of [C] to give [C] 7 88 Exmple ABC Deprtmentl Store lotion A nd the sles of jens re given y mke (in rows) nd qurters (in olumns) s shown elow
14 where the rows represent the sle of Levis, Newport nd Pepe jens respetively nd the olumns represent the qurter numer: 1,,, nd 4. Find the per qurter sles of store A if the following re the pries of eh jens. Levis = $. Newport = $40.19 Pepe = $.0 The nswer is given y multiplying the prie mtrix y the quntity of sles of store , so the per qurter sles of store A would e given y [C] A. The prie mtrix is k 1 ik kj 11 k 1 1 k k 1 $118.8 Similrly $147.8 $ $ Therefore, eh qurter sles of store C A in dollrs is given y the four olumns of the row vetor Rememer sine we re multiplying 1 mtrix y 4 mtrix, the resulting mtrix is 1 4 mtrix. Wht is the slr produt of onstnt nd mtrix? If is n n mtrix nd k is rel numer, then the slr produt of k nd is nother n n mtrix ], where k. [B
15 Exmple Let.1 1 Find [ A].1 [ A] Wht is liner omintion of mtries? If [ A1 ],[ A ],...,[ Ap ] re mtries of the sme size nd k 1, k,..., k p re slrs, then k A ] k [ A ]... k p[ A 1[ 1 p ] is lled liner omintion of [A ],[ A ],...,[ A 1 p ] Exmple If A ],[ A 1.1 ] 1,[ [ 1 A 0 ].. then find [ A1 ] [ A ] 0.[ A ] [ A1 ] [ A ] 0.[ A ] Wht re some of the rules of inry mtrix opertions? Commuttive lw of ddition If nd [B] re m n mtries, then [ A] [ B] [ B] [ A] ] Assoitive lw of ddition
16 If, [B] nd [C] re ll m n mtries, then [ B] [ C] [ A] [ B] [ ] [ A] C Assoitive lw of multiplition If, [B] nd [C] re m n, n p nd p r size mtries, respetively, then [ B][ C] [ A][ B] [ ] [ A] C nd the resulting mtrix size on oth sides of the eqution is m r. Distriutive lw If nd [B] re m n size mtries, nd [C] nd [D] re n p size mtries [ C] [ D] [ A][ C] [ A][ ] [ A] D [ A] [ B] [ C] [ A][ C] [ B][ C] nd the resulting mtrix size on oth sides of the eqution is Exmple m p. Illustrte the ssoitive lw of multiplition of mtries using 1 1 [ A], [ B], [ C] 9 0 [ B][ C] [ A]([ B][ C]) [ A][ B] 9 0
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