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Applied Linear Algebra I

by: August Feeney

Applied Linear Algebra I MTH 261

Marketplace > Portland Community College > Math > MTH 261 > Applied Linear Algebra I
August Feeney
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This 10 page Class Notes was uploaded by August Feeney on Monday October 19, 2015. The Class Notes belongs to MTH 261 at Portland Community College taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/224650/mth-261-portland-community-college in Math at Portland Community College.


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Date Created: 10/19/15
MTH 261 Mr Simonds class Matrices and vectors a few introductory definitions An m xn matrix is a rectangular array of numbers with m rows and n columns m and n are called the whensoils of the matrix Matrices are most commonly denoted by capitol letters or single subscripted capitol letters The numbers entries of a matrix are denoted by double subscripted lower case letters When a matrix is explicitly written out it is delineated by square brackets or parentheses Abstractly you will see things like A ay 1 where ax represents the entry in the row and f column When the number of rows or columns goes above 9 we separate the is and 39s by commas we won39t be going there Example 7 77 8 72 0 Consider the matrix B 74 11 9 a What are the dimensions of the matrix 718 762 3 22 71 14 g Y 3 b What are entries b32 b23 b and b t 9 L313 39 ll ILLJJ BJL f J IJl J b 7 O I lkvv hu 31 A matrix with only one row is called a row yecfor and a matrix with only one column is called a column vector Example Which of the following are vectors n 3 71 Gil41718 5 0 v 612 4 G 3 7 Page 1 of 10 MTH 261 Mr Simonds class Matrices a couple of simple arithmetic operations Muffx IOU an Two matrices with The exact same dimensions can be added or subtracted thusly avibvlavibv If either of the corresponding dimensions ofntwo matrices is not the same the matrices can neither be added nor subtracted Scalar mul gica on A matrix can be multiplied by a scaldnumber thus ly thy kav Example 3 71 4 8 72 71 72 6 Consider the matrices A 5 2 B and C 75 71 6 3 72 72 0 77 Find A B B 7 2C and C At The fransgose of the m x n matrix M is then x m matrix Mt that results from swapping the rows A 4 b to 44 Le and columns of M JJJJ MI39J39V A39VLquot Jwisj i L B lc jril cl l3 3 H 7quotLL 1Iz L co q c c In w H 3 la cw 93311 vJf 1 Which of the following are column vectors 0 n 3 1 471s iv 5 0 v 612 II Page 2 of 10 MTH 261 Mr Simonds class Matrices and vectors ramping up the vocabulary A sum of scalar multiples of two or more matrices is called a rmer combquotaffair of the matrices whose scalar multiples are being summed For example the row vector 18 77 is a linear combination of the vectors 4 71 and 73 2 because 18 7734 il7273 2 When a matrix B can be written as a linear combination of a set of matrices AH12 Ak we say that B is linearly dependent upon AIA2AC If the matrix B cannot be written as a linear combination of a set of matrices AN12 Ak we say that B is inear x independent of AA2Ak When having discussions about linear dependence and independence it is almost always assumed that the matrices under discussion all have the same dimensions Example 5 73 71 0 Establish that the matrix C2 0 is linearly independent of the matrices A and 4313 l DC CLJW quotquotquotl l Unlurg llt 1 9 aquot A L6LUx led l S o C K 1 l Q l I 614 l ku CPA34 lXJ A one 5 amp Oar77 bl 6f Luj li f budlla quot s A M l is OQNJ cu0 A set of two or more matrices is said to be 39Ileal39 d2 wideIf if any matrix in the set is linearly dependent upon the remaining matrices in the set ie if any matrix in the set can be written as a linear combination of the remaining matrices in the set we say the set in linearly dependent A set of matrices is said to be IIIearz Integer039er if there is not any matrix in the set that can be written as a linear combination of the remaining matrices in the set Page 3 of 10 MTH 261 Mr Simonds class Example Consider The column vecTors A 3720t B4072t and C7960t a EsTablish ThaT B is linearly independenfof The vecTors A and C 3mm Kauzc KKE 4 Z T mmjgxrs lt1 0 on L0 b EsTablish never The less ThaT The seT ABC is linearly dependent UL 3 q 14 LAM u Example The concest of linear combinaTions linear dependence and linear independence can be applied To seTs of objecTs oTher maTrices For example we can app The definiTions To The seT of funcTions C g3A 408 whose domains are all real numbers Show ThaT The seT 111sinx2 1 lncosx2 1 111tan2 x2 is linearly dependenT L L 1 1L mum 1aquot 39l k X 44 2 L Ehquot K xkn mom q 1 AUW W 4 00169441 Q L 5 39J and mm L f L Page 4 of 10 MTH 261 Mr Simonds class The seT of all linear combinaTions of The members of a seT is called re seal of The seT Example Show ThaT The m x n zero mafrx he maTriX where every enTry is zero is in The span of every seT of m x n maTrices funde HW I is gA IA V quot39J Akg OOmwwwOW Example 76 3 7 Show ThaT The span of The seT AB where A 4 3 and B 8 5 I conTains no 6 maTrices where every enTry is posiTive furruh Lul39 tux17 wJ39A roJlJrlu TL ho 34 652 gt0 WW gun NJ It i Txvo My tum Yo 3KA M 175 gt szx MJ 21wlt a KA13 we r4 Q CwA r Ji china Page 5 of 10 MTH 261 Mr Simonds class Example Suppose Tha r A is linear dependen r on B and C bu r A is bo rh linearly independen r of B and linearly independen r of C Show Tha r B mus l be linearly dependenT on A and C A K G ULLFL mug 31M 94 llt 6 0 x B C K K Example 2 1 11 ShowThaTThecolumnsof ThemaTriX 73 1 724 arelinearly dependen r 5 6 17 ull ll lh 07 39quotf L39m S II La h UlV quot 395 39 M Q J k JL 39 59 45 K M C quotquotquotquot vrv pan4 L lt I ll 1 I a aquot Jl RM L gt 2KJZII 9 QKl JJl39 h 32 4391 J f l DEI 3K 4 1kf W l4 quot1 Kl01 35 ilt7 BAOKJ39JD 1L 44 1H X 393 3 BF JluLK UM a 04750 lt Q 3 39 33quot4 7 l Page 6 of 10 MTH 261 Mr Simonds class Gaussian elimination an introduction Introductory Example 6xy736 a Use the elimination method to solve the system of equations 2x73y8 x j 39 652 9 x 3j 37 EQx Szx ath 393 3El gt L 7j J7 0 rj JL 5M EfrenWE 2 Cx4j 17 Uj CU gt j L 3 c H N 3 34 zes b Rewrite the system from part a as an camelfed matrix and solve the system using this abbreviated representation of the system C IlJL 1 3 58 la 7 HQL c I NC I 1 3 398 Bk 39 Lat I o oquot o 2 x EL 7 bx 1 10 a Jam 3lt 3gt Page 7 of 10 MTH 261 Mr Simonds class The Gaussian eliminaTion process for solving sysTems of linear equaTions is predicaTed upon The Three eemenfarz row ogera o s The Three elemenTary row operaTions are i InTerchanging Two rows of The maTriX ii Replacing one row of The maTriX wiTh a non zero mulTiple of iTself iii Replacing one row of a maTriX wiTh iTself added To a mulTiple of anoTher row of The maTriX Two maTrices are said To be row eguValem if one maTriX can be Transformed inTo The oTher via a series of elemenTary row operaTions The firsT non zero enTry of any row of a maTriX is called re glyof enfrz of ThaT row A maTriX is said To be in row echelon forI7 if The maTriX saTisfies each of The following Two properTies i Every row ThaT conTains noThing buT zeros occurs aT The boTTom of The maTriX i39 The pivoT enTry of any non zero row appears To The righT of The pivoT enTry in The row direchy above iT The process of Transforming a maTriX inTo a row equivalenT maTriX of echelon form is called The aussan eimna on graces Example Use The meThod of Gaussian eliminaTion To find an echelon form for each of The following maTrices 2 75 73 723 x a 75 L 72 77 1 3 1 3 1 rJ L3 l 3 l 3 y I 39L 7 K 9 R r I 2 7 l 3 3 2 5 7 L3 RLA Laura I 3 3 0396 3 Z Rs ks 13921 0 r l 3 l 3 9K IL 3 7 nq f 3 ILQZ gb Fy LnL L Page 8 of 10 MTH 261 Mr Simonds class 1 72 0 1 4 b 72 3 72 2 73 39 0 71X20 711 5 74 6 L 10 I I z o I vil 3 1 1 3 R1 RT1RI 0 39o39 2 039I 2 92 0 I l 3939r L 39 Io RV V o agKL I 2 u 1 39 In 0 quotf T WhaT does The echelon form of This maTriX Tell you abouT The soluTion seT To The sysTem of equaTions modeled by The maTriX sTaTed in parT b Lo ylux LIA fXab39l39b DA Jo 4 quot LJl l L quotz w J M gt 031 Page 9 of 10 MTH 261 Mr Simonds class Example 2 75 73 723 WriTe down The sysTem implied by The maTriX 75 1 72 77 and use The echelon form 1 3 1 3 maTriX found in parT a of The IasT example To find The soluTion To The sysTern szfj 3139 J3 x 4 3 3 I lj 13931quot 7 4 139 5 7 s 37 3 z 211 3 11 gt6 9 71 J 73973 7 ILJ 3lt Y gt 3 5quot 3643I473 x51 PracTice HW Sedion 11 pp 15 19 1a 3 6 7 9 13 14 15 27 34 Page 10 of lO


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