MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function : ln, R
Proprtis: ln * ln ln ln ln lim lim ln ln ln ln ln ln LINER LGER Vctors: Gomtricall spaking a vctor is an orintd sgmnt, which mans that two points and an orintation dfins uniqul a vctor. Th lngth of th vctor is th lngth of th sgmnt. In a -dimnsional sstm of coordinats XOY, givn two points s,t th origin and u,v th nd point and orintation from to w can build vctor and its lngth is u s v t If X and Y ar points in a n-dimnsional sstm of coordinats thn, for X,,..,n and Y,,..,n, th lngth of th vctor XY is: XY n i i i lgbraicall spaking, a vctor is undrstood as having th origin at th origin of th sstm O,..,. This mans that is nough to giv th nd point of th vctor in ordr to dscrib an uniqu vctor X ; th orintation is assumd to b from th origin to th nd point. Now, having vctor X and Y w can dfin th sum and diffrnc of thm as: X ± Y ±, ±,.., n ± n a * X a *, a *,.., a * n th last qualit rprsnts th product btwn a scalar a numbr and a vctor. Plas notic that both th abov oprations hav as rsult anothr vctor!!
W dfin th transposd vctor X ' as. If th vctor is givn vrticall column. n vctor, its transposd will b horizontal row or lin vctor. Whn it coms to th product, thr ar two tps of products btwn two vctors. THE SCLRDOT-in US PRODUCT: Th scalar product of two n-dimnsional vctors is a scalara numbr!!! NOT a vctor!! Th scalar product can b computd onl btwn two qual dimnsional column and row vctors. Suppos: X,,.. n Y. n X Y n i i * i THE VECTORCROSS-in US PRODUCT: for w talk about th cross product w nd to talk about matrics. MTRICES: matri is a tabl of qual dimnsional column or row vctors. a a an a a an Matri can b sad it is formd with column vctors am am amn a a an a a an.. amam amn a a an, a a an,.. am am amn. or with row vctors - aij mans lmnt of matri a placd at th intrsction of row i and column j.
Notic that th numbr of rows is not ncssaril qual with th numbr of columns. Matri is said to b of dimnsion mn to b rad m b n; m rprsnts th numbr of rows, n th numbr of columns. If mn, th matri is calld a squar matri. On can add, subtract onl matrics of th sam dimnsion, and is don lmnt b lmnt lik in th cas of vctors. On can also multipl a scalar with a matri, and is don b multipling th scalar with ach of th lmnts of th matri. Th transposd matri is obtaind b changing th column vctors into row vctors or th row vctors into column vctors. Eampl: ' ;.. ' ;.. π π You can notic that both and hav th sam lmnts on th diagonal top-lft to bottom-right. Th diagonal is calld th first diagonal and th sum of its lmnts is calld th TRCE of th matri. gnral formula is: n i aii Trac transposing a matri bcoms a matri. Instad of ou ma s T. Ths ar th most common notations I hav sn so far. In ordr to comput th product btwn matrics ou nd to tak car of on important rul: th numbr of columns of th first matri should qual th numbr of rows of th scond matri. Which mans if is mn, has to b np, in ordr to comput. Th rsult is going to b anothr matri of dimnsion mp. Eampl: don t gt scard!!..yt ;. C You comput th valu of cij b calculating th scalar product btwn th i-th row vctor of matri and th j-th column vctor of matri. Now w can talk about th vctor product btwn two vctors. ctuall thr is not much to talk about. Think to th vctors as matrics. First on is a matri m and scond p th rsult is a matri mp or m and m cas which givs ou a matri or a scalar.
Lik in th cas of ral numbrs, thr is a mn matri calld th null or zro matri having all lmnts dnotd O such that /-O for an mn matri, and a matri I calld th idntit or unit nn matri having on th first diagonal and in rst such that II for an squar nn matri.. O...... I..... It is VERY IMPORTNT to undrstand that du to th wa of computing th product of two matrics, THE MTRIX MULTIPLICTION IS NOT COMMUTTIVE!!! That mans that is not qual not lik in th cas of numbr multiplication. So ou ma sa O and I pla th sam rol and pla in th cas of numbrs, rspctivl. Thr ar two mor notions to b mntiond about matrics. On is th dtrminant of a matri dt. Onl for squar matrics on can comput thir dtrminant. It is a numbr and sinc thr ar a lot of softwar up thr to comput th dtrminant of a matri for ou, I will show ou how to comput it for a and matri. For : Dta*a-a*a For : Dt b*b*bb*b*bb*b*b -b*b*b-b*b*b-b*b*b You can imagin how it will look for a matri Th othr notion is th invrs of a matri. matri has a invrs onl if its dtrminant is diffrnt from notic that mans th matri has to b squar too! Th invrs of a matri is a matri such that I. So is for matri lik what th invrs of a numbr is for that numbr a* a. Wh do w nd matrics? On of th most important applications of matrics is in solving sstms of quations. Suppos w hav a IG sstm to solv n quations with n variabls. It will tak das to solv b substitution, unlss ou alrad know a fastr wa or somon ls is doing for ou. If nithr th cas, pa attntion: Lt s sa w hav th unknowns variabls,,.., n: a* a *... an * n b a* a *... an * n b... an* an *... ann * n bn whr all aij and bi ar ral numbrs.
If ou look carfull on ach lft sid of th abov quations w hav th scalar product btwn th corrsponding row vctor of th matri aij and th column vctor of th variabls a a. an.. So, that maks th ntir lft part of th sstm th n b b product *X whr X.. Now if w dnot, thn w can rwrit th sstm. n bn as a matri quation: *X. So, what? Wll this is all, bcaus th solution is X * but onl if ists which mans dta not and not *!!! rmmbr th matri multiplication is not commutativ. Isn t it, this asir?