MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity, thik first of wht it ll es i the cses = = 3, d or s we ofte do i clss t first. My of the hoework questios re lso forulted for such sll vlues of! I prctice, however, d c be huge - tht's where coputers coe i hdy!! Let's first cocetrte o the lgebr hlf of the ter "Lier Algebr". A Vector is ordered list of ubers (rel or cople) = v v = v, v R C = = ll ordered lists of rel ubers, ll ordered lists of cople ubers. The set of right hd vlues d the set of vribles i syste of lier equtios i vribles c be idetified with vectors for eple, but y others occur: b,,,,,, b b = i, = i, R R The set of sple vlues v = π π 7π of the fuctio y = cos t 0,,,,, π s 4 4 show bove is vector i R 9 ; CD is huge vector of this sort obtied by splig udio sigl 44, 000 ties per secod. 0 The list of 8 roots of uity v = ( + i) i ( i) s show bove o the uit circle i the cople ple is vector i C 8. We shll eet vectors like this whe we get to eigevectors of trices. The 'lgebr' of vectors refers to the dditio, subtrctio, d sclr ultiplictio of vectors. u, v R k R v R We c dd/subtrct vectors i by d for the sclr ultiple of i d i by u v u ± v u ± v = u ± v u ± v =, ± u v u v v kv kv = k v kv = ; v kv i other words, clcultios re doe copoet-wise. There re siilr defiitios of dditio d sclr ultiplictio for C
vectors i C R R 3 = (, ) = (,, 3 ) 3 correspods to the oes you lered for vectors i the ple d 3-spce., ecept tht ow sclr ultiplictio by cople ubers is llowed. Whe tryig to iterpret cocepts d results grphiclly it's ofte coveiet to idetify vectors i or with poits i the ple or i -spce. The dditio/subtrctio/sclr ultiplictio just defied for vectors i colu for the But wht's the poit of thikig of vectors i colu for? Eple: give vectors = i, d =, =,, = i, R R the the opertios of dditio d sclr ultiplictio for vectors show tht so the usul syste of lier equtios i vribles c be rewritte s the + + + + + + =, + + + + + + Vector Equtio: + + + = b =. b A row of colu vectors where MATRIX with rel or cople ubers s etries c be thought of both s Arry of ubers d s 3 3 A = = [ ] 3 =, =,, = R C R re colu vectors i or. The set of ll trices with rel etries will be deoted by d those with cople etries by C. To keep fro drowig i ottio, it's coo to write tri s A = [ jk ] isted of writig out A R R C C ll the etries i. Notice tht the set of ll trices cosistig of oe colu of rel etries is just other wy of thikig of, while is just other wy of thikig of. The ter lgebr of trices refers to dditio, subtrctio, d ultiplictio of trices. I future lecture we'll lso
discuss the ide of the iverse of tri. A, B R We dd/subtrct trices i etry-by-etry: A ± B = [ jk ] ± [ b jk ] = [ jk ± b jk ]. A ± B A ± B A, B I prticulr, is defied oly whe re, d the lso is. For eple, 3 4 4 6 [ ] + [ ] = [ ], 3 4 4 0 3 4 3 3 [ ] [ ] = [ ] 3 4 4 5 8 0 We defie the sclr ultiple of i d i R by Thus ech etry i ka = k[ jk ] = [k jk ]. prticulr, the sclr ultiple A is ultipled by ; i of tri lso is. For eple, A ka k R A k 3 3 6 9 3[ ] = [ ]. 3 4 9 3 To itroduce ultiplictio, let's begi with the product of tri d vector: Mtri-vector Rule: if with colus i d is vector i R, the A = [ ],,, R A = [ ] = + + +. But wht's the poit of this defiitio? Well, if is the coefficiet tri of the usul syste of lier equtios i vribles, d if A is writte i colu for A = [ ], the so the usul syste of lier equtios i vribles c be rewritte s the A + + + A = [ ] + + + = + + + =, + + + Mtri Equtio: b A = b =. Thus the lgebr of vectors d trices provides three differet wys of writig syste of lier equtios i ukows: s ugeted tri, vector equtio d sigle tri equtio. I other words, by usig vectors d + = =
+ y = b = b trices we hve de thigs just s copct s the equtios d studied i high school, yet we c hdle systes i hudreds or thousds of vribles. The et theore sys tht ech provides wy of iterpretig d solvig syste of lier equtios. Fudetl Theore: if d re vectors i, the ech of the followig the syste of lier equtios with ugeted tri, the vector equtio, the tri equtio whe, hs the se solutio set.,,, b R + + + = b A = b A = [ ] [ b ] I prctice, to solve give syste of lier equtios it's probbly quickest d esiest to write the ssocited ugeted tri i Reduced Row Echelo For, especilly if oe hs electroic wy of rrivig t this for. Noetheless, the other two wys of iterpetig systes of lier equtios will becoe very iportt coceptully. Filly, to defie products of trices quite geerlly, we ow siply thik of tri i colu for d the use the Mtri-vector rule: A p B p B = [ b b ] AB A B Mtri-Colu Rule: if is d is tri writte i colu for, the the product of d is the tri AB defied by AB = A[ b b ] = [A b A A b ]. A R p B R p b j B R p Ab j R B AB R Notice tht the restrictios: i d i re eeded so the colus of re i d the tri-vector product is defied s vector i. Sice there re colus i the product thus hs colus ech i. Hece AB is i R. Eple: copute Solutio: Write while AB whe 4 3 A = [ ], B = [ ]. 3 4 6 B = [ b b 3 ]. The 4 9 A b = [ ] [ ] = 4[ ] [ ] = [ ], 3 4 3 4 8 d Thus A = [ ] [ ] = [ ] + [ ] = [ ], 3 4 3 4 4 3 A b 3 = [ ] [ ] = 3[ ] + 6[ ] = [ ]. 3 4 6 3 4 5 9 AB = [A b A A b 3 ] = [ ]. 8 4 5 The 'lgebr' prt of the ter 'lier lgebr' hs thus bee eplored. But wht bout the 'lier' prt. Well, oe of the key
ides uderlyig wht we've doe is the fct tht trices d vectors hve fudetl property tht you et origilly just u, v, b u + bv i j f() + bg() for vectors i the ple, sy: give vectors i the ple d sclrs, the the Lier Cobitio is gi vector i the ple which c be defied both i ters of its d copoets s well s grphiclly i ters of the prllelogr d trigle lws for ddig vectors. But you y recll tht you lso et lier cobitios fuctios whe delig with properties of liits, derivtives d itegrls. Wht we've just see is tht the otio of lier R C R C cobitio kes good sese for vectors i d s well s for trices i d. Lierity is fudetl cocept occurig everywhere i thetics d its pplictios. Lter we shll forlize these ides bstrctly by itroducig the otio of Vector spce (or Lier Spce s it is lso clled). of