Online Technology Guide Elementary Linear Algebra, 6e
Contents
1. 2 and B 1 2 2 2 Keystrokes for TI 83 Enter the matrices into matrix A and matrix B To find the transition matrix use the following keystrokes MATRX gt ALPHA B MATRX gt 7 MATRX 2 C MATRX 1 ENTER Keystrokes for TI 83 Plus Enter the matrices into matrix A and matrix B To find the transition matrix use the following keystrokes MATRX ALPHA B nd MATRX MATRX 2 MATRX 1 Keystrokes for TI 84 Plus Enter the matrices into matrix A and matrix B To find the transition matrix use the following keystrokes MATRX B MATRX 7 MATRX 2 MATRX 1 Keystrokes for TI 86 Enter the matrices into matrix A and matrix B To find the transition matrix use the following keystrokes MATRX F4 F5 MORE F1 2nd M1 F2 EN ENTER Keystrokes for TI 89 Enter the matrices into matrix A and matrix B To find the transition matrix use the following keystrokes MATH 4 4 MATH 4 7 alpha B alpha A O O ENTER 24 Online Technology Guide Keystrokes for TI 92 and Voyage 200 Enter the matrices into matrix A and matrix B To find the transition matrix use the following keystrokes MATH 4 4 2nd MATH 4 7 8 O A O O ENTER Programming Syntax for MATLAB Enter the matrices into matrix B and matrix BPRIME rref BPRIME B Hit the return or enter key Programming Syntax for Maple wi2. A 0 ENTER 20 Online Technology Guide Keystrokes for TI 92 and Voyage 200 Enter the matrix into matrix A To find the determinant use the following keystrokes 2nd MATH 4 2 A Programming Syntax for MATLAB Enter the matrix into matrix A det A Hit the return or enter key Programming Syntax for Maple with linalg Hit the return or enter key Enter the matrix into matrix A det A Hit the return or enter key Programming Syntax for Mathematica Enter the matrix into matrix A Det A Hit shift enter Programming Syntax for Derive Enter the matrix into matrix A Det A Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 3 4 page 153 EXAMPLE 2 Finding Eigenvalues and Eigenvectors 1 4 Find the eigenvalues and corresponding eigenvectors of the matrix A al Keystrokes for TI 83 TI 83 Plus and TI 84 Plus These graphing utilities cannot find eigenvalues Keystrokes for TI 86 Enter the matrix into matrix A To find the eigenvalues use the following keystrokes MATRX F3 F4 2nd M1 F1 ENTER Keystrokes for TI 89 Enter the matrix into matrix A To find the eigenvalues use the following keystrokes MATH 4 9 alpha A O ENTER Keystrokes and Programming Syntax for Selected Examples 21 Keystrokes for TI 92 and Voyage 200 Enter the matrix into matrix A To find the eigenvalues use the follow
3. 4 7 2 5 8 3 6 O c a Inverse a Out 5 1 0 0 0 1 O 0 0 1 d 2 a Out 6 112 4 6 8 10 12 14 16 O Online Technology Guide e RowReduce a Out 7 1 0 0 10 1 0 0 0 1 Detla Out 8 27 Enter the matrix b 7 16 7 You can solve the system of linear equations ax b in different ways a LinearSolve a b Out 10 1 0 2 b c Table Append al i b i i 1 3 and then RowReduce c Out 11 1 2 3 7 4 5 6 16 7 8 0 7 In 12 RowReduce c MatrixForm Out 12 MatrixForm 1 0 0 1 0 1 0 0 0 0 1 2 Enter the matrices A 1 2 3 4 and B 2 0 3 5 and perform the following matrix operations Notice that Mathematica distinguishes between upper and lowercase letters using the uppercase letter B to label the matrix above is not the same as using the lowercase letter b a A B Out 15 113 2 10 91 b A B Out 16 1 2 6 1 c A B Matrix multiplication is not A B Out 17 4 10 6 20 What do the following commands do a Tablel0 i 1 3 3 1 5 b IdentityMatrix 5 c Table Random Integer 100 100 4 6 MatrixForm d Table 1 i j 1 i 1 6 3 1 6 Computer Software Programs and Graphing Utilities 9 6 Describe the effects of the following commands on the matrix a 1 2 3 4 5 6 7 8 O a
4. ENTER Part c alpha V 2 alpha U ENTER Keystrokes for TI 92 and Voyage 200 Store the vectors in U and V To find each vector use the following keystrokes Part a U V ENTER Part b 2 Part c V 2 U ENTER Programming Syntax for MATLAB Enter the vectors into vector u and vector v ut v Hit the return or enter key 2 u Hit the return or enter key v 2 u Hit the return or enter key Programming Syntax for Maple with inalg Hit the return or enter key Enter the vectors into vector u and vector v evalm u v Hit the return or enter key evalm 2 u Hit the return or enter key evalm v 2 u Hit the return or enter key Programming Syntax for Mathematica Enter the vectors into vector u and vector v utv Hit shift enter 2u Hit shift enter v 2u Hit shift enter Keystrokes and Programming Syntax for Selected Examples 23 Programming Syntax for Derive Enter the vectors into vector u and vector v uty Hit the return or enter key Choose Simplify from the toolbar Then choose Basic 2u Hit the return or enter key Choose Simplify from the toolbar Then choose Basic v 2u Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 4 7 page 257 EXAMPLE 5 Finding a Transition Matrix Find the transition matrix from B to B for the bases for R listed below B 3 2 4
5. Transpose Norm Power Eigenvalues LU decomposition Identity matrix Random matrix Concatenate matrices Condition number Complex matrices Vectors xX Xx x gt IS A AAA E ie AAA gt lt Pal gt lt II AA AAA gt lt gt Pal Oo o o o l d o d d do d do d d do l do lo od For other graphing utility models see your user s manual 12 Online Technology Guide Technology Pitfalls Many matrix computations in linear algebra can be performed by modern graphing utilities and computer software programs However you should be aware that these devices are not perfect because of roundoff error that is the graphing utility or computer must approximate fractions by decimals either by rounding off the last digit or by truncating For example evaluate z using your graphing utility On the TI 86 you will get 0 166666666667 which is not correct because the graphing utility has rounded the last digit up To analyze this more closely evaluate 2 E 3 0 5 Do you get the expected answer of 0 Normally this kind of rounding will not affect your answers to problems in linear algebra but it can happen Here are a couple of simple examples 3 H 1 Calculate the determinant of the matrix A f A on the TI 86 Then calculate the greatest integer of the determinant int det A Again roundoff error has produced an error 2 The Hilbert matrix of order 6 is 1 1 1 1 1 133 4 5 6 i ia 4 2 i 2 3 4 5 6 7 toda a4
6. z a 1 b x 1 y 0 z 2 c You can Expand the equation a x y z b Then Solve the Expression by choosing solution variables x y and z and choosing Solve Ix 1 y 0 z 2 Enter the matrices A 1 2 3 4 and B 2 0 3 5 and perform the following matrix operations a soa B Lo a b A B bs i Computer Software Programs and Graphing Utilities 11 c oA B 4 10 Cs 20 5 What do the following commands do a VECTOR VECTOR 0 j 1 5 k 1 3 b Identity_Matrix 5 c VECTOR VECTOR Random 100 j 1 6 k 1 4 d VECTOR VECTOR 1 j k 1 j 1 6 k 1 6 6 Describe the effects of the following commands on the matrix a 1 2 3 4 5 6 7 8 0 a Element Element a 2 3 b Element a 2 c Element a 3 7 You can obtain more information about a particular topic by choosing Contents from the help menu 8 You can exit Derive by choosing Exit from the file menu Graphing Utilities The following chart lists various graphing utilities and their built in matrix capabilities For example the first row indicates that all the graphing utilities can perform matrix addition Matrix operation Voyage TI 92 TI 89 TI 86 TI 84 TI 83 TI 83 200 Plus Plus X X Addition Scalar multiplication Multiplication Elementary row operations Reduced row echelon form Determinant Inverse
7. 1 x x Hit shift enter Programming Syntax for Derive Fit x a x b 1 0 2 1 3 3 Hit the return or enter key Choose Simplify from the toolbar Then choose Basic 30 Online Technology Guide Section 5 4 page 332 EXAMPLE 10 Application to Astronomy Table 5 2 shows the mean distances x and the periods y of the six planets that are closest to the sun The mean distance is given in terms of astronomical units where the Earth s mean distance is defined to be 1 0 and the period is provided in years Find a model for these data Source CRC Handbook of Chemistry and Physics TABLE 5 2 Planet Mercury Venus Earth Mars Jupiter Saturn Distance x 0 387 0 723 1 0 1 523 5 203 9 541 Period y 0 241 0 615 1 0 1 881 11 861 29 457 Keystrokes for TI 83 TI 83 Plus and TI 84 Plus Enter the data into lists L1 and L2 To find the power model use the following keystrokes STAT ALPHA A Eno L1 nd L2 ENTER Keystrokes for TI 86 Use STAT to enter the data There should be a 1 in each row of the f Stat column To find the power model use the following keystrokes STAT ED MORE F1 ENTER Keystrokes for TI 89 Enter the data into cl and c2 using the Data Matrix Editor To find the power model use the following keystrokes while in the Data Matrix Editor E O 8 L Gipha c 1 4 Calpha c 2 ENTER ENTER Keystrokes for TI 92 and Voyage 200 Enter the data
8. TI 83 Plus and TI 84 Plus Enter the data into lists L1 and L2 To find the least squares regression line use the following keystrokes STAT gt 4 nd L1 nd L2 ENTER Keystrokes for TI 86 Use STAT to enter the data There should be a 1 in each row of the fStat column To find the least squares regression line use the following keystrokes STAT Keystrokes and Programming Syntax for Selected Examples 29 Keystrokes for TI 89 Enter the data into cl and c2 using the Data Matrix Editor To find the least squares regression line use the following keystrokes while in the Data Matrix Editor 65 G 5 Calpha c 1 1 Calpha c 2 ENTER ENTER Keystrokes for TI 92 and Voyage 200 Enter the data into cl and c2 using the Data Matrix Editor To find the least squares regression line use the following keystrokes while in the Data Matrix Editor 6105040 10 O 2 ENTER ENTER Programming Syntax for MATLAB x 1 2 3 Hit the return or enter key y 01 3 Hit the return or enter key polyfit x y 1 Hit the return or enter key Programming Syntax for Maple with Statistics Hit the return or enter key xvalues vector 1 2 3 Hit the return or enter key yvalues vector 0 1 3 Hit the return or enter key Fit a x b xvalues yvalues x Hit the return or enter key Programming Syntax for Mathematica data 1 0 2 1 3 3 Hit shift enter Fit data
9. al 2 3 b al 2 c Transpose a 3 7 You can obtain more information about a particular topic such as determinants by typing Det 8 You can exit Mathematica by choosing Exit from the file menu Introduction to Derive The purpose of this introduction is to illustrate some basic Derive commands for linear algebra It is suggested that you complete the following series of simple examples while actually working at the computer terminal 1 Enter the matrix 1 2 3 a 4 5 6 7 8 0 by using the Author Expression feature i e a 1 2 3 4 5 6 7 8 0 and hitting the return or enter key Be sure to separate the entries in each row by commas and place each row of the matrix within brackets Derive should return the following matrix 1 2 3 a 4 5 6 7 8 0 2 Perform the following elementary operations on the matrix a by typing the given expression and Simplifying it You should obtain the indicated results a an 1 16 8 1 9 9 79 l4 _7 2 9 9 9 al 2 _1 9 9 9 b a 1 4 7 2 9 8 Online Technology Guide c ar ar 1 1 0 0 0 1 0 0 0 1 d a 2 2 4 6 8 10 12 14 16 0 e Row_Reduce a 1 0 0 0 1 0 0 0 1 det a 27 Enter the column matrix b 7 16 7 You can solve the system of linear equations ax b in several ways a You can row reduce the augmented matrix a b Row_Reduce a b 1 0 0 1 0 1 0 0 0 0 1 2 b You can Simplify the equation x y
10. into cl and c2 using the Data Matrix Editor To find the power model use the following keystrokes while in the Data Matrix Editor F5 50380010 O 2 ENTER ENTER Programming Syntax for MATLAB x 0 387 0 723 1 0 1 523 5 203 9 541 Hit the return or enter key y 0 241 0 615 1 0 1 881 11 861 29 457 Hit the return or enter key polyfitdog x log y 1 Hit the return or enter key Keystrokes and Programming Syntax for Selected Examples 31 Programming Syntax for Maple with Statistics Hit the return or enter key xvalues vector 0 387 0 723 1 0 1 523 5 203 9 541 Hit the return or enter key yvalues vector 0 241 0 615 1 0 1 881 11 861 29 457 Hit the return or enter key Fit a x b xvalues yvalues x Hit the return or enter key Programming Syntax for Mathematica lt lt Statistics NonLinearFit Hit shift enter data 0 387 0 241 0 723 0 615 1 0 1 0 1 523 1 881 5 203 11 861 9 541 29 457 Hit shift enter NonlinearFit data a xb x fa b Hit shift enter Programming Syntax for Derive Derive cannot calculate power models Section 5 5 page 336 EXAMPLE 1 Finding the Cross Product of Two Vectors Provided that u i 2j k and v 3i j 2k find a u x v Keystrokes for TI 83 TI 83 Plus and TI 84 Plus These graphing utilities cannot find the cross product Keystrokes for TI 86 Enter the vectors in
11. 18 EXAMPLE 4 Row Echelon Form 1 2 2 f 0 0 0 0 0 1 2 4 Keystrokes for TI 83 Enter the matrix into matrix A To rewrite the matrix in reduced row echelon form use the following keystrokes MATRX gt ALPHA B MATRX ENTER ENTER Keystrokes for TI 83 Plus Enter the matrix into matrix A To rewrite the matrix in reduced row echelon form use the following keystrokes MATRX ALPHA B nd MATRX Keystrokes for TI 84 Plus Enter the matrix into matrix A To rewrite the matrix in reduced row echelon form use the following keystrokes MATRIX ALPHA B Ena MATRIX Keystrokes for TI 86 Enter the matrix into matrix A To rewrite the matrix in reduced row echelon form use the following keystrokes MATRX F4 F5 ALPHA A ENTER 16 Online Technology Guide Keystrokes for TI 89 Enter the matrix into matrix A To rewrite the matrix in reduced row echelon form use the following keystrokes MATH 4 4 alpha A O ENTER Keystrokes for TI 92 and Voyage 200 Enter the matrix into matrix A To rewrite the matrix in reduced row echelon form use the following keystrokes MATH 4 4 A Programming Syntax for MATLAB Enter the matrix into matrix A rref A Hit the return or enter key Programming Syntax for Maple with inalg Hit the return or enter key Enter the matrix into matrix A rref A Hit the return or enter key Pro
12. 2 3 4 and B 2 0 3 5 and perform the following matrix operations Notice that MATLAB distinguishes between upper and lowercase letters using the uppercase letter B to label the matrix above is not the same as using the lowercase letter b Online Technology Guide a A B 2 ans 9 bd A B 2 ans 6 1 c A B 4 10 ans 6 20 5 What do the following commands do a zeros 3 5 b eye 5 c rand 6 d hilb 6 6 Describe the effects of the following commands on the matrix a 1 2 3 4 5 6 7 8 0 a a 2 3 b a 2 c a 3 7 You can obtain more information about MATLAB by typing help 8 You can exit MATLAB by typing exit Introduction to Maple The purpose of this introduction is to illustrate some basic Maple commands for linear algebra It is suggested that you complete the following series of simple examples while actually working at the computer terminal Once you have signed on to Maple you should type in your commands to the right of the prompt gt Begin by loading the linear algebra package that comes with Maple by typing with linalg and hitting the return or enter key Notice that all commands in Maple terminate with a semicolon 1 Enter the matrix 1 2 3 a 4 3 6 7 8 0 by typing a matrix 1 2 3 4 5 6 7 8 0 and hitting the return or enter key Be sure to separate the entries in each row by commas and place each row of the matrix within brackets Map
13. 4 3 4 5 6 T 8 H 1 1 1 1 1 L 45 6 7 8 9 a Ll 5 6 7 8 9 10 AM LA 6 7 8 9 10 11 This matrix is notorious for causing poor results in matrix computations Try calculating inv H H to see if you obtain the identity matrix You might find it interesting to discover other examples for which your graphing utility gives erroneous results In general you should always be aware that such errors are possible and accept the outputs of your graphing utility with some skepticism p Keystrokes and Programming Syntax for Selected Examples 13 Keystrokes and Programming Syntax for Selected Examples Selected examples in your text can be solved using a variety of graphing utilities and com puter software programs Keystrokes and programming syntax for these utilities programs are provided on the following pages for use with a variety of graphing utilities MATLAB Maple Mathematica and Derive Section 1 1 page 7 EXAMPLE 7 Using Elimination to Rewrite a System in Row Echelon Form Solve the system x 2y 3z 9 x Sy 4 2x 5y 5z 17 Keystrokes for TI 83 Enter the system into matrix A To rewrite the system in row echelon form use the following keystrokes MATRX gt ALPHA A MATRX ENTER ENTER Keystrokes for TI 83 Plus Enter the system into matrix A To rewrite the system in row echelon form use the following keystrokes MATRX ALPHA A nd MATRX Keystrokes for TI 84 Plus Enter the sy
14. Online Technology Guide for Elementary Linear Algebra 6e Larson Falvo Computer Software Programs and Graphing Utilities Introduction to MATLAB o oooocoooon nett nent e eens Introduction to Maple secuestradas idad a Introduction to Mathematica 2 0 0 0 eee nee eens Introduction to Derive io ii daa Graphing Uthtes coo dad aia Technology Pitfalls 0 0 0 eee arara a aora a arasina aranana ne een Eia eana Keystrokes and Programming Syntax for Selected Examples Section ll Example Torta A A E natn Section 1 2 Example east N E E Section k2 Explorer E EE rita Section 2l Example idad pes Section 2 3 Example Jere ita a a o Secon dd Examples cet ost oul antennae ee section 3 4 Example Zi 12 4454425 9s Sees ooh SoubS yey ee eb sore tase hie section4 1 Example 4 22iuctescae Io oieoia toes Pos SSCHOM A T Example Jsi ere iia Section Example lacra es section 5 1 Technology Note vicosrosrsorirsrrrr rr rip vet Section dull Example Atreides ii rs Section 3 2 EXAMPLE Pra tir ips O E aye ENA section 34 Example lO coco rra Section 9 90 Example li dcir ds is Section 7 1 Example O revocada iii ira es Online Technology Guide Computer Software Programs and Graphing Utilities To help you become acquainted with the rudiments of using a computer software program or graphing utility the following pages contain certain introductory instructions for the following popular systems MATLAB Maple Mathematica and Derive as well
15. as information about various graphing utility models Introduction to MATLAB The purpose of this introduction is to illustrate some basic MATLAB commands for linear algebra It is suggested that you complete the following series of simple examples while actually working at the computer terminal Once you have signed on to MATLAB you should type in your commands to the right of the prompt gt gt 1 Enter the matrix 1 2 3 a 4 gt 6 7 8 0 by typing a 1 2 3 4 5 6 7 8 0 and hitting the return or enter key Be sure to separate the entries in each row by spaces and to terminate the rows with semicolons MATLAB should return the following matrix 1 2 3 a 4 5 6 7 8 0 2 Perform the following elementary commands on the matrix a You should obtain the indicated results a inv a 1 7778 0 8889 0 1111 ans 1 5556 0 7778 0 2222 0 1111 0 2222 0 1111 b a 1 7 ans 2 5 8 3 6 0 c a inv a 1 0000 O 0 0000 ans 0 0000 1 0000 0 0 0000 0 0000 1 0000 Computer Software Programs and Graphing Utilities 3 d 2 a 2 4 6 ans 8 10 12 14 16 0 e rref a 1 0 0 ans 0 1 0 0 0 1 f det a ans 27 g rank a ans 3 h diag a 1 ans 5 0 Enter the matrix b 7 16 7 You can solve the system of linear equations ax b in different ways a a b 1 ans 0 2 b e a b and then rref c 1 2 3 7 c 4 5 6 16 1 0 0 1 ans 1 0 0 0 0 1 2 Enter the matrices A 1
16. atrix into matrix A To perform the elementary row operations use the following keystrokes MATRIX gt ALPHA F 2 nd MATRIX 10 103 Keystrokes for TI 86 Enter the matrix into matrix A To perform the elementary row operations use the following keystrokes MATRX F4 MORE F5 2 nd MATRX E ED 1 O 3 ENTER Keystrokes for TI 89 Enter the matrix into matrix A To perform the elementary row operations use the following keystrokes MATH 4 alpha J 4 2 O alpha A O 1 O 3 0 ENTER Keystrokes for TI 92 and Voyage 200 Enter the matrix into matrix A To perform the elementary row operations use the following keystrokes nd MATH 4 D 40 20 W 0 1 0 3 ENTER Programming Syntax for MATLAB Enter the matrix into matrix A AB 2 AC AG Hit the return or enter key Keystrokes and Programming Syntax for Selected Examples 15 Programming Syntax for Maple with linalg Hit the return or enter key Enter the matrix into matrix A A addrow A 1 3 2 Hit the return or enter key Programming Syntax for Mathematica Enter the matrix into matrix A A 3 2A 11 Al Hit shift enter A MatrixForm Hit shift enter Programming Syntax for Derive Enter the matrix into matrix A Subtract_Elements A 3 1 2 Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 1 2 page
17. e the following keystrokes MATRX F3 F4 2nd M1 F1 ENTER M3 M1 Keystrokes for TI 89 Enter the matrix into matrix A To find the eigenvalues and eigenvectors use the following keystrokes MATH 4 9 alpha A O ENTER MATH 4 alpha A alpha A O ENTER Keystrokes and Programming Syntax for Selected Examples 33 Keystrokes for TI 92 and Voyage 200 Enter the matrix into matrix A To find the eigenvalues and eigenvectors use the following keystrokes MATH 4 9 A O ENTER 2nd MATH 4 A A O ENTER Programming Syntax for MATLAB Enter the matrix into matrix A V D eig A Hit the return or enter key Programming Syntax for Maple with linalg Hit the return or enter key Enter the matrix into matrix A eigenvalues A Hit the return or enter key eigenvectors A Hit the return or enter key Programming Syntax for Mathematica Enter the matrix into matrix A Eigensystem A MatrixForm Hit shift enter Programming Syntax for Derive Enter the matrix into matrix A Eigenvalues A Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Derive cannot find eigenvectors
18. gramming Syntax for Derive Enter the vectors into vector u and vector v u v Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 5 2 page 297 EXAMPLE 7 Using the Inner Product on C O 1 Calculus Use the inner product defined in Example 5 and the functions f x x and g x x in C O 1 to find b d f 8 Keystrokes for TI 83 TI 83 Plus and TI 84 Plus To find d f g use the following keystrokes End y 90 QIO Xt A OC O At 000 1 ENTER Keystrokes for TI 86 To find d f g use the following keystrokes 2nd y CALC 9 F5 O GAR O GAR C2 0 09 O GCR OOO 1 28 Online Technology Guide Keystrokes for TI 89 TI 92 and Voyage 200 To find d f g use the following keystrokes End y ES 2 O 49 O 09 020 0 2 0 4 000 1 0 0 ENTER Programming Syntax for MATLAB MATLAB cannot calculate inner products Programming Syntax for Maple sqrt int x x2 42 x 0 1 Hit the return or enter key Programming Syntax for Mathematica Sqrt Integrate x x 2 2 x 0 1 Hit shift enter Programming Syntax for Derive sqrt int x x 2 2 x 0 1 Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 5 4 page 328 EXAMPLE 7 Solving the Normal Equations Find the solution of the least squares problem Ax b 1 1 0 i 2 K 1l C1 3 3 1 Keystrokes for TI 83
19. gramming Syntax for Mathematica Enter the matrix into matrix A RowReduce A MatrixForm Hit shift enter Programming Syntax for Derive Enter the matrix into matrix A Row_Reduce A Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 2 1 page 52 EXAMPLE 5 Matrix Multiplication ja 2 Oli alla 1 2H 2 222 Keystrokes for TI 83 Enter the matrices into matrix A and matrix B To multiply the matrices use the following keystrokes MATRX 1 MATRX 2 ENTER Keystrokes and Programming Syntax for Selected Examples 17 Keystrokes for TI 83 Plus Enter the matrices into matrix A and matrix B To multiply the matrices use the following keystrokes MATRX 1 MATRX 2 Keystrokes for TI 84 Plus Enter the matrices into matrix A and matrix B To multiply the matrices use the following keystrokes MATRIX 1 MATRIX 2 Keystrokes for TI 86 Enter the matrices into matrix A and matrix B To multiply the matrices use the following keystrokes MATRX F1 F1 F2 ENTER Keystrokes for TI 89 Enter the matrices into matrix A and matrix B To multiply the matrices use the following keystrokes alpha A x alpha B ENTER Keystrokes for TI 92 and Voyage 200 Enter the matrices into matrix A and matrix B To multiply the matrices use the following keystrokes A 63 B ENTER Programming Syntax for MATLAB Enter the matrice
20. ing keystrokes MATH 4 9 A Programming Syntax for MATLAB Enter the matrix into matrix A eig A Hit the return or enter key Programming Syntax for Maple with linalg Hit the return or enter key Enter the matrix into matrix A eigenvalues A Hit the return or enter key Programming Syntax for Mathematica Enter the matrix into matrix A Eigenvalues A Hit shift enter Programming Syntax for Derive Enter the matrix into matrix A Eigenvalues A Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 4 1 page 184 EXAMPLE 4 Vector Operations in R Provided that u 1 0 1 and v 2 1 5 in R find each vector a ut v b 2u c v 2u Keystrokes for TI 83 TI 83 Plus and TI 84 Plus Store the vectors in lists L1 and L2 To find each vector use the following keystrokes Part a 2nd L1 2nd L2 ENTER Part b 2 nd L1 ENTER Part c nd L2 2 nd L1 ENTER Keystrokes for TI 86 Enter the vectors into vector U and vector V To find each vector use the following keystrokes Part a VECTR CD CD F2 ENTER Part b VECTR 2 Part c VECTR ED F2 2 D ENTER 22 Online Technology Guide Keystrokes for TI 89 Store the vectors in U and V To find each vector use the following keystrokes Part a alpha U alpha V ENTER Part b 2 alpha U
21. ith linalg Hit the return or enter key Enter the vector into vector v norm v 2 Hit the return or enter key Programming Syntax for Mathematica Enter the vector into vector v lt lt LinearAlgebra MatrixManipulation VectorNorm N v 2 Hit shift enter Programming Syntax for Derive Enter the vector into vector v Abs v Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 5 1 page 281 Technology Note Find the unit vector for v 3 4 Keystrokes for TI 83 TI 83 Plus and TI 84 Plus These graphing utilities cannot find the unit vector Keystrokes for TI 86 Enter the vector into vector V To find the unit vector use the following keystrokes VECTR M1 Keystrokes for TI 89 Store the vector in V To find the unit vector use the following keystrokes MATH 4 L 1 Galpha V O ENTER 26 Online Technology Guide Keystrokes for TI 92 and Voyage 200 Store the vector in V To find the unit vector use the following keystrokes MATH 4 L 1 LV 0 ENTER Programming Syntax for MATLAB Enter the vector into vector v v norm v Hit the return or enter key Programming Syntax for Maple with linalg Hit the return or enter key Enter the vector into vector v evalm v norm y 2 Hit the return or enter key Programming Syntax for Mathematica Enter the vector into vector v lt lt LinearAlgebra MatrixManipulation v Vecto
22. le should return the following matrix Computer Software Programs and Graphing Utilities 5 1 2 3 a 4 5 6 7 8 0 2 Perform the following elementary commands on the matrix a You should obtain the indicated results a inverse a _16 8 _1 9 9 9 l4 _7 2 9 9 9 e 2 _l 9 9 9 b transpose a 1 4 7 2 5 8 3 6 0 c multiply a inverse a 1 0 0 0 1 0 0 0 1 d scalarmul a 2 2 4 6 8 10 12 14 16 0 e rref a 1 0 0 0 1 0 0 0 1 f det a 27 g rank a 3 h diag a inverse a 1 2 3 0 0 0 4 5 6 0 0 0 7 8 0 0 0 0 0 0 0 9 9 g 0 0 0 F 5 0 0 0 9 9 Online Technology Guide Enter the column matrix b transpose matrix 7 16 7 You can solve the system of linear equations ax b in different ways a linsolve a b 1 0 2 b c augment a b and then rref c 1 2 3 T c 4 5 6 16 7 8 0 7 1 0 0 1 0 1 0 0 0 0 1 2 Enter the matrices A matrix 1 2 3 4 and B matrix 2 0 3 5 and perform the following matrix operations Notice that Maple distinguishes between upper and lowercase letters using the uppercase letter B to label the matrix above is not the same as using the lowercase letter b a matadd A B b al 0 9 b matadd A B 1 1 ral 6 1 c multiply A B gt 6 20 What do the following commands do a matrix 3 5 0 b band 1 5 c randmatrix 4 6 d hilbert 6 Describe the effects
23. of the following commands on the matrix a matrix IEL 2 3 4 5 6 7 8 0 a submatrix a 2 2 3 3 b submatrix a 2 2 1 3 c submatrix a 1 3 3 3 You can obtain more information about a particular topic such as eigenvalues by typing eigenvalues You can exit Maple by choosing Exit from the file menu Computer Software Programs and Graphing Utilities 7 Introduction to Mathematica The purpose of this introduction is to illustrate some basic Mathematica commands for linear algebra It is suggested that you complete the following series of simple examples while actually working at the computer terminal Once you have signed on to Mathematica you should type in your commands to the right of the prompt 1 Enter the matrix 1 2 3 a 4 3 6 7 8 0 by typing a 1 2 3 4 5 6 7 8 0 and evaluating the expression cell Be sure to separate the entries in each row by commas and place each row of the matrix within braces Mathematica should return the following matrix You can obtain a more natural matrix display by typing MatrixForm Out 1 1 2 3 14 5 6 7 8 0 In 2 MatrixForm Out 2 MatrixForm 1 2 3 4 5 6 7 8 0 2 Perform the following elementary commands on the matrix a You should obtain the indicated results a Inverse a MatrixForm Out 3 MatrixForm _ 16 8 _1 9 9 9 l4 _7 2 9 9 9 _l 2 _1 9 9 9 b Transpose a Out 4 1
24. owing keystrokes 000 1 ENTER Programming Syntax for MATLAB Enter the matrix into matrix A inv A Hit the return or enter key Programming Syntax for Maple with inalg Hit the return or enter key Enter the matrix into matrix A inverse A Hit the return or enter key Keystrokes and Programming Syntax for Selected Examples 19 Programming Syntax for Mathematica Enter the matrix into matrix A Inverse A MatrixForm Hit shift enter Programming Syntax for Derive Enter the matrix into matrix A A 1 Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 3 1 page 127 EXAMPLE 4 The Determinant of a Matrix of Order 4 Find the determinant of I 2 3 0 1 2 poe 0 0 2 0 3 3 4 0 Keystrokes for TI 83 Enter the matrix into matrix A To find the determinant use the following keystrokes MATRX 3 1 MATRX 1 ENTER Keystrokes for TI 83 Plus Enter the matrix into matrix A To find the determinant use the following keystrokes MATRX 1 MATRX 1 Keystrokes for TI 84 Plus Enter the matrix into matrix A To find the determinant use the following keystrokes MATRX 1 MATRX 1 Keystrokes for TI 86 Enter the matrix into matrix A To find the determinant use the following keystrokes MATRX M1 Keystrokes for TI 89 Enter the matrix into matrix A To find the determinant use the following keystrokes MATH 4 2 alpha
25. rNorm N v 2 Hit shift enter Programming Syntax for Derive Enter the vector into vector v v Abs v Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 5 1 page 283 EXAMPLE 4 Finding the Dot Product of Two Vectors Find the dot product of u 1 2 0 3 and v 3 2 4 2 Keystrokes for TI 83 TI 83 Plus and TI 84 Plus These graphing utilities cannot find the dot product Keystrokes for TI 86 Enter the vectors into vector U and vector V To find the dot product use the following keystrokes VECTR F3 F4 2nd M1 F1 G F2 ENTER Keystrokes for TI 89 Store the vectors in U and V To find the dot product use the following keystrokes MATH 4 alpha L 3 alpha U alpha V O ENTER Keystrokes and Programming Syntax for Selected Examples 27 Keystrokes for TI 92 and Voyage 200 Store the vectors in U and V To find the dot product use the following keystrokes End MATH 4 1 3 U VY O ENTER Programming Syntax for MATLAB Enter the vectors into vector u and vector v dot u v Hit the return or enter key Programming Syntax for Maple with linalg Hit the return or enter key Enter the vectors into vector u and vector v dotprod u v Hit the return or enter key Programming Syntax for Mathematica Enter the vectors into vector u and vector v u v Hit shift enter Pro
26. s into matrix A and matrix B A B Hit the return or enter key Programming Syntax for Maple with inalg Hit the return or enter key Enter the matrices into matrix A and matrix B multiply A B Hit the return or enter key Programming Syntax for Mathematica Enter the matrices into matrix A and matrix B A B MatrixForm Hit shift enter Programming Syntax for Derive Enter the matrices into matrix A and matrix B A B Hit the return or enter key Choose Simplify from the toolbar Then choose Basic 18 Online Technology Guide Section 2 3 page 76 EXAMPLE 3 Finding the Inverse of a Matrix Find the inverse of the matrix il 0 A 1 Q 1 6 2 3 Keystrokes for TI 83 Enter the matrix into matrix A To find the inverse use the following keystrokes MATRX 1 x 1 ENTER Keystrokes for TI 83 Plus Enter the matrix into matrix A To find the inverse use the following keystrokes MATRX 1 Keystrokes for TI 84 Plus Enter the matrix into matrix A To find the inverse use the following keystrokes MATRIX 1 Keystrokes for TI 86 Enter the matrix into matrix A To find the inverse use the following keystrokes MATRX F1 F1 nd 3 ENTER Keystrokes for TI 89 Enter the matrix into matrix A To find the inverse use the following keystrokes Gha A O 1 ENTER Keystrokes for TI 92 and Voyage 200 Enter the matrix into matrix A To find the inverse use the foll
27. stem into matrix A To rewrite the system in row echelon form use the following keystrokes MATRIX ALPHA A 2nd MATRIX Keystrokes for TI 86 Enter the system into matrix A To rewrite the system in row echelon form use the following keystrokes MATRX F4 F4 ALPHA A ENTER Keystrokes for TI 89 Enter the system into matrix A To rewrite the system in row echelon form use the following keystrokes MATH 4 3 alpha A O ENTER Keystrokes for TI 92 and Voyage 200 Enter the system into matrix A To rewrite the system in row echelon form use the following keystrokes End MATH 4 3 A MATLAB Maple Mathematica and Derive These computer software programs cannot produce row echelon form Row echelon form is not unique 14 Online Technology Guide Section 1 2 page 16 EXAMPLE 2 Elementary Row Operations Add 2 times the first row to the third row to produce a new third row Original Matrix 1 2 4 3 c 0 3 2 1 2 1 gt 2 Keystrokes for TI 83 Enter the matrix into matrix A To perform the elementary row operations use the following keystrokes MATRX gt ALPHA F 2 G MATRX 1 1 3 ENTER Keystrokes for TI 83 Plus Enter the matrix into matrix A To perform the elementary row operations use the following keystrokes nd MATRX ALPHA F 2 nd MATRX 1 O 1 3 ENTER Keystrokes for TI 84 Plus Enter the m
28. th linalg Hit the return or enter key Enter the matrices into matrix B and matrix BPRIME rref augment BPRIME B Hit the return or enter key Programming Syntax for Mathematica Enter the matrices into matrix B and matrix BPRIME lt lt LinearAlgebra MatrixManipulation RowReduce AppendRows BPRIME B MatrixForm Hit shift enter Programming Syntax for Derive Enter the matrices into matrix B and matrix BPRIME Row_Reduce Append_Columns BPRIME B Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 5 1 page 279 EXAMPLE 1 The Length of a Vector in R a Find the length of v 0 2 1 4 2 Keystrokes for TI 83 TI 83 Plus and TI 84 Plus These graphing utilities cannot find the length of a vector Keystrokes for TI 86 Enter the vector into vector V To find the length use the following keystrokes VECTR F3 F3 nd M1 D ENTER Keystrokes for TI 89 Store the vector in V To find the length use the following keystrokes MATH 4 alpha H 1 alpha V O ENTER Keystrokes and Programming Syntax for Selected Examples 25 Keystrokes for TI 92 and Voyage 200 Store the vector in V To find the length use the following keystrokes MATH 4 4 1 V O ENTER Programming Syntax for MATLAB Enter the vector into vector v norm v Hit the return or enter key Programming Syntax for Maple w
29. to vector U and vector V To find the cross product use the following keystrokes VECTR M1 ED F2 ENTER Keystrokes for TI 89 Store the vectors in U and V To find the cross product use the following keystrokes MATH 4 L 2 alpha U G alpha V O ENTER Keystrokes for TI 92 and Voyage 200 Store the vectors in U and V To find the cross product use the following keystrokes MATH 4 L 2 0 O ENTER 32 Online Technology Guide Programming Syntax for MATLAB Enter the vectors into vector u and vector v cross u v Hit the return or enter key Programming Syntax for Maple with inalg Hit the return or enter key Enter the vectors into vector u and vector v crossprod u v Hit the return or enter key Programming Syntax for Mathematica Enter the vectors into vector u and vector y Cross u v Hit shift enter Programming Syntax for Derive Enter the vectors into vector u and vector v Cross u v Hit the return or enter key Choose Simplify from the toolbar Then choose Basic Section 7 1 page 429 EXAMPLE 6 Finding Eigenvalues and Eigenvectors Find the eigenvalues of 1 0 0 0 1 eel 5 10 1 0 2 0 1 0 0 3 Keystrokes for TI 83 TI 83 Plus and TI 84 Plus These graphing utilities cannot find eigenvalues or eigenvectors Keystrokes for TI 86 Enter the matrix into matrix A To find the eigenvalues and eigenvectors us
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