Introduction to Linear Algebra using MATLAB Tutorial on Material Covered in ENG EK 127 Relevant to Linear Algebra.

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1 Introduction to Liner Algebr using MATLAB Tutoril on Mteril Covered in ENG EK 7 Relevnt to Liner Algebr By Stormy Attwy Reference: Stormy Attwy, MATLAB: A Prcticl Introduction to Progrmming nd Problem Solving, pp.45+x, Burlington, MA, Elsevier Inc., 9. MATLAB Bsics Windows nd Prompt Vribles nd Assignment Sttements Constnts Opertors nd Precedence Built-in functions, Help Types Vectors nd Mtrices in MATLAB Creting Vector nd Mtrix Vribles Row Vectors Column Vectors Mtrices Specil Mtrix Functions Referring to Elements Dimensions Vectors nd Mtrices s Function Arguments Mtrix Definitions Mtrix Opertions Arry Opertions (term-by-term) Mtrix Multipliction Inverse Mtrix Augmenttion Vector Opertions: Dot Product nd Cross Product Introduction to Liner Algebr Systems of Equtions Mtrix Form x Systems Elementry Row Opertions Guss Elimintion Guss-Jordn Elimintion Reduced Row Echelon Form (RREF) RREF to solve Ax=b for x RREF to find inverse The solve Function in the Symbolic Mth Toolbox Copyright August by Stormy Attwy Pge

2 MATLAB Bsics... Windows nd Prompt... Vribles nd Assignment Sttements... Constnts... 4 Opertors nd Precedence... 4 Built-in functions, Help... 5 Types...5 Vectors nd Mtrices in MATLAB... 6 Creting Vector nd Mtrix Vribles... 6 Row Vectors... 6 Column Vectors... 7 Mtrices... 7 Specil Mtrix Functions... 7 Referring to Elements... 8 Dimensions... 9 Vectors nd Mtrices s Function Arguments... Mtrix Definitions... Mtrix Opertions... Arry Opertions (term-by-term)... Mtrix Multipliction... Inverse... 4 Mtrix Augmenttion... 5 Vector Opertions: Dot Product nd Cross Product... 5 Intro to Liner Algebr: Systems of Equtions... 6 Mtrix Form... 6 x Systems... 7 Elementry Row Opertions... 9 Guss Elimintion... 9 Guss-Jordn Elimintion... Reduced Row Echelon Form (RREF)... Guss, Guss-Jordn, RREF Exmple... The solve Function in the Symbolic Mth Toolbox... 5 Copyright August by Stormy Attwy Pge

3 MATLAB Bsics Windows nd Prompt MATLAB cn be used in two bsic modes. In the Commnd Window, you cn use it interctively; you type commnd or expression nd get n immedite result. You cn lso write progrms, using scripts nd functions (both of which re stored in M-files). This document does not describe the progrmming constructs in MATLAB. When you get into MATLAB, the configurtion my vry depending on settings nd the Version number. However, you should hve these bsic windows: the Commnd Window: this is the min window nd is usully on the right the Workspce Window: shows current vribles the Current Directory Window: shows files, by defult in the work directory (the Current Directory is set using the pull-down menu bove the Commnd Window) the Commnd History Window: shows commnds tht hve been entered in the Commnd Window In the Commnd Window, you should see >> which is the prompt. Any commnd cn be entered t the prompt. Vribles nd Assignment Sttements In order to store vlues, either in n M-file or the Commnd Window, vribles re used. The simplest wy to put vlue into vrible is to use n ssignment sttement. An ssignment sttement tkes the form vrible = expression where the nme of the vrible is on the left-hnd side, the expression is on the righthnd side, nd the ssignment opertor is the =. Vribles do not hve to be declred in MATLAB. Vribles tht re defined cn be seen in the Workspce Window. In the Commnd Window, when n ssignment sttement is entered t the prompt, MATLAB responds by showing the vlue tht ws stored in the vrible. For exmple, >> mynum = 5 + mynum = 8 >> Here, the user entered mynum = 5 + nd MATLAB responded tht it hd stored in the vrible mynum the result of the expression, 8. Putting semicolon t the end of sttement will suppress the output from tht sttement, mening tht the commnd will be crried out, but MATLAB will not show the result. Here is n exmple of initilizing vrible count to, but suppressing the output, nd then dding one to the vlue of the vrible (nd not suppressing tht output): Copyright August by Stormy Attwy Pge

4 >> count = ; >> count = count + count = MATLAB hs defult vrible, clled ns. Any time n expression is entered in the Commnd Window, without ssigning it to vrible, MATLAB will ssign the result to the vrible ns. >> 7- ns = 4 There re commnds tht cn be used to see wht vribles hve been creted in given session, nd to delete vribles. The commnds who nd whos will disply the current vribles (whos gives more informtion). The cler commnd cn be used to delete some or ll vribles. Constnts There re built-in constnts in MATLAB, including: pi.459. i, j inf infinity NN stnds for not number ; e.g. the result of / (Techniclly, these re built-in functions tht return the vlues shown.) Opertors nd Precedence There re mny opertors in MATLAB, which cn be used in expressions. Some of the rithmetic opertors include: + ddition - negtion, subtrction * multipliction / division (divided by) \ division (divided into e.g. \ is 4) ^ exponentition (e.g. ^ is 9) There is hierrchy, or set of precedence rules, for the opertors. For exmple, for the opertors shown here, the precedence is (from highest to lowest): () prentheses ^ exponentition - negtion *, /, \ ll multipliction nd division +, - ddition nd subtrction Copyright August by Stormy Attwy Pge 4

5 Built-in functions, Help MATLAB hs mny built-in functions. It hs mny mthemticl functions, e.g. bs for bsolute vlue, tn for tngent, etc. The functions re grouped together logiclly in wht re clled help topics. The help commnd cn be used in MATLAB to find out wht functions re built-in, nd how to use them. Just typing help t the prompt will show list of the help topics, the beginning of which is displyed here: >> help HELP topics: mtlb\generl mtlb\ops mtlb\lng mtlb\elmt mtlb\elfun mtlb\specfun mtlb\mtfun - Generl purpose commnds. - Opertors nd specil chrcters. - Progrmming lnguge constructs. - Elementry mtrices nd mtrix mnipultion. - Elementry mth functions. - Specilized mth functions. - Mtrix functions - numericl liner lgebr. Typing help nd the nme of help topic (the mtlb\ is not necessry) will show the functions tht re contined in tht prticulr grouping. For exmple, the beginning of the list of opertors nd specil chrcters is shown here: >> help ops Opertors nd specil chrcters. Arithmetic opertors. plus - Plus + Typing help nd the nme of function will explin the use of the function. There re mny other functions in Toolboxes with re purchsed seprtely. Types There re mny built-in types in MATLAB. These re clled clsses ; you cn see the clss of vribles when using the whos commnd or in the Workspce window. By defult, the type of numericl expressions nd vribles is double, which is double precision type for flot, or rel, numbers. Even when using n integer expression, the defult type is double. There is lso the flot type single. For integers, there re mny built-in types: int8, int6, int, int64; the number in these types represents the number of bits tht cn be stored in vrible of tht type. There re lso unsigned integer types, for exmple, uint8 is used in true color imge mtrices. Other types include chr for single chrcters (e.g. x ) or chrcter strings (e.g. hi ), nd logicl for the result of reltionl, or Boolen, expressions. Copyright August by Stormy Attwy Pge 5

6 All of these type nmes re lso nmes of functions tht cn be used to convert vrible or expression to tht type. For exmple: >> cler >> myvr = 5 + 4; >> intvr = int(myvr); >> whos Nme Size Bytes Clss intvr x 4 int rry myvr x 8 double rry Vectors nd Mtrices in MATLAB MATLAB is written to work with vectors nd mtrices; the nme MATLAB is short for Mtrix Lbortory. A mtrix looks like tble with rows nd columns; n m by n (or m x n) mtrix hs m rows by n columns (these re the dimensions of the mtrix). Vectors re specil cse in which one of the dimensions is : row vector is single row, or in other words it is by n ( row by n columns), nd column vector is m by (m rows by column). A sclr is n even simpler cse; it is by mtrix, or in other words, single vlue. As seen from the whos commnd, the defult in MATLAB is to tret single vlues s x mtrix. Creting Vector nd Mtrix Vribles Row Vectors Row vector vribles cn be creted in severl wys. The simplest method is to put the vlues tht you wnt in the vrible in squre brckets, seprted by either spces or comms: >> rowvec = [ 5 7] rowvec = 5 7 This cretes x 4 row vector rowvec, or in other words, row vector with four elements. The colon opertor cn be used to crete row vector tht itertes from the strting to ending vlue with defult step of one; in this cse, the squre brckets re not necessry: >> itvec = :7 itvec = Copyright August by Stormy Attwy Pge 6

7 A step vlue cn lso be specified: >> stepvec = :: stepvec = Column Vectors There re two bsic methods for creting column vector: either by putting the vlues in squre brckets, seprted by semicolons, or by creting row vector nd then trnsposing it. The trnspose opertor in MATLAB is the postrophe. For exmple: >> colvec = [;7;] colvec = 7 >> rowvec = [:5 44]; >> cvec = rowvec cvec = Mtrices Mtrix vribles cn be creted by putting the vlues in squre brckets; the vlues within the rows re seprted by either spces or comms, nd the rows re seprted by semicolons. One thing is very importnt: there must lwys be the sme number of vlues in every row. The colon opertor cn be used to iterte within the rows. >> mt = [:; 5 ; 6:-:4] mt = Specil Mtrix Functions There re mny built-in functions in MATLAB tht crete specil mtrices. Some of them re listed here: zeros(m,n) cretes n m x n mtrix of ll zeros zeros(n) cretes n n x n mtrix of zeros ones(m,n) cretes n m x n mtrix of ll ones ones(n) cretes n n x n mtrix of ones Copyright August by Stormy Attwy Pge 7

8 eye(n) cretes n n x n identity mtrix (ll zeros but ones on the digonl) mgic(n) cretes n n x n mgic mtrix (sum of ll rows, columns, nd digonl re the sme) rnd(n) cretes n n x n mtrix of rndom rel numbers, ech in the rnge from to rnd(m,n) cretes n m x n mtrix of rndom rel numbers, ech in the rnge from to Note: some versions of MATLAB lso hve function to crete mtrix of rndom integers, clled either rndint or rndi. Use help to see whether it is vilble in your version nd if so, how to use it. Referring to Elements Once you hve creted vector or mtrix vrible, it is necessry to know how to refer to individul elements nd to subsets of the vector or mtrix. In this section, we will use the following vribles to illustrte: >> vec = 4::4 vec = >> mt = [:; 5 ; 6:-:4] mt = For vector (row or column), referring to n individul element is done by giving the nme of the vrible nd in prentheses the number of the element (which is clled the index or the subscript). In MATLAB, the elements of vector re numbered from to the length of the vector. For mtrix, both the row nd the column index must be given in prentheses, seprted by comm (lwys the row index nd then the column index). >> vec() ns = 8 >> mt(,) ns = The colon opertor cn be used to iterte through subscripts. For exmple, this sys the third through fifth elements of the vector (which cretes nother row vector): >> vec(:5) ns = 8 Copyright August by Stormy Attwy Pge 8

9 Using just the colon by itself for row or column index mens ll of them. For exmple, this sys ll rows within the second column, or in other words, the entire second column: >> mt(:,) ns = 5 There is lso specil keyword end which cn be used to refer to the lst element in vector, or the lst row nd/or column in mtrix (for exmple, below, the lst element in the first row): >> mt(,end) ns = These methods of referring to elements cn be used to modify elements nd in some cses to delete them. To delete element(s), the empty vector [] is ssigned. This exmple cretes vector, modifies vlue in it, nd then deletes two elements: >> newvec = [ ]; >> newvec(4) = 49 newvec = >> newvec(:) = [] newvec = Note: this cn be done with mtrices lso, s long s every row in the mtrix hs the sme number of vlues. This mens, for exmple, tht you could not delete n individul element from mtrix, but you could delete n entire row or column. Dimensions There re severl functions tht re used to determine the number of elements in nd the dimensions of vribles. Assuming vector vrible vec nd mtrix vrible mt, numel(vec) returns the number of elements in the vector numel(mt) returns the totl number of elements in the mtrix (the product of the number of rows nd the number of columns) length(vec) returns the number of elements in the vector length(mt) returns either the number of rows or the number of columns in the mtrix, whichever is lrger size(mt) returns the number of rows nd the number of columns of the mtrix size(vec) returns the number of rows nd the number of columns of the vector Copyright August by Stormy Attwy Pge 9

10 The size function brings up couple of unique nd importnt concepts in MATLAB: A function cn return more thn one vlue It is possible to hve vector contining more thn one vlue on the left-hnd side of n ssignment sttement. This cn be used to store the vlues tht re returned from function. For exmple, to store the number of rows in mtrix in vrible r nd the number of columns in vrible c, vector with the vribles r nd c is on the left-hnd side of n ssignment sttement tht uses the size function to return both of these vlues: >> mt = [7 5; 4:7] mt = >> [r c] = size(mt) r = c = 4 In generl, it is NOT good prctice to ssume the dimensions of vector or mtrix. For vector vrible, typiclly either numel or length is used to determine the number of elements in the vector. For mtrix, the ssignment sttement shown bove with size is generlly used since it is frequently useful to hve the number of rows nd columns stored in seprte vribles. There re lso mny functions in MATLAB tht cn chnge the orienttion or dimensions of mtrix: reshpe(mt,m,n) reshpes the mtrix s n m x n mtrix (the number of elements must be the sme) by filling the new mtrix with the vlues from the old one column t time fliplr(mt) flips the columns from left to right flipud(mt) flips the rows up nd down Vectors nd Mtrices s Function Arguments In MATLAB, n entire vector or mtrix cn be pssed s n rgument to function, nd the function will evlute the function on every element, nd return s result vector or mtrix with the sme dimensions s the input rgument. For exmple, >> v = :5; >> exp(v) ns = Copyright August by Stormy Attwy Pge

11 >> mt = [- 4 ; - ] mt = >> bs(mt) ns = 4 For some functions, however, MATLAB will operte column-wise on mtrix rguments. For exmple, there is function sum tht will sum the elements in vector. For mtrix, however, it will sum ech individul column. Using the vribles v nd mt shown bove, >> sum(v) ns = 5 >> sum(mt) ns = - Mtrix Definitions There re mny definitions tht relte to mtrices, nd MATLAB hs relevnt functions for mny of them. Two mtrices re sid to be equl to ech other if they hve the sme dimensions, nd ll corresponding elements re equl. In MATLAB, there is function isequl tht will receive two mtrix rguments nd will return logicl for true if they re equl, or logicl for flse if not. A mtrix is squre if the number of rows is the sme s the number of columns. There re definitions tht pply only to squre mtrices. The digonl of squre mtrix is the set of elements from the upper left corner to the lower right; these re the elements for which the row nd column indices re the sme. The trce of squre mtrix is the sum of the elements on the digonl. A digonl mtrix is mtrix for which ll of the elements tht re not on the digonl re zeros. There re severl functions relted to squre mtrices: trce(squremt) returns the trce of squre mtrix dig(squremt) returns the digonl of squre mtrix s vector dig(vec) cretes digonl mtrix by creting mtrix of ll zeros nd putting the vector vec on the digonl An upper tringulr mtrix is squre mtrix is mtrix tht hs ll zeros below the digonl. A lower tringulr mtrix is squre mtrix tht hs ll zeros bove the digonl. MATLAB hs functions triu(mt) nd tril(mt) tht will receive mtrix Copyright August by Stormy Attwy Pge

12 rgument n will crete n upper or lower tringulr mtrix, respectively, by replcing elements with zeros s pproprite. The digonl of mtrix is sometimes clled the min digonl. There cn be subdigonls. For exmple, tridigonl mtrix is mtrix tht hs min digonl nd sub-digonl directly below the min digonl nd nother directly bove it. Mtrix Opertions Mthemticl opertions cn be performed on mtrices; MATLAB hs built-in opertors nd/or functions for them. Since MATLAB is written to work on mtrices, loops re not necessry for mtrix opertion. For exmple, sclr multipliction mens multiplying every element in mtrix by sclr vlue. In MATLAB, this is ccomplished simply using the * opertor: >> mt = reshpe(:8,,4) mt = >> mt * 5 ns = Arry Opertions (term-by-term) Mtrix opertions re clled rry opertions if they operte on two mtrices the two mtrices hve the sme dimensions the opertions re performed term-by-term; in other words, on corresponding elements For rry ddition nd subtrction, the + nd - opertors re used. For exmple, for rry ddition: 8 A + B = C = >> A = [:5; 8:] A = >> B = reshpe(:6,,) B = Copyright August by Stormy Attwy Pge

13 >> C = A + B C = For ny opertion tht is bsed on multipliction (this mens multipliction, division, nd exponentition), the rry opertor hs dot (. ) in front of it. For exmple, using the mtrices A nd B bove: >> timesb = A.* B timesb = >> risedtob = A.^ B risedtob = Mtrix Multipliction Mtrix multipliction is VERY different from the rry multipliction defined bove. If mtrix A hs dimensions m x n, then in order to be ble to multiply it by mtrix B, B must hve dimensions n x something; we ll cll the column dimension p. In other words, in order to multiply A * B, the number of rows of B hs to be the sme s the number of columns of A. We sy tht the inner dimensions must gree. The resulting mtrix, C, will hve s its dimensions m x p (the outer dimensions ). So, [A] mxn [B] nxp = [C] mxp This just explins the dimensions; it does not yet describe how the elements in the mtrix C re derived. Every element in C is the result of summing the products of corresponding elements in the rows of A nd the columns of B. Any given element in C, C ij, is defined s C ij = n ik b k kj For exmple, 8 A * B = C * = The first element in C, C, is found by multiplying corresponding elements in the first row of A nd the first column of B, nd summing these, e.g. * + 4* + 5* = 6. In MATLAB, mtrix multipliction is ccomplished using the * opertor. Copyright August by Stormy Attwy Pge

14 >> A = [:5; 8:] A = >> B = reshpe(:6,,) B = >> C = A * B C = Note tht for squre mtrices, multiplying mtrix A by n identity mtrix I with the sme size results in the mtrix A (so, multiplying mtrix by I is similr to multiplying sclr by ; it doesn t chnge it). >> A = mgic() A = >> A * eye() ns = Inverse The definition of the inverse of mtrix is tht when multiplying mtrix by its inverse, the result is the identity mtrix. Mthemticlly, we would write this [A] [A - ] = [I]. MATLAB hs built-in function to find n inverse, inv. >> A = [ ; 4] A = 4 >> inv(a) ns = >> A * inv(a) ns = Copyright August by Stormy Attwy Pge 4

15 Mtrix Augmenttion Mtrix ugmenttion mens tking mtrix nd ugmenting by dding more columns, or nother mtrix, to it. Note tht the dimensions must be correct in order to do this. In MATLAB, this is clled conctenting nd cn be done for either vector or mtrix by putting the two together in squre brckets. Here is n exmple in which mtrix A is ugmented with n identity mtrix with the sme size of A; note tht cn be obtined with eye(size(a)): >> A = [ ; 4] A = 4 >> [A eye(size(a))] ns = 4 Vector Opertions: Dot Product nd Cross Product As long s the dimensions re correct, some of the definitions nd opertions given on mtrices bove re lso vlid for vectors, since vectors re just specil cse of mtrices. There re some opertions, however, tht re only vlid for vectors, including the dot product nd cross product. For two vectors A nd B tht hve the sme length, the dot product is written A B nd is defined s n Ai B i i = A B + A B + A B + + A n B n where n is the length of the vectors. In MATLAB, there is function dot to ccomplish this. The cross product AxB is defined only if A nd B re vectors of length, nd cn be written using the following mtrix multipliction: A A B A x B = A A B = [A B -A B, A B -A B, A B -A B ] A A B MATLAB hs built-in function cross for the cross product. Copyright August by Stormy Attwy Pge 5

16 Intro to Liner Algebr: Systems of Equtions A system of liner lgebric equtions is of the form x + x + x +. + n x n = b x + x + x +. + n x n = b x + x + x +. + n x n = b m x + m x + m x +. + mn x n = b m where the s re the coefficients, the x s re the unknowns, nd the b s re constnt vlues. Using MATLAB, there re two bsic methods for solving such set of equtions: By putting it in mtrix form Using the solve function which is prt of the Symbolic Mth Toolbox Mtrix Form Becuse of the wy tht mtrix multipliction works, the system of equtions shown bove cn be written s the product of mtrix of the coefficients A nd column vector of the unknowns x: m m A x = b n x b n x b n x = b m mn x n b m We hve mtrix multipliction eqution of the form Ax = b, nd we wnt to solve for the unknowns x. This cn be ccomplished s follows: A x = b A - A x = A - b I x = A - b x = A - b So, the solution cn be found s product of the inverse of A nd the column vector b. Copyright August by Stormy Attwy Pge 6

17 In MATLAB, there re two wys of doing this, using the built-in inv function nd mtrix multipliction, nd lso using the \ opertor: >> A = [ 4 ; - ; 4] A = 4-4 >> b = [ ] b = >> x = inv(a) * b x = >> A\b ns = x Systems The simplest system is x system, with just two equtions nd two unknowns. For these systems, there is simple definition for the inverse of mtrix, which uses the determinnt D of the mtrix. For coefficient mtrix A defined generlly s A =, the determinnt D is defined s. The inverse A - = D Copyright August by Stormy Attwy Pge 7

18 For exmple, for the system x + x = - x + 4x = This would be written in mtrix form s 4 x x = The determinnt D = *4 -* = -. The inverse A - = 4 / = / x / So the solution is: = = x / / 5/ MATLAB hs built-in function det to find the determinnt of mtrix. >> A = [ ; 4] A = 4 >> b = [-;] b = - >> det(a) ns = - >> inv(a) ns = >> x = inv(a) * b x = Copyright August by Stormy Attwy Pge 8

19 Elementry Row Opertions Although finding the inverse of x mtrix is strightforwrd, s is using it to find the unknowns, it is not so simple for lrger systems of equtions. Therefore, we resort to other methods of solution. Once the system is in mtrix form, some of these methods involve trnsforming mtrices using wht re clled Elementry Row Opertions, or EROs. The importnt thing to understnd bout these EROs is tht using them to modify mtrix does not chnge wht the solution to the set of equtions will be. There re EROs:. Scling: multiplying row by sclr (mening, multiplying every element in the row by the sme sclr). This is written sr i r i, which indictes tht row i is modified by multiplying it by sclr s.. Interchnge: interchnging the loctions of two rows. This is written s r i r j which indictes tht rows i nd j re interchnged.. Replcement: replcing ll of the elements in one row with tht row plus or minus sclr multiplied by nother row. This is written s r i +/- sr j r i These EROs form the bsis for the methods described next. Guss Elimintion The Guss Elimintion method is method for solving mtrix eqution Ax=b for x. The process is:. Strt by ugmenting the mtrix A with the column vector b.. Perform EROs to trnsform this ugmented mtrix to upper tringulr form.. Use bck-substitution to solve for the unknowns in x. For exmple, for x system, the first step is to ugment the mtrix [A b]: b b Then, EROs re pplied to get the ugmented mtrix into n upper tringulr form (which, for x mtrix mens finding n ERO tht would chnge to ): b b Here, the primes indicte tht the vlues my hve been chnged. Putting this bck into the eqution form, we hve x x = b b Copyright August by Stormy Attwy Pge 9

20 So, we now use the lst line which sys x = b to solve first for x : x = b / nd then go bck to the first line which sys to solve for x : x + x = b x = (b x ) / Guss-Jordn Elimintion The Guss-Jordn Elimintion method strts exctly the sme wy s the Guss Elimintion method, but insted of bck-substitution to solve for x, EROs re used to get the ugmented mtrix into digonl form.. Strt by ugmenting the mtrix A with the column vector b.. Perform EROs to trnsform this ugmented mtrix to upper tringulr form (forwrd elimintion).. Use bck elimintion (perform more EROs) to get to digonl form 4. Solve for the unknowns in x. For exmple, for x mtrix, the mtrix is first ugmented: b b b EROs re then performed using forwrd elimintion to get it to upper tringulr form: b b b At this point, the Guss method would then use bck-substitution to solve first for x, then x, then x. Insted, the Guss-Jordn method continues with bck elimintion to get it to digonl form: Copyright August by Stormy Attwy Pge

21 b b b Then, the unknowns cn be found esily, in ny order using these equtions: x i = b i / ii Reduced Row Echelon Form (RREF) Reduced Row Echelon Form (RREF) tkes the Guss-Jordn elimintion method one step further by performing scling EROs on ll rows so tht the ii coefficients on the digonl ll become ones. RREF to solve Ax=b for x To use the RREF method to solve the mtrix eqution Ax = b for x, the mtrix A is ugmented with b, nd then the Guss-Jordn method is followed. The finl step is to scle ll rows. For exmple, for x mtrix, b b b b b b Then, the solution is simply the lst column. MATLAB hs function rref to ccomplish this. RREF to find inverse This technique nd function cn lso be used to find the inverse of mtrix A. The procedure is: Augment A with n identity mtrix the sme size s A Follow the procedure bove using EROs to reduce the left-side to n identity mtrix; the right side will be the mtrix inverse. For exmple, for x mtrix, strt with [A I]: nd then reduce this to: Copyright August by Stormy Attwy Pge

22 r r r r r r r r r which is [I A - ]. Guss, Guss-Jordn, RREF Exmple Since the Guss, Guss-Jordn nd RREF methods to solve Ax = b for x ll begin the sme wy, we will demonstrte ll with one exmple. The following x system of equtions: x + x + x = x = x + x = cn be written in mtrix form: x x x = To solve for x, we strt with the ugmented mtrix [A b] nd perform forwrd elimintion by finding EROs to get it to upper tringulr form: r / r r 4 / / 8/ r -4/ r r / Now, for the Guss method, we use bck substitution: -/x = so x = x = x + x + x = x + - = x = - x = - Insted for Guss-Jordn, we continue with bck elimintion: Copyright August by Stormy Attwy Pge

23 / r + r r r -r r / Once in digonl form, we cn solve: x = - so x = - x = -x = so x = For RREF, scle ll rows: / r r -r r So now the lst column is x. We cn lso use RREF to find A - nd solve tht wy. Strt with the ugmented mtrix [A I]: r r r r r 4 r + 4r r 4 r - r r 4 So A - = nd x = = 4 4 Copyright August by Stormy Attwy Pge

24 In MATLAB, to solve Ax=b we begin by ugmenting [A b]: >> A = [ ; ; ]; >> b = [:]; >> Ab = [A b] Ab = Performing n ERO in MATLAB cn be ccomplished by using ssignment sttements to modify rows. For exmple this ERO r / r r 4 / / 8/ is done with: >> Ab(,:) = Ab(,:) - / * Ab(,:) Ab = MATLAB hs function rref to reduce: >> rref(ab) ns = - In MATLAB, we cn find the inverse of mtrix using rref nd then check tht using the inv function: >> rref([a eye(size(a))]) ns = >> inv(a) ns = Copyright August by Stormy Attwy Pge 4

25 The solve Function in the Symbolic Mth Toolbox If you hve the Symbolic Mth Toolbox, the solve eqution cn lso be used to solve sets of equtions. The solution is returned s structure. Structures re similr to vectors in tht they cn store multiple vlues. However, insted of hving elements, structures hve fields tht re nmed. While indexing is used to refer to elements of vector, the dot opertor is used to refer to the fields in structure vrible. As n exmple, we will solve the following x system of equtions tht ws used in the previous section: x + x + x = x = x + x = For simplicity, we will chnge the x,x,x to x,y,z, nd pss the three equtions s strings to the solve function: >> result = solve(*x+*y+z=, y=, x+*y=) result = x: [x sym] y: [x sym] z: [x sym] This stores the unknowns in fields clled x, y, nd z within the result vrible. The dot opertor is used to see the ctul vlues, which re stored s symbolic expressions. The double function cn convert them to numbers: >> result.x ns = - >> x = double([result.x result.y result.z]) x = - Copyright August by Stormy Attwy Pge 5