Fan Fiction

c v... + c v, c R. Linear combination of a set of vectors in R N is a location the vectors can take us.

. Closed Uder Additio: For ech x, y i the set, x+y lso i the set.. Closed Uder Sclr Multiplictio: For ech x i the set, kx is lso i the set ( k R ) 3. Subspce: A subset of vector spce tht is lso vector
of 7
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
. Closed Uder Additio: For ech x, y i the set, x+y lso i the set.. Closed Uder Sclr Multiplictio: For ech x i the set, kx is lso i the set ( k R ) 3. Subspce: A subset of vector spce tht is lso vector spce. A oepty subset S of vector spce V is subspce iff x, y S,, b R, x + by S (MUS SHOW 0 is i the set to verify o-eptiess) 4. ier Cobitio: et { be subset of vectors i vector spce V. A lier cobitio is vector of the for v,..., v k } c v... + c v, c R. ier cobitio of set of vectors i R N is loctio the vectors c tke us. + k k i 5. Sp: he sp of set of vectors cotied i vector spce V is the set of ll possible lier cobitios of tht set. i.e. the set of ll possible loctios tht set of vectors i R N c tke us. 6. ier rsfortios: how vector spces couicte with ech other. et V d W be vector spces. A lier trsfortio is Fuctio : VW tht stisfies properties: ) (v +v )=(v )+ (v ) v,v V ) (cv)=c(v) v V, c R (V- Doi W- Codoi) 7. Doi d Co-doi: Give lier trsfortio f: R N R M represeted by A x, the doi of f is R N d codoi is R M 7b. Ige / Colu Spce of A: he sp of the colus of A. It cosists of ll the vlues the fuctio tkes i its codoi. { v : w s.t. Aw = v} or { y Y : Ax = y for soe x X } A SE! 7c. erel(a) / Null Spce: he set of vectors x (i the doi X) s.t. Ax = 0 (the trsfortio returs the 0 vector i the codoi). 8. ier depedece: A set of vectors i vector spce V is lierly depedet if t lest oe of the vectors c be writte s lier cobitio of the other vectors (i.e. hs redudcy). ht is, set of vectors S = { v,..., vk } V is lierly depedet iff c v... + c v = 0, c R hs t lest oe c 0 (t lest oe o-trivil solutio). + k k i 9. Bsis: A subset B of vector spce V is bsis if. B is lierly idepedet d. Sp B = V (show Sp(B) = MA (R) ) 9. MA (R): Set of x trices with rel etries. 0. Diesio: Diesio of vector spce V is the uber of eleets i y bsis.. Syetric Mtrix: If A = A, the A is sid to be syetric.. Idetity Mtrix (I ): here exists x idetity trix for ll iteger such tht I x = x 3. -: et f: xy be fuctio, the f is - (ective) if for ech y Y there exists t ost oe x X such tht f(x) = y. 4. Oto: et f: xy be fuctio, the f is oto (surjective) if for ech y Y there exists t lest oe x X such tht f(x) = y. i for ll x i R N. (For A x, I A = AI =A) 5. Rk-Nullity: For y x trix A, di(kera) + di(ia) = ullity of A + rk(a) = # of depedet vrs/colus + # of ledig vrs/li idep colus = # of totl vrs. Rk: # of ledig s i rref(a). Properties of Rk:. Rk(A) =, Rk(A) = for ll x trix A.. If Rk(A) = the syste is cosistet o 0 row. (But c hve either uique solutio or ifiitely y solutios). 3. If Rk(A) = the syste hs t ost solutio. (hs 0 solutio if icosistet, i.e. whe with icos row). 4. If Rk(A) the syste hs either 0 (if icosistet) or ifiitely y solutios (if cosistet, but there s free vrs). 5. If Rk(A) = =, the rref(a) = I (squre trix, ivertible). Solutios of ier Systes: A lier syste is sid to be. Cosistet if there exists t lest solutio (. exctly if ll vr s re ledig d rk = =. b. ifiite if there is t lest free vr Icosistet (i.e. o solutios) iff the rref for of A i ugeted for cotis row [ ] ypes of Mtrices: (NOE: is lie tht psses through the origi) k 0, k R 0 k 3. Reflectio (of x bout i b. Sclig (by k):. Projectio (of 4. Rottio (of x i 5. Rottio w/ Sclig (of x i b R ):, + b = Note: x u u u u oto i u A( x) = ref ( x) = proj ( x) ( x proj ( x)) = proj ( x) x R ):, = is uit vector prllel to. uu u u Note: Ref = Proj I (hese refer to trices. Oce you hve the projectio trix we c get reflectio trix) cosθ siθ siθ cosθ cosθ siθ k 0 b b siθ cosθ b k R ): or where b 0 + b = + b R ): or where, d 6. Sheer: ) Horizotl Sheer: b) Verticl Sheer:, k R Sp:. Sp of ozero vector is lie.. Sp of ozero, lierly idepedet vectors is ple. (ll i R N ) Ige: = r cosθ, b = r siθ Ax = b hs solutio s log s b I(A). i.e. b ust be cotied i the set of ll possible li. cobitios of the colus of A. Ige d erel: Are both lier subspces closed uder dditio, closed uder sclr ultiplictio, cotis 0 vector (i R N / R M ). For lier trsfortio fro R M R N, represeted by A x, I(A) is subset of the codoi R N d er() is subset of the doi R M. Diesio of I d er: Di(erA) = # of vectors i the erel/null spce (or # of li. depedet colus i A). Di(er A) = # of li. id. colus of A = rk(a). Bsis of IA d era: Siply row-reduce A (i iplied ugeted trix with b = 0 ) to fid the lierly dep. d idepedet colus. Rk(BA) Rk(B) d Nullity(BA) Nullity(B) Bc I(BA) is cotied i I(B) d er(a) is cotied i er(ba). Whe is er(a) = {0}? ) for x squre trix A, ker(a) = {0} iff A is ivertible (i.e. colus re li id, etc see below) b) Geerlly, for x trix A, er(a) = {0} iff rk(a) =. (iplies , sice = rk(a) ) ier rsfortio: Every lier trsf. fro fiite di. VS to fiite di. VS c be represeted by trix et : VÆW be lier trsfortio betwee the vector spces V d W. he trix represetig is give by A= (e ), (e ),..., (e ) i stdrd bsis Æ Ae = first colu of A, Ae = secod colu of A etc Or [ ] A= [ (v ), (v ),..., (v )] where ll the vectors re i stdrd coordites. Mtrix Multiplictio Properties x squre trix A. Associtive: A( BC ) = ( AB )C, ( ka) B = k ( AB ). Distributive: ( A + B )C = AC + BC 3. Rrely Couttive: AB BA (AI=IA) 3. Idetity: Give ivertible trix x A there exists A- s.t. A-A = I 4. Ivertibility: (BA)-=B-A- exists whe A, B both ivertible. 5. Bx Ax = I A = B, B = A, AB = B A = I A, B ivertible by ierity: Mtrix product is lier. A(C+D) = AC+AD, (A+B)C = AC+BC, (ka)b = k(ab) = A(kB) give k sclr. 7. Mtrix i Sutio For: Ech etry i trix product is dot product, so Bx Axp = C xp, c = bik kj k = 8. Mtrices C be Prtitioed (the ultiplied): Prtitioed trices c be ultiplied s though the subtrices (o the lhs) re sclrs. 9. Mtrix Multiplictio is Fuctio Copositio: If ( x ) = Ax = y, S ( y ) = By = z, the C = Bx Aqxp = S ( ( x )) = z (d trix-vector ultiplictio is lier trsfortio) (co-doi of A, q, ust be se s doi of B,. ht is, q =.) Subspce d Idepedet Vectors d Spig Vectors: A subspce V of RN with di(v) =. We c fid t ost lierly idepedet vectors i V. J0 cid.0c ( )6( )6( )6( )6( )66( )6 [(. 06(W(, qeed idle(o)-6 Orthogolity/Orthoorlity Orthoorl Vectors: A set of vectors re orthoorl if they re ll uit vectors d orthogol to oe other For u, u,, u i R N. u = u = = u = [egth/mgitude/nor v = v v ]. u i. u j = 0 (if j i ) [u, v perpediculr/orthogol iff u. v = 0] [u, v orthogol iff u + v = u + v ] 3. u i. u j = (if i = j) [u is uit vector if its legth is, i.e. v = v. v = ] 4. Orthool Vectors re lierly idepedet [Prove this w. successive dot products] 5. Orthoorl vectors u, u,, u for bsis of R N [Ay lierly idepedet vectors i R N fors bsis] ANY VECOR X IN R N IS A INEAR COMBINAION OF P V (X) AND X FOR SOME SUBSPACE V OF R N Give y vector x i R N d y (rbitrry) subspce V of R N, we c express x s su of perp d projectio o to V: [ x = x + x x is the orth. Proj. of x oto V d x is perpediculr/orthogol copleet to V] Orthogol Projectios: x (Note: Orthogol Projectios re lier d re NO orthogol trsfortios). Give subspce V of R N with orthoorl bsis u, u,, u : x = P V (x) = (u.x)u + (u.x)u + + (u.x)u Note: Give y orthoorl bsis u, u,, u of R N, x = (u.x)u + (u.x)u + + (u.x)u [Reeber tht the coordites of vector i y bsis is just the costts of the lier cobitio of the bsis eleets.]. Give x i R N d subspce V of R N, x Proj V (x) or x [ x = x + x x = x iff x = 0] [Cuchy Iequlity. he stteet is equlity iff x is i V] 3. Give x d y i R N d subspce V of R N, x. y x y [Cuchy-Scwrtz Iequlity. he stteet is equlity iff x d y re prllel] y Derivtio: x ( x u) u = x u = x = x y (u is uit vector o lie sped by y) y y Note kv = k v, so (x.u)u = x.u 4. Mtrix of orthogol projectio of x i R N oto subspce V of R N c be costructed s Proj V (x) = QQ where Q is orthogol trix coposed of orthoorl bsis of V. 5. Orthogol projectio of x i R N oto subspce V of i R N c be thought of s the vector i V closest to x x (Give subspce V of R N, the orthogol copleet V of R N ) N Orthogol Copleet: Def: Orthogol copleet of V is V = { x R : v x = 0 v V}. V = er( ) for ( x)=p V (x) [ V ={ x: v. x = 0 for ll v i V = i()] x i er()]. is subspce of R N V [ V = er() for which projects oto V which is subspce of R N ] 3. = { 0 V V } [If x i V d V, the x is orthogol to itself, i.e. x. x = 0 x = 0 ] Di ( V ) + Di( V ) = 4. [Di(V) = Rk(), Di(V ) = Di(er()) = Nullity()] 5. V ) ( = V [ V ={ x:v.x = 0 for v i V} ) (V ={ u: x.u = 0 for x i V }u i V] 6. [I( A )] = er( A ) [[I( A)] ={ x: v.x = 0 for ll v i V} = {x: v x = 0 for ll v i V}] for y trix A s.t. i(a) = subspce V [ = er(a ) ] Orthogol rsfortios Properties of Orthogol Mtrix A. A orthogol IFF he trs (x)=ax preserves legth [Defiitiol ~ orthogol trs if preserves legth] he trs (x)=ax preserves orthogolity [if x,y orth, (x)+ (y) = (x+y) = x+y = x + y = (x) + (y) ] he trs (x)=ax preserves dot product [if x,y orth, Ax. Ay)=(Ax) (Ay)=x A Ay= x y=x. y] Colus of A for orthoorl bsis of R N A A = I N A = A - EXAMPES: Rottios d Reflectios re trsfortios tht preserve legth. Properties: If A,B orthogol d k costt, the AB, ka, orthogol, QR Fctoriztio d Gr-Schidt. Gr-Schidt lgorith represets CHANGE OF BASIS fro old bsis to ew orthoorl bsis U of V.. Give y x trix M with lierly idepedet colus the there exists orthogol trix Q d digol trix R such tht: M = QR (his represettio is UNIQUE) est Squres Solutio/Approxitio For syste Ax = b tht is icosistet (i.e. b is ot i the i(a) = subspce V), the solutio vector x c be pproxited by the vector x * i R N such tht Ax * (i V) is closest to b. (See orthogol projectio bove) x * is the lest squres solutio of the syste Ax = b for y x trix A b Ax * b Ax for ll x i R N Ax * = Proj V (b), where V = i(a) b - Ax * ( =b b = b ) i V = ( i( A)) = er( A ) A (b - Ax * ) = 0 A b = A Ax * ( orl equtio ) if A ivertible, the A A ivertible (see below), d we hve uique lest squres solutio: x * = (A A) - A b Note: If Ax = b is cosistet, the the lest squres solutio is the EXAC solutio (sice the error would be 0, d its orthogol projectio oto V is itself). rspose: A. (A+B) = A +B. (AB) = B A. (A ) - = (A - ) if A ivertible [AA - =I (AA - ) =(I ) (A - ) A =I (A ) - = (A - ) ] 3. rk(a) = rk(a ) for y A 4. er(a) = er(a A) for y x trix A. [er(a)cer(a A), er(a A)c er(a)] 5. If er(a) = {0} the A A is ivertible for y x trix A [er(a A) = er(a) = {0}] 6. Det(A) = Det(A ) for squre trix A 7. Dot Product: v u v = u 8. For Orthogol Mtrices: A A = I A - = A 9. For Mtrix of Orthogol Projectio (of x oto subspce V): P V (x) = QQ [Colus of Q = orthoorl bsis of V] 0. Qudrtic Fors: q x x Ax x ( ) = = Ax Deterit Clcultig Deterit b. For x trix A = c d b A = d e g h det A = d bc c. For 3x3 trix f = ( ei fh) b( di fg) + c( dh eg) i 3. plce Expsio: We c f9id det. of x trix A by plce expsio dow y colu or cross y row of A Expsio cross ith row: det( A) = j= ( ) i+ j det( A ) for fixed row i i+ j Expsio cross jth colu: det( A) = ( ) det( A ) for fixed colu j 4. Guss Eliitio: Usig the tisyetric d ultilier properties, we c reduce A ito sipler trix B such tht Det(B) is esy to copute, the ultiply by the pproprite costts d sig chges to bck out Det(A) = (-) S k k k r Det(B) [Swp rows s ties d divide rows by sclrs k k r to go fro A to 5. Geoetric / Colus of A: For ivertible x trix A = [v v ] Det(A) = Det(QR) = Det(Q) Det(R) = Det(R) = v v v3... v fro Gr-Schidt = Volue of -diesiol prlellopepid fored by v v colus of A [see below for geerl cse] 6. Product of Eigevlues: For x trix A (digolizeble) with eigevlues, listed with their lgebric ultiplicities λ, λ,... λ, the Det(A) = λ λ... λ Properties. Det(A ) = Det (A) [So everythig tht is true for rows is true for colus]. If A hs row/colu of 0 s the Det (A) = 0 [Rows/Colus re lier depedece] 3. If A hs ideticl rows/colus the Det(A) = 0 [Rows/Colus re lier depedece] 4. Atisyetric: If we switch colus of A, Det(A) chges sig. [As result of the sig p i the det. expsio] 5. Multilier:. Sclrs c be fctored out of col/row, so if we fctored out k fro colu, the det(a) = (/k)det(a ) b. We c brek up row/colu to copute A 6. Det (I ) = 7. Addig ultiple of oe col/row to other does ot chge the Det (A) [We c row reduce A w/o chgig the det] 8. Det (AB) = Det(A) Det(B) 9. If A ivertible, det( A ) = [A ivertible AA - = I det(a)det(a - )=] det( A) 0. For upper/lower trigulr A, Det(A) = product of A s digol etries. [Expd o row/col tht hs ll 0 s]. For prtitioed trix (if we c prtitio A ito 4 squre trices, ot ecessrily of the se size), the A B det( M ) = det = det( A)det( D) 0 D [Det(A)Det(D)-Det(B)Det(C) does ot geerlly hold for sq trices A,B,C,D]. For siilr trices A d B, Det(A) = Det(B) [B=S - AS, Det(B) = Det(S - )Det(A)Det(S) = Det(A)] 3. For orthogol trix A, Det(A) = or - [A A = I, = Det(A A) = Det(A )Det(A) = Det(A) ] 4. For x trix A (digolizeble) with eigevlues, listed with their lgebric ultiplicities λ, λ,... λ, the Det(A) = λ λ... λ i= Geoetric Iterprettio. Prllelepipeds i R N : Give lierly idepedet vectors v v i R N, the -diesiol prllelepipeds defied by these vectors hs the volue, V(v v ) defied recursively by V(v ) = v d V(v v ) = V(v v - ) v this iics the forul (bse)(height) ( v is the orth. Copleet of v oto the subspce V sped by v v -, s defied by Gr-Schidt lgorith). Volue of prllelepiped i R N : For x trix A = [v v ], the volue of the prllelepiped defied by lierly idepedet vectors v v : V ( v,..., v ) det( A A) = Pf: A = QR A A = (QR) (QR)=R Q QR=R R Det(A A)=Det(R R)=Det(R )Det(R) = Det(R) = (r r r ) = ( ) v v... v = V ( v,..., v ) I prticulr, if =, the V ( v,..., v ) = det( A A) = det( A )det( A) = det( A) det( ) 3. Expsio Fctor: = A Digol Mtrices. QR Fctoriztio: Ay x trix c be represeted Syetric Mtrices A = A (Skew syetric: A = -A). A + A lwys syetric. [(A+A ) = A + A = A + A ] Siilr Mtrices: A & B siilr there exists chge of bsis trix S such tht B = S - AS. Se Deterit: Det B = Det S - AS = Det A. I(A) =. I( A) = [ er( A )]. I(A) = I(AA ) Proof: I( A ) = [ er( A )] = [ er( AA )] = [ er( AA ) ] = [I( AA ) ] = I( AA 3. I(A) c I(AB) )
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks