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Solution Manual For Introduction To Linear Algebra, 5th Edition

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    Marie Vanisko Douglas B. Meade Introduction to Linear Algebra a FIFTH EDITION : Lee W.Iohnson ® R. Deon Rips m Jimmy I. fmold SoC ee ela Ee Ee

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    Downloaded from StudyXY.com ® + StudyXY Sd Ye. o> \ | iF ’ pr E \ 3 S Stu dy Anything This ContentHas been Posted On StudyXY.com as supplementary learning material. StudyXY does not endrose any university, college or publisher. Allmaterials posted are under the liability of the contributors. wv 8) www.studyxy.com

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    BASIC OPERATIONS 1 _— Maple Technology Resource Manual Maple is a popular and powerful software tool used for the numeric, symbolic, and graphic analysis of mathematical problems. Unlike most traditional computational software, a computer algebra system (CAS) such as Maple is designed to perform exact symbolic manipulations of mathematical objects, including vectors and matrices. The purpose of this supplement is to introduce the basic Maple commands needed to start using Maple as a tool for exploring the linear algebra topics developed in this text. Maple, like any other computer algebra system (CAS), is designed to perform symbolic computations. That is, Maple can manipulate and process mathematical objects such as expressions, equations, sets, vectors, and matrices using the rules of algebra thereby avoiding many of the numerical issues that are involved in most floating-point numeric computations. Maple can also perform numeric computations, but the focus of this supplement is on the exact symbolic computations related to linear algebra. 1 Basic Operations Launch Maple. When Maple opens a large window will appear. This window provides an environment for creating, modifying, and viewing Maple documents, called worksheets or .mws files. A Maple worksheet can contain text, input, output, and graphics (including animations). Worksheets are platform-independent ASCII files that are easily transferred between users via e-mail or the WWW. Maple can be used for simple computations, such as the following: > 8 * 17 + 49; 185 > alpha := sin( Pi/3 ); 1 a=3 V3 If, instead of a semi-colon, a colon is used to terminate a Maple command, the output of the command is suppressed: > beta := sqrt( 1 + 7/10*alpha ): If neither a semi-colon nor a colon is used to terminate a Maple command, an error message will be generated: > beta”2 Warning, premature end of input ‘When this occurs, it is not necessary to re-type the command. Simply position the cursor at the end of the command, type the appropriate terminal character and re-execute the command. Note that assignments are made using “:=" and “=” is used to construct equations: > beta = evalf( beta ); 1 75 V100 +353 = 1267366475 Maple has an extensive online help database. Type help(topic); or the shorthand ?topic to get help on a specific topic. For a first look at Maple’s linear algebra capabilities, enter and execute the command ?linalg. The Help pull-down menu in the menu bar of Maple’s graphical user interface provides search tools for accessing Maple’s help documents when the specific topic or Maple command name is not known. — StudyXY

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    2 MAPLE TECHNOLOGY RESOURCE MANUAL In order to have a means of accessing previous results it is a good idea to assign results to unique (and meaningful) names. Maple names are case-sensitive (e.g., the constant 7 is Pi, not pi nor PI). See ?names for additional information about valid names in Maple. Exercises for §1 2. Evaluate tan 14.32. Hint: See ?math for examples showing the use 3. Evaluate ( [sin 4 1)* using exact of some of Maple’s basic mathematical arithmetic, using 32-, 16-, 8-, 4, 2-, and functions. 1-digit floating-point arithmetic. (Note 1. Evaluate 3.132 + | cos v/7|. that the results are not always the same as rounding this number to the specified number of digits.) 2 Entering Matrices Suppose we wish to enter the matrix 1-10 — 1 1 A=|1 3 1]. 0 ee 1 Prior to entering the matrix it is necessary to use the command with( linalg ); to load Maple’s linalg package. To enter A type the command: > A := matrix( 3, 3, [ 1, -1, 0, 1/2, 3, 1/3, 0, exp(Pi), 1] ); 1-10 1 1 A=|3 3 3 0 e™ 1 The first two arguments of this matrix command specify the number of rows and columns in the matrix. If the matrix dimensions ere not stated explicitly, extra square brackets (“[”, |”) must be used to separate the rows: > Ac:=matrix( [ [ 1, -1, 03, [ 1/2, 3, 1./3 1, [ 0, exp(Pi), 111); 1 -1 0 1 A= 5 3 .3333333333 0 em 1 See ?matrix for a full description of the syntax of the matrix command. Notice that the (2,3) entry of A has been converted to a floating-point approximation, but not the other fraction or the exponential function. In general, Maple preserves exact values until a floating- point number appears in an expression or until the evalf command is used. A floating-point version of this matrix, with five (5) significant digits, is obtained with: > evalf( evalm( A ), 5); 1. -1 0. .50000 3. .33333 0. 23.141 1. The default number of significant digits is contained in the Maple name Digits. Unless explicitly changed, all floating-point computations are performed with ten (10) significant digits. That is, at the start of a Maple session, the assignment Digits:=10: is executed. — StudyXY

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    THE GAUSSELIM AND GAUSSJORD COMMANDS 3 The row and column indices can be used to access individual elements of a matrix. For example, to negate the (2,1) element of A, type: > A[2,1] := - A[2,1]; -1 Ag i= 5 To see the current value of a matrix, use the evalm command: > evalm( A ); 1-1 0 -1 5 3 .3333333333 0 e" 1 Exercises for §2 2. Obtain the floating-point version of the 1. Create the matrix Q by changing the matrix Q created in Exercise 1 (1,3) entry of A from 0 to 2. 3 The gausselim and gaussjord Commands The Maple commands gausselim and gaussjord use elementary row operations to transform a matrix to echelon form and reduced echelon form, respectively. To illustrate, consider the problem of solving the system zy +229 +223 + x4 = 3 2z7 + 4p + 3x3 +44 = 6 The pair of linear equations can be entered in Maple as: > eql := x[1] + 2xx[2] + 2%x[3] + x[4] = 3: > eq2 := 2xx[1] + 4xx[2] + 3*x[3] + 4*x[4] = 6: The augmented coefficient matrix is: > A := genmatrix( [eql,eq2], [x[i] $ i=1..4], flag ); 12213 A= | 24346 The reduced echelon form of the augmented matrix is: > Ar := gaussjord( A); 120 53 a=} 01 -2 H Thus, the original system is equivalent to Bx = b with: > B := submatrix( Ar, 1..2, 1..4); > b := convert( submatrix( Ar, 1..2, [5] ), vector ); [tr 20 5 B= E 01 —2 b:=[3,0] — StudyXY

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    4 MAPLE TECHNOLOGY RESOURCE MANUAL The two linear equations in the reduced system are: > sol := geneqns( B, x, b); sol :={z1 +222 +524 =3, 23 — 24 = 0} From which the solutions are seen to be: > solve( sol, {x[11,x[3]1}); {#1 =—229 — 524 +3, 23 = 224} The linsolve command returns the general solution when the system has an infinite number of solutions and NULL (the empty response) when the system of equations is inconsistent. Exercises for §3 2. Solve the system in Exercise 31 (Section 1. Solve the system in Exercise 30 (Section 1.2). 1.2). 4 Surgery on Matrices Maple provides convenient tools for the manipulation of individual elements and submatrices (blocks) of existing matrices to create new matrices. To illustrate, consider the 3 x 4 matrix: > B :=matrix(3, 4, [ [ 1,3, -5,0], > [9,6,-2,11, > [o,4, 4,111): To extract an individual element of B, say by 3, use: > b23 := B[2,3]; b23 := —2 To extract the 2 x 2 block from the upper-right corner of B, type: > Bl := submatrix( B, [1,2], [3,4] ); -5 0 NET) This command instructs Maple to define a new matrix, B1, made from rows 1 and 2 of B and columns 3 and 4 of B. An equivalent way to achieve the same result is: > Bl := submatrix( B, 1..2, 3..4); -5 0 NET] The range notation lo .. hi is an abbreviation for all integers beginning with lo and ending with hi. Similarly, a matrix @ could be assigned to be the upper-right 2 x 3 block of B with the command: > Q := submatrix( B, [1,2], 2..4); 3 -5 0 Q:= | 6 -2 1 To define P to consist of columns 1, 3, and 4 of rows 1 and 2 of B, use: > P := submatrix( B, [1,2], [1,3,4]); 1 -5 0 P= 9 -2 1 — StudyXY

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    ELEMENTARY ROW OPERATIONS ON MATRICES 5 Note that the range notation could not be used to select the nonconsecutive columns 1, 3, and 4. (The definition of P could use [ 1, $(3..4) | as the third argument to the submatrix command.) To form a 2 x 4 matrix from the upper-left and lower-right 2 x 2 blocks of B, construct the two blocks separately and then place them side-by-side with the commands: > C1 := submatrix( B, 1..2, 1..2 ): > C2 := submatrix( B, 2..3, 3..4 ): > C := augment( C1, C2); [13-21 C= 96 41 The augment command creates a composite matrix by placing the arguments side-by-side provided each argument has the same number of rows. To stack C1 on top of C2, use: > C3 := stackmatrix( C1, C2); 13 9 6 C3 := 21 41 . 13 Exercises for §4 Lig Consider the matrix 0 8 —6 1 342-8 2034 7 0 085 ~6 13 —7 A= 2 343 7. -1 -1 06 4 13 4 4 433 -7 3 00 8 T1437 Use matrix surgery to create each of the 06 4 following matrices. 5 Elementary Row Operations on Matrices Gauss and Gauss-Jordan elimination can be completed in one step with the gausselim and gaussjordan commands, respectively. However, to obtain a thorough understanding of these algorithms it is necessary to practice deciding on the elementary row operation to be performed at each step. The three Maple commands that implement the three elementary row operations are swaprow, mulrow, and addrow. To illustrate, consider the transformation of A to reduced echelon form where 0213 A=12 316]. 3222 — StudyXY

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    6 MAPLE TECHNOLOGY RESOURCE MANUAL The original matrix is: > A := matrix( 3, 4, [0,2,1,3,2,3,1,6,3,2,2,2]1): First, to have a non-zero pivot in the first row, interchange rows 1 and 2 of A: > Al := swaprow( A, 1, 2); 2316 Al:=[0 2 1 3 322 2 The next step is to create a leading 1 in the first row, multiply row 1 of Al by 1/2: > A2 := mulrow( Al, 1, 1/A1[1,1] ); 3 1 12 = A2 = 22 ’ “10 2 13 3 2 2 2 To eliminate the 3 in the (1,3) position, add -3 times row 1 to row 3 (of A2): > A3 := addrow( A2, 1, 3, -A2(3,1] ); 3 1 1 3 3 3 A43=10 2 1 3 —5 Note that in each step the result of the row operation is assigned to a new name. While this is not required, it is recommended so that the state at each step is available in the event it is necessary to investigate a different sequence of steps or to correct an error. Maple Command swaprow( A, i, j ) | interchange rows 4 and j of matrix A mulrow( A, i, r) multiply row 7 of matrix A by 7 addrow( A, j, i, r ) | replace row i of matrix A with the sum of row i and 7 times row j Exercises for §5 3. Use elementary row operations to put the 1. Complete the reduction of matrix A to following matrix into reduced row eche- lon form. reduced row echelon form. 2. C lete Exercise 44 in Section 1.2 or 2 . Complete Exerc in Section 1.2. B=]1 1 1 1 3 2 6 0 6 Graphing Curves To graph the Maple expression: > F := cos(3%x+1) + sin(2¥x-3); F:=cos(3z +1) +sin(2z — 3) for 0 < z < 10, use — Study

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    GRAPHING CURVES 7 > plot( F, x=0..10 ); 18 7 A AY B / | / | i | os | i J | \ SET NP BIS BN PU ST of 1 7 i i ERAN / esq) \VARRVi \/ NERY REI i. Sal i RR \ To add a plot of y = sinz to this plot, modify the command to: > plot( [ F, sin(x) J, x=0..10 ); ne A A IA i [ [4 p [ fh Ad [A] [ERE 0.51 Por [A BE [ER JE Ii TERE gl i \ lal PR a |e | AE AN AR BY BRAN 4 Se RAR VAR VAR I BVA HE : vod fr) | RN LJ SY, V4 An alternate method for combining plots is to use the display command from the plots package. For example, to add the plots of y = sin and y = cosz on the interval [0,27] to the first graph (on the larger interval [0,10] ) use: > pi :=plot( F, x=0..10 ): > p2 := plot( [cos(x),sin(x)], x=0..2%Pi, color=[GREEN,BLUE] ): > display( [p1,p2] J); 1s A JA A i I fA ARS TN JA [ \ ZN ET ON RE R SP BNP BS la | 10 NN] \ 7 Vo VANS 0 \ J os; | Ad NJ | \/ LLY XY / dy or vod DRY y — Study

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    8 MAPLE TECHNOLOGY RESOURCE MANUAL Note that the output from the definitions of p1 and p2 has been supressed. To view one of these plots, say p2, use either display( p2 ); or p2;. Plots can be enhanced with the inclusion of a title, a legend, axis labels, or different line color or line style, etc.. The online help topic ?plot,options provides a complete listing of all available options. To identify the coordinates of a point in the plot window, for example, the first intersection of the graphs of y = sinz and y = cosz in the plot p2, position the cursor on the desired point and click the (left) mouse button. The coordinate of this point will appear in the left-most box on the context bar. These coordinates can be copied to the desired destination. Exercises for §6 a) Create a single plot with the graphs of f 1. Consider the two functions and g. . b) Note that if discont=true is omitted from f(z) = cos2z +sinz the plot command used to create the plot and in a) there appear to be seven places where the two graphs intersect. Explain g(z) =0.0l1tanz N why there can be only five points where on the interval 0 < z <9. The objective the two graphs intersect. What causes of this problem is to estimate the points the two “extraneous” points? where the graphs of these functions in- tersect. c) Estimate the coordinates of each of the five intersection points. 7 Matrix Operations Let A and B be m x n matrices and let s be a scalar. The Maple command for matrix addition is evalm( A + B ) and the command for scalar multiplication is evalm( s * A). To add the scalar s to each element of a matrix A, use the command evalm( A +s). If Ais m x n and Bis n x p, the matrix product AB is obtained with the command evalm( A &* B). As seen previously, if b is an n-dimensional column vector, the command linsolve( A, b ) returns the set of all solutions to Ax = b. To prepare a table of function values for 2 z° + 3z + 1) cosz soy = EET ems + sinz for z = 1.0, 1.2, 1.4, ..., 29.8, 30.0, use the Maple commands: > f= x => (x"2+3*x+1)*cos(x)/(2+sin(x)); 2, fiz (z +3z+ 1) cos(z) 2+ sin(z) > X:= [1.41/56 $i =10..145 ]: > Y :=map( f, X ): > hdr := [[x, fx 1, [ --==—, --—-—- 11: > pts := zip( (x,y)->[x,y]1, X, ¥Y ): > stackmatrix( hdr, pts ): The output of this stackmatrix command is a 148 x 2 matrix. In the interest of conserving space, this output has been suppressed. The 146 data points can be plotted with the command: — StudyXY

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    TRANSPOSE, NORM, AND INVERSE 9 > plot( pts J; 400 A 200 nooo i i Fan [HY io { oo of Mus | ob asi a ° = Jn TT Vo RE A OB Vou —200 Vd Vol Vl | iY) Vol —a00 by Exercises for §7 3. Create a table of values of the function 1. Use Maple to work Exercises 31-41 in fo) = tan (1 +2?) Section 1.5. x)= (1+z3)itz 2. Use Maple’s linsolve command to solve at the points z = 2.0, 2.1, ..., 5.0. the system in Exercise 27 in Section 1.1. 8 Transpose, Norm, and Inverse To define B= A", type B := evalm( transpose( A ) );. Maple’s norm command computes the norm of a matrix or vector. To obtain the Euclidean norm of a vector, be sure to include 2 as the second argument, e.g., norm( x, 2 ). To find the inverse of a square matrix 4, use the Maple command inverse( A ). The inverse is calculated using Gaussian-Jordan elimination when A is larger than 4 x 4; for smaller matrices, determinants and Cramer’s rule are used. If any entry of A is a floating-point number, then Maple computes A ! as a floating-point matrix. In all other cases, e.g., all entries of A are integers or rational numbers, Maple will compute an exact representation for A 1. If A is singular, or numerically singular, Maple will report this with an error message. — StudyXY

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    10 MAPLE TECHNOLOGY RESOURCE MANUAL Exercises for §8 a) Execute the Maple commands inverse(A); 1. Use Maple to find the inverse of the ma- and inverse(evalf(evalm(A))):. Note that A © . in both cases Maple reports that the ma- trix in Exercise 20 of Section 1.9. ew trix is singular. 2. Use Maple to find the inverse of the ma- . trix in Exercise 27 of Section 1.9. b) Repeat a) for the matrix 5. 3. To illustrate Maple’s handling of singu- ©) Repeat a) for the matrix C for i = —1, y . —2,...,—20. (Hint: Use the Maple com- lar matrices and the differences between mand inverse(eval(evalm(C),i=-1)):.) exact and floating-point arithmetic, con- Ben sider the matrices d) Repeat «c¢) when the entries of 36 9 4 C are floating-point numbers. A= 19 ,B= 12 ] s (Hint: Use the Maple command inverse(evalf(eval(evalm(C),i=-1)));.) c= 1 2+10° e) Explain the different results in parts c) 1 2 : and d). f) (Optional.) Compare these results with those obtained with MATLAB . 9 Special Matrices To define the zero matrix, use the Maple command matrix( m, n, 0 ) where m and n are the number of rows and columns, respectively, in A. Likewise, matrix( 2, 3, 1 ) creates a 2 x 3 matrix in which every entry is 1. More generally, B := matrix( m, n, f ); creates an m x n matrix B with entries bij = f(i,7). To illustrate, the 3 x 3 matrix in which the entries alternate between 1 and 0, starting with 1, can be created with the command B := matrix( 3, 3, (i,j)->i+j+1 mod 2 );. The 5 x 5 identity matrix is obtained with matrix( 5, 5, (i,j)—>if i=j then 1 else 0 end if ). A random m x n matrix can be created with the Maple command randmatrix( m, n ). By default, the entries in this matrix range from -99 to 99, that is a two-digit integer. To limit the values to integers between a and b, inclusive, add a third argument: randmatrix( m, n, entries=rand(a..b) ). Random matrices are frequently used to test a conjecture or simulate a series of random events, such as traffic flow in a street network. Exercises for §9 b) Use the gaussjord command to solve each 1. Create a 4 x 5 matrix consisting entirely of the 10 systems. of sevens. c) Using pencil and paper (or linsolve), 5 write each solution in standard form. : What can you conclude if m < n? (Hint: a) Use the randmatrix command to generate See Theorem 4, Section 1.3.) 10 different homogeneous m x n systems with integer entries between -20 and 20. Make sure at least 6 of the matrices m < n. — StudyXY

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    WORKING WITH DATA IN MAPLE 11 10 Working with Data in Maple Maple has a number of tools for working with lists of data. The (minimal) interpolating polynomial for a collection of data points can be found with the interp command. Also, the stats package contains a command, fit, that can determine the best least-squares fit of data to a given family of functions. To illustrate the use of these commands, consider the table of data points: z|1 2 3 4 5 6 y|4 12 17 33 45 52 The lists of z and y values are: > xi :=[8$1..61]; 21 :=[1,2,3,4,5,6| > yi :=[4, 12, 17, 33, 45, 52 1; ul = [4, 12,17, 33, 45, 52) The corresponding data points are: > pts := zip( (x,y)->[x,y], x1, yi); pts = [[1, 4], [2, 12], [3, 17], [4, 33], [5, 45}, [6, 52]] For future reference, construct a plot of the data points: > pt_plot := pointplot( pts ): The best least-squares linear fit to the data is: > 1s_fit := fit[leastsquare[ [x,y], y=a*x+b, {a,b} 1 1( [x1,y1] ); 71 25 Is_fit y= ro The following graph provides a visual confirmation of this result: > 1s_plot := plot( rhs(ls_fit), x=0..7 ): > display( [ pt_plot, 1ls_plot ] ); so ~~ Pp = A BS EP BE ao a The best least-squares quintic fit to these 6 data points is the polynomial interpolator: > fit[ leastsquarel [x,y], y=a*x"5+b*x~4+c*x"3+d*x"2+exx+f, {a,b,c,d,e,f } 1] > C [x1, y1] J; 43, 79,359 , 1661 , 5903 ¥=1557 pr tgE To z=" 30 z—93 — StudyXY

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    12 MAPLE TECHNOLOGY RESOURCE MANUAL but it is simpler to obtain this interpolating polynomial directly from the data points: > interp_poly := interp( x1, yi, x J); 43 79 359 1661 5903 interp_poly = — 5 — 2 pd 4 227 13 2 — interp_poly 150° Bot 3 x 5 z° 4 30 z — 93 The best linear fit and interpolating polynomial are easily compared in a graph: > ip_plot := plot( interp_poly, x=0..7, color=GREEN ): > display( [ pt_plot, ls_plot, ip_plot] ); 100 ‘ ° es Note that the data consists of integers and the coefficients computed for the best least-squares linear fit and interpolating polynomial are rational; no floating-point approximations have been em- ployed. To see floating-point approximations to these polynomials, change one or more data value to a floating-point number or use evalf( interp_poly );. Exercises for §10 c) Plot the data points, best least-squares 1. Consider the following table of data fit, and interpolator on the same graph. 2. Repeat 1. with the table of data z| 1 2 3 4 5 y [33 52 67 90 108 z|{-3 -2 -1 0 1 2 3 a) Find the best least-squares fit to the y {33 7.8 105 12.2 109 82 3.1 data. b) Find the polynomial interpolator for the data. 11 Procedures The Maple worksheet provides a convenient environment for developing and archiving sequences of Maple commands that will be executed many times. It is also possible to read Maple commands from a text file; see 7read for a complete explanation. To further facilitate the implementation of new algorithms, blocks of Maple commands can be grouped together to define a procedure. (An interesting fact about Maple is that the vast majority of the Maple commands are written in Maple — as procedures.) For example, suppose it is necessary to compare best least-squares polynomial fits to a given set of data points for polynomials of different degrees. The following procedure, named Isplot, [+s

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    PROCEDURES 13 three arguments — the list of « values, the list of y values, and the degree of the polynomial to be used in the least-squares fit — and returns a plot of the data points and the best least-squrare polynomial of the requested degree: > 1splot := proc( X, Y, n) > local a, ls_plot, p, pts, pt_plot, x, y; > global 1ls_fit; > p := add( alli*xx™i, i=0..n ); > a i= { coeffs( p, x) }; > 1s_fit := fit[ leastsquarel[ [x,y], y=p, a 1 1( [X,Y] ); > pts := zip( (x,y)->[x,y], X, Y ); > pt_plot := pointplot( pts, color=RED ); > 1s_plot := plot( rhs(Is_fit), x=min(op(X))-1 .. max(op(X))+1, > color=GREEN ); > return display( [ pt_plot, 1ls_plot ] ); > end proc: This procedure closely follows the approach presented in the previous section. The first line defines Isplot to be a Maple procedure and specifies the arguments. Line 2, beginning local, defines the names that are to remain local to this procedure; the next line defines Is fit to be a global variable. Local variables do not assume any value at the start of the procedure and their final values are not accessible once the procedure terminates. Conversely, global variables retain their value after the procedure terminates and can be accessed from elsewhere in the Maple session. (For additional information on local and global variables see ?procedure; see ?parameter for parameter passing in Maple procedures.) The next two lines define p to be the general polynomial of degree n and a to be the set of coefficients in p. The next four lines implement the algorithm discussed in Section 10. Note that the polynomial is plotted over an interval that extends one unit on either side of the smallest and largest z value without assuming that these values are sorted. The final command instructs Maple to return the combined plot as the result of the Isplot command. Now, the best least-squares cubic fit to the data in Section 10 can be graphed via the single command: > 1splot( x1, yi, 3); so = / “0 , 20 , se 0 . 5 g = ER 5 s > Because the best least-squares fit is a global variable, it can be accessed as if it had been returned by the procedure: > 1s_fit; _% 3355 MB, 55 4 VCE TER Tm 108 Another way to define a procedure is to use Maple’s “arrow” notation (see ?->). This form is par- ticularly suited to the implementation of mathematical functions and commands that are expressed — StudyXY

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    14 MAPLE TECHNOLOGY RESOURCE MANUAL without the need for local variables or intermediate results. For example, the scalar triple product of the vectors u, v, and w, u- (v x w), can be represented as: > sc_triple_prod := ( u, v, w ) -> dotprod( u, crossprod( v, w) ); sc_triple_prod := (u, v, w) — dotprod(u, crossprod(v, w)) Consider the vectors: > a :=vector( [ 1,2, 3] ): > b :=vector( [ 2, 3, 1] ): > ¢ :=vector( [ 1, 2, 1] ): The scalar triple product of a, b, and c is > sc_triple_prod( a, b, c J; 2 Exercises for §11 2. Use the procedure Isplot with a value of n - 3) i 1. Write a Maple procedure that calculates 0 that the best least-squares polynomial . approximation of degree n or less looks the vector triple product u x (v x w). 5 Test function using the vectors reasonable for the data points (z;,y;) = est your function using (ini), i = 1,2,...,20. (Hint: There is 1 9 0 10 need to consider n > 5.) a=|1|,b=|-2],e=]|1], 2 1 4 — StudyXY

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