Review of Linear Algebra

Transcription

1 PGE 30: Forulto d Soluto Geosstes Egeerg Dr. Blhoff Sprg 0 Revew of Ler Alger Chpter 7 of Nuercl Methods wth MATLAB, Gerld Recktewld Vector s ordered set of rel (or cople) uers rrged s row or colu sclr lowercse Greek () vector lowercse ro (u,v,,,) tr uppercse ro (A,B,C)

2 Vector Opertos Addto d Sutrcto volve correspodg eleets of dfferet vector wth the se uer of eleets Multplcto Sclr volves ultplcto of ever eleet the sclr Vector Trspose coverts row vector to colu vector or vce vers 3 Ler Coto volves sclr ultplcto d vector ddto u v w u v u v w u v u v w u v u v w 4

3 Vector Ier Product s operto etwee two vectors tht hve the se uer of eleets ALWAYS results sclr MUST e row vector tes colu vector C use trspose for two colu vectors Vector Nors copre the sze (gtude) of vector For eple, t kes sese tht Ut vector hs gtude of Zero vector hs gtude of 0 Asolute Vlue s esure of gtude for sclrs Nor s esure of gtude for vectors 6

4 Vector Nors Wht s the veloct u=3+4 Wht s the speed? u Vector Nors Euclde Nors D d 3D re geoetrc legths l l l c z l 0 4 l 3 3 D 3D 8 l c 4 9 3

5 Vector Nors However, Euclde Nors c e defed ultdesos C e epressed ters of the er product T 9 Three Vector Nors re coo d useful d elog to the fl of p-ors L s oe of the p-ors p p p p p L (p = )or of vector s L or (p = ) or or of vector s,,, 0

6 Propertes of Vector Nors () 0 0 () (3) Trgle Ieqult Orthogol Vectors cos u T v u v I D we s two vectors re perpedculr/orthogol f = 90 Two vectors re orthogol f d ol f the dot product s zero

7 Orthoorl Vectors re ut vectors tht re orthogol Ut vector hs gtude of A vector c e coverted to ut vector dvdg ts L or ^ u u u 3 Mtrces re rrs of rel or cople uers Upper Cse Ro Letters (e.g. A, B, C) MATLAB dfferece etwee tr d vector s the # of rows d colus Rows d Colus of tr re vectors Mtr s vewed s collecto of rows or colu vectors 4

8 Mtr Opertos Addto d Sutrcto s perfored eleet eleet Multplcto sclr ust ultples the sclr ever eleet Mtr Trspose coverts ech row to colu d vce vers A B, ;,,, A B T, ;,,, c B A C, ;,,,, 5 Multplcto of trces d vectors s essetl operto uercl ler lger Mtr A tes colu vector c e vewed s: Ler coto of colu of colus of A Seres of er products volvg the rows of A A 6

9 The Vector-Mtr Product A= produces vector fro ler coto of the colu vectors of A [ ] [ ] = [ ] 7 A ltertve vew s oted cosderg the tr s collecto of rows Dot product of ech row (trsposed) of A tes the colu vector 8

10 Vector-Mtr Multplcto s dfferet fro Mtr-Vector Multplcto Mtr-Vector ultplcto hs colu vector o rght. The result s colu vector Vector-Mtr ultplcto hs row vector o left d the result s row vector MATLAB uses * d tkes cre of the result for us 9 The product of trces s other tr The colu vew provdes thetcl sght The row vew s esest to perfor wth hd clcultos The MATLAB * opertor tkes cre of the detls [ r] [r ] = [ ] r c c c c r r r 0

11 Colu Vew s ler coto A*B = C Worr out ONE colu of B t te d get oe colu of C t te A () c () r Perfor ler coto to fd colu of ew tr d repet Colu Vew s ler coto A*B = C Worr out ONE colu of B t te d get oe colu of C t te 3 3 c c c3 c c c c3 c3 c 33 Perfor tr-vector product of ever colu of B to detere ew colu of C A( ) c( ) Col of B creted col of C We hve lred lered how to do tr-vector products

12 The Row Vew s lttle eser to uderstd Perfor the dot product of row Mtr A d colu of Mtr B,,, r,, r, c, 3 The Row Vew s lttle eser to uderstd Perfor the DOT PRODUCT of row Mtr A d colu of Mtr B 3 3 c c c3 c c c c3 c3 c 33 c 33 A product volvg trces d vectors requres tht the operds hve equl er desos Order Mtters! AB BA 4