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

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. 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
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. 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) )
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