Online Technology Guide for Elementary Linear Algebra, 6e Larson/Falvo

Transcription

1 Online Technology Guide for Elementary Linear Algebra, e Larson/Falvo Computer Software Programs and Graphing Utilities Introduction to MATLAB Introduction to Maple Introduction to Mathematica Introduction to Derive Graphing Utilities Technology Pitfalls Keystrokes and Programming Syntax for Selected Examples Section., Example Section., Example Section., Example Section., Example Section., Example Section., Example Section.4, Example Section 4., Example Section 4., Example Section., Example Section., Technology Note Section., Example Section., Example Section.4, Example Section.4, Example Section., Example Section., Example

2 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 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 >>.. Enter the matrix by typing a [ ; 4 ; ] 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.. Perform the following elementary commands on the matrix a. You should obtain the indicated results. (a) inv a (b) (c) a a 4 a 4 a*inv a ans ans ans

3 Computer Software Programs and Graphing Utilities (d) *a (e) (f ) (g) (h). Enter the matrix b [; ; ]. You can solve the system of linear equations ax b in different ways: (a) a\ b (b) rref a det a rank a diag a ans ans ans ans ans c [a b] and then rref c c 4 rref c ans 4 ans 4 4. Enter the matrices A [ ; 4] and B [ ; ] 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.

4 4 Online Technology Guide (a) A B ans (b) A B ans (c) A * B. What do the following commands do? (a) zeros, (b) eye() (c) rand() (d) hilb(). Describe the effects of the following commands on the matrix a [ ; 4 ; ]. (a) a(, ) (b) a(, :) (c) a(:, ). You can obtain more information about MATLAB by typing help.. 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 >. 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 ;.. Enter the matrix a 4 ans 4 by typing a: matrix [[,, ], [4,, ], [,, ]] ; 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. Maple should return the following matrix.

5 a: 4 Computer Software Programs and Graphing Utilities. Perform the following elementary commands on the matrix a. You should obtain the indicated results. (a) inverse(a); 4 (b) transpose(a); 4 (c) multiply(a, inverse(a)); (d) scalarmul(a, ); 4 4 (e) rref(a); (f) det(a); (g) rank(a); (h) diag(a, inverse(a)); 4 4

6 Online Technology Guide. Enter the column matrix b: transpose matrix [[,, ]] ;. You can solve the system of linear equations ax b in different ways: (a) linsolve(a, b); (b) c: augment a, b ; and then rref(c); 4. Enter the matrices A: matrix [[, ], [, 4]] ; and B: matrix [[, ], [, ]] ; 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) matadd A, B,, ; (c) multiply A, B ;. What do the following commands do? (a) matrix,, ; (b) band [], ; (c) randmatrix 4, ; (d) hilbert ;. Describe the effects of the following commands on the matrix [[,, ], [4,, ], [,, ]] ;. (a) (b) (c) c: 4 4 submatrix a,..,.. ; submatrix a,..,.. ; submatrix a,..,.. ; a: matrix. 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.

7 Computer Software Programs and Graphing Utilities 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.. Enter the matrix a 4 by typing a {{,, }, {4,, }, {,, }} 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[] In[] %//MatrixForm Out[]//MatrixForm 4. Perform the following elementary commands on the matrix a. You should obtain the indicated results. (a) Inverse[a] //MatrixForm Out[]//MatrixForm 4,,, 4,,,,, (b) Transpose[a] Out[4], 4,,,,,,, (c) a.inverse[a] Out[],,,,,,,, (d) *a Out[], 4,,,,, 4,,

8 Online Technology Guide (e) RowReduce[a] Out[],,,,,,,, (f) Det[a] Out[]. Enter the matrix b {,, }. You can solve the system of linear equations ax b in different ways: (a) LinearSolve[a, b] Out[],, (b) c Table[Append[a[[i]], b[[i]]], {i,, }] and then RowReduce[c] Out[],,,, 4,,,,,,, In[] RowReduce[c]//MatrixForm Out[]//MatrixForm 4. Enter the matrices A {{, }, {, 4}} and B {{, }, {, }} 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[],,, (b) A B Out[],,, (c) A.B Matrix multiplication is not A * B. Out[] 4,,,. What do the following commands do? (a) Table[, {i,, }, {j,, }] (b) IdentityMatrix[] (c) Table[Random[Integer,{, }], {4}, {}]//MatrixForm (d) Table[/ i j, {i,, }, { j,, }]

9 Computer Software Programs and Graphing Utilities. Describe the effects of the following commands on the matrix a {{,, }, {4,, }, {,, }}. (a) (b) (c) a[[, ]] a[[]] Transpose[a] [[]]. You can obtain more information about a particular topic, such as determinants, by typing?det.. 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.. Enter the matrix a 4 by using the Author/Expression feature i.e., a: [[,, ], [4,, ], [,, ]] 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. a: 4. Perform the following elementary operations on the matrix a by typing the given expression and Simplifying it. You should obtain the indicated results. (a) a^ 4 (b) a` 4

10 Online Technology Guide (c) a * a^ (d) a * 4 (e) Row_Reduce(a) (f) det(a). Enter the column matrix b:,,. 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) (b) You can Simplify the equation [[x], [ y], [z]] a^ * b. x y z (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. 4. Enter the matrices A : [[, ], [, 4]] and B : [[, ], [, ]] and perform the following matrix operations. (a) A B (b) A B x, y, z 4

11 Computer Software Programs and Graphing Utilities (c) A * B. What do the following commands do? (a) VECTOR VECTOR, j,,, k,, (b) Identity_Matrix() (c) VECTOR(VECTOR(Random(), j,, ), k,, 4) (d) VECTOR(VECTOR / j k, j,,, k,,. Describe the effects of the following commands on the matrix a: [[,, ], [4,, ], [,, ]]. (a) (b) (c) 4 Element Element a,, Element a, Element a`, `. You can obtain more information about a particular topic by choosing Contents from the help menu.. 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- TI- TI- TI-4 TI- TI- Plus Plus Addition x x x x x x x Scalar multiplication x x x x x x x Multiplication x x x x x x x Elementary row operations x x x x x x x Reduced row-echelon form x x x x x x x Determinant x x x x x x x Inverse x x x x x x x Transpose x x x x x x x Norm x x x x Power x x x x x x x Eigenvalues x x x x LU-decomposition x x x x Identity matrix x x x x x x x Random matrix x x x x x x x Concatenate matrices x x x x x x x Condition number x Complex matrices x x x x Vectors x x x x *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 using your graphing utility. On the TI- you will get., which is not correct, because the graphing utility has rounded the last digit up. To analyze this more closely, evaluate.. Do you get the expected answer of? 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.. Calculate the determinant of the matrix A on the TI-. Then calculate the greatest integer of the determinant, int det A. Again, roundoff error has produced an error.. The Hilbert matrix of order is H 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

13 Keystrokes and Programming Syntax for Selected Examples Section., page Keystrokes and Programming Syntax for Selected Examples Selected examples in your text can be solved using a variety of graphing utilities and computer 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. EXAMPLE Using Elimination to Rewrite a System in Row-Echelon Form Solve the system. x y z x y 4 x y z Keystrokes for TI- Enter the system into matrix A. To rewrite the system in row-echelon form, use the following keystrokes. MATRX ALPHA [A] MATRX ENTER ENTER Keystrokes for TI- Plus Enter the system into matrix A. To rewrite the system in row-echelon form, use the following keystrokes. nd [MATRX] ALPHA [A] nd [MATRX] ENTER ENTER Keystrokes for TI-4 Plus Enter the system into matrix A. To rewrite the system in row-echelon form, use the following keystrokes. nd [MATRIX] ALPHA [A] nd [MATRIX] ENTER ENTER Keystrokes for TI- Enter the system into matrix A. To rewrite the system in row-echelon form, use the following keystrokes. nd [MATRX] F4 F4 ALPHA [A] ENTER Keystrokes for TI- Enter the system into matrix A. To rewrite the system in row-echelon form, use the following keystrokes. nd [MATH] 4 alpha [A] ENTER Keystrokes for TI- and Voyage Enter the system into matrix A. To rewrite the system in row-echelon form, use the following keystrokes. nd [MATH] 4 A ENTER MATLAB, Maple, Mathematica, and Derive These computer software programs cannot produce row-echelon form. (Row-echelon form is not unique.)

14 4 Online Technology Guide Section., page EXAMPLE Elementary Row Operations Add times the first row to the third row to produce a new third row. (c) Original Matrix 4 Keystrokes for TI- To perform the elementary row operations, use the following keystrokes. MATRX ALPHA [F], MATRX,, ENTER Keystrokes for TI- Plus To perform the elementary row operations, use the following keystrokes. nd [MATRX] ALPHA [F], nd [MATRX],, Keystrokes for TI-4 Plus To perform the elementary row operations, use the following keystrokes. nd [MATRIX] ALPHA [F], nd [MATRIX],, Keystrokes for TI- To perform the elementary row operations, use the following keystrokes. nd [MATRX] F4 MORE F, nd [MATRX] F F,, Keystrokes for TI- To perform the elementary row operations, use the following keystrokes. nd [MATH] 4 alpha [J] 4, alpha [A],, ENTER Keystrokes for TI- and Voyage To perform the elementary row operations, use the following keystrokes. nd [MATH] 4 J 4, A,, ENTER Programming Syntax for MATLAB A(, :) *A(, :) A(, :) ENTER ENTER ENTER

15 Keystrokes and Programming Syntax for Selected Examples Section., page Programming Syntax for Maple with(linalg); A: addrow(a,,, ); Programming Syntax for Mathematica A A A A // MatrixForm Programming Syntax for Derive Subtract_Elements(A,,, ) EXAMPLE 4 Row-Echelon Form (f) 4 Keystrokes for TI- To rewrite the matrix in reduced row-echelon form, use the following keystrokes. MATRX ALPHA [B] MATRX ENTER ENTER Keystrokes for TI- Plus To rewrite the matrix in reduced row-echelon form, use the following keystrokes. nd [MATRX] ALPHA [B] nd [MATRX] ENTER ENTER Keystrokes for TI-4 Plus To rewrite the matrix in reduced row-echelon form, use the following keystrokes. nd [MATRIX] ALPHA [B] nd [MATRIX] ENTER ENTER Keystrokes for TI- To rewrite the matrix in reduced row-echelon form, use the following keystrokes. nd [MATRX] F4 F ALPHA [A] ENTER

16 Online Technology Guide Section., page Keystrokes for TI- To rewrite the matrix in reduced row-echelon form, use the following keystrokes. nd [MATH] 4 4 alpha [A] ENTER Keystrokes for TI- and Voyage To rewrite the matrix in reduced row-echelon form, use the following keystrokes. nd [MATH] 4 4 A ENTER Programming Syntax for MATLAB rref(a) Programming Syntax for Maple with(linalg); rref(a) Programming Syntax for Mathematica RowReduce[A] // MatrixForm Programming Syntax for Derive Row_Reduce(A) EXAMPLE Matrix Multiplication (c) Keystrokes for TI- Enter the matrices into matrix A and matrix B. To multiply the matrices, use the following keystrokes. MATRX MATRX ENTER

17 Keystrokes and Programming Syntax for Selected Examples Keystrokes for TI- Plus Enter the matrices into matrix A and matrix B. To multiply the matrices, use the following keystrokes. nd [MATRX] nd [MATRX] ENTER Keystrokes for TI-4 Plus Enter the matrices into matrix A and matrix B. To multiply the matrices, use the following keystrokes. nd [MATRIX] nd [MATRIX] ENTER Keystrokes for TI- Enter the matrices into matrix A and matrix B. To multiply the matrices, use the following keystrokes. nd [MATRX] F F F ENTER Keystrokes for TI- Enter the matrices into matrix A and matrix B. To multiply the matrices, use the following keystrokes. alpha [A] alpha [B] ENTER Keystrokes for TI- and Voyage Enter the matrices into matrix A and matrix B. To multiply the matrices, use the following keystrokes. A B ENTER Programming Syntax for MATLAB Enter the matrices into matrix A and matrix B. A*B Programming Syntax for Maple with(linalg); Enter the matrices into matrix A and matrix B. multiply(a, B); Programming Syntax for Mathematica Enter the matrices into matrix A and matrix B. A.B // MatrixForm Programming Syntax for Derive Enter the matrices into matrix A and matrix B. A * B

18 Online Technology Guide Section., page EXAMPLE Finding the Inverse of a Matrix Find the inverse of the matrix. A Keystrokes for TI- To find the inverse, use the following keystrokes. MATRX x ENTER Keystrokes for TI- Plus To find the inverse, use the following keystrokes. nd [MATRX] x ENTER Keystrokes for TI-4 Plus To find the inverse, use the following keystrokes. nd [MATRIX] x ENTER Keystrokes for TI- To find the inverse, use the following keystrokes. nd [MATRX] F F nd x ENTER Keystrokes for TI- To find the inverse, use the following keystrokes. alpha [A] ENTER Keystrokes for TI- and Voyage To find the inverse, use the following keystrokes. A ENTER > > Programming Syntax for MATLAB inv(a) Programming Syntax for Maple with(linalg); inverse(a);

19 Keystrokes and Programming Syntax for Selected Examples Section., page Programming Syntax for Mathematica Inverse[A] // MatrixForm Programming Syntax for Derive A ^ EXAMPLE 4 The Determinant of a Matrix of Order 4 Find the determinant of A 4. Keystrokes for TI- To find the determinant, use the following keystrokes. MATRX MATRX ENTER Keystrokes for TI- Plus To find the determinant, use the following keystrokes. nd [MATRX] nd [MATRX] ENTER Keystrokes for TI-4 Plus To find the determinant, use the following keystrokes. nd [MATRX] nd [MATRX] ENTER Keystrokes for TI- To find the determinant, use the following keystrokes. nd [MATRX] F F nd [M] F ENTER Keystrokes for TI- To find the determinant, use the following keystrokes. nd [MATH] 4 alpha [A] ENTER

20 Online Technology Guide Section.4, page Keystrokes for TI- and Voyage To find the determinant, use the following keystrokes. nd [MATH] 4 A ENTER Programming Syntax for MATLAB det(a) Programming Syntax for Maple with(linalg); det(a); Programming Syntax for Mathematica Det[A] Programming Syntax for Derive Det(A) EXAMPLE Finding Eigenvalues and Eigenvectors Find the eigenvalues and corresponding eigenvectors of the matrix Keystrokes for TI-, TI- Plus, and TI-4 Plus These graphing utilities cannot find eigenvalues. Keystrokes for TI- To find the eigenvalues, use the following keystrokes. nd [MATRX] F F4 nd [M] F ENTER Keystrokes for TI- To find the eigenvalues, use the following keystrokes. nd [MATH] 4 alpha [A] ENTER A 4.

21 Keystrokes and Programming Syntax for Selected Examples Section 4., page 4 Keystrokes for TI- and Voyage To find the eigenvalues, use the following keystrokes. nd [MATH] 4 A ENTER Programming Syntax for MATLAB eig(a) Programming Syntax for Maple with(linalg); eigenvalues(a); Programming Syntax for Mathematica Eigenvalues[A] Programming Syntax for Derive Eigenvalues(A) EXAMPLE 4 Vector Operations in R Provided that u,, and v,, in R, find each vector. (a) u v (b) u (c) v u Keystrokes for TI-, TI- Plus, and TI-4 Plus Store the vectors in lists L and L. To find each vector, use the following keystrokes. Part (a) nd [L] nd [L] ENTER Part (b) nd [L] ENTER Part (c) nd [L] nd [L] ENTER Keystrokes for TI- Enter the vectors into vector U and vector V. To find each vector, use the following keystrokes. Part (a) nd [VECTR] F F F ENTER Part (b) nd [VECTR] F F ENTER Part (c) nd [VECTR] F F F ENTER

22 Online Technology Guide Keystrokes for TI- Store the vectors in U and V. To find each vector, use the following keystrokes. Part (a) alpha [U] alpha [V] ENTER Part (b) alpha [U] ENTER Part (c) alpha [V] alpha [U] ENTER Keystrokes for TI- and Voyage Store the vectors in U and V. To find each vector, use the following keystrokes. Part (a) U V ENTER Part (b) U ENTER Part (c) V U ENTER Programming Syntax for MATLAB Enter the vectors into vector u and vector v. u v * u v * u Programming Syntax for Maple with(linalg); Enter the vectors into vector u and vector v. evalm(u v); evalm * u ; evalm v * u ; Programming Syntax for Mathematica Enter the vectors into vector u and vector v. u v u v u

23 Keystrokes and Programming Syntax for Selected Examples Section 4., page Programming Syntax for Derive Enter the vectors into vector u and vector v. u v u v u EXAMPLE Finding a Transition Matrix Find the transition matrix from B to for the bases for R listed below. B B,, 4, and B,,, Keystrokes for TI- Enter the matrices into matrix A and matrix B. To find the transition matrix, use the following keystrokes. MATRX ALPHA [B] MATRX MATRX, MATRX Keystrokes for TI- Plus Enter the matrices into matrix A and matrix B. To find the transition matrix, use the following keystrokes. nd [MATRX] ALPHA [B] nd [MATRX] nd [MATRX] [MATRX] ENTER Keystrokes for TI-4 Plus Enter the matrices into matrix A and matrix B. To find the transition matrix, use the following keystrokes. nd [MATRX] ALPHA [B] nd [MATRX] nd [MATRX] [MATRX] ENTER Keystrokes for TI- Enter the matrices into matrix A and matrix B. To find the transition matrix, use the following keystrokes. nd [MATRX] F4 F MORE F nd [M] F, ENTER Keystrokes for TI- Enter the matrices into matrix A and matrix B. To find the transition matrix, use the following keystrokes. nd [MATH] 4 4 nd [MATH] 4 alpha [B], alpha [A] ENTER F ENTER,, nd nd

24 4 Online Technology Guide Section., page Keystrokes for TI- and Voyage Enter the matrices into matrix A and matrix B. To find the transition matrix, use the following keystrokes. nd [MATH] 4 4 nd [MATH] 4 B, A Programming Syntax for MATLAB Enter the matrices into matrix B and matrix BPRIME. rref([bprime B]) Programming Syntax for Maple with(linalg); Enter the matrices into matrix B and matrix BPRIME. rref(augment(bprime, B)); Programming Syntax for Mathematica Enter the matrices into matrix B and matrix BPRIME. <<LinearAlgebra`MatrixManipulation` RowReduce[AppendRows[BPRIME, B]] // MatrixForm Programming Syntax for Derive Enter the matrices into matrix B and matrix BPRIME. Row_Reduce(Append_Columns(BPRIME, B)) ENTER EXAMPLE The Length of a Vector in R n (a) Find the length of v,,, 4,. Keystrokes for TI-, TI- Plus, and TI-4 Plus These graphing utilities cannot find the length of a vector. Keystrokes for TI- Enter the vector into vector V. To find the length, use the following keystrokes. nd [VECTR] F F nd [M] F ENTER Keystrokes for TI- Store the vector in V. To find the length, use the following keystrokes. nd [MATH] 4 alpha [H] alpha [V] ENTER

25 Keystrokes and Programming Syntax for Selected Examples Keystrokes for TI- and Voyage Store the vector in V. To find the length, use the following keystrokes. nd [MATH] 4 H V ENTER Programming Syntax for MATLAB Enter the vector into vector v. norm(v) Programming Syntax for Maple with(linalg); Enter the vector into vector v. norm(v, ); Programming Syntax for Mathematica Enter the vector into vector v. <<LinearAlgebra`MatrixManipulation` VectorNorm[N[v], ] Programming Syntax for Derive Enter the vector into vector v. Abs(v) Section., page, Technology Note Find the unit vector for v, 4. Keystrokes for TI-, TI- Plus, and TI-4 Plus These graphing utilities cannot find the unit vector. Keystrokes for TI- Enter the vector into vector V. To find the unit vector, use the following keystrokes. nd [VECTR] F F nd [M] F ENTER Keystrokes for TI- Store the vector in V. To find the unit vector, use the following keystrokes. nd [MATH] 4 alpha [L] alpha [V] ENTER

26 Online Technology Guide Section., page Keystrokes for TI- and Voyage Store the vector in V. To find the unit vector, use the following keystrokes. nd [MATH] 4 L V ENTER Programming Syntax for MATLAB Enter the vector into vector v. v/norm(v) Programming Syntax for Maple with(linalg); Enter the vector into vector v. evalm(v/norm(v, )); Programming Syntax for Mathematica Enter the vector into vector v. <<LinearAlgebra`MatrixManipulation` v/vectornorm[n[v], ] Programming Syntax for Derive Enter the vector into vector v. v/abs(v) EXAMPLE 4 Finding the Dot Product of Two Vectors Find the dot product of u,,, and v,, 4,. Keystrokes for TI-, TI- Plus, and TI-4 Plus These graphing utilities cannot find the dot product. Keystrokes for TI- Enter the vectors into vector U and vector V. To find the dot product, use the following keystrokes. nd [VECTR] F F4 nd [M] F, F ENTER Keystrokes for TI- Store the vectors in U and V. To find the dot product, use the following keystrokes. nd [MATH] 4 alpha [L] alpha [U], alpha [V] ENTER

27 Keystrokes and Programming Syntax for Selected Examples Section., page Keystrokes for TI- and Voyage Store the vectors in U and V. To find the dot product, use the following keystrokes. nd [MATH] 4 L U, V ENTER Programming Syntax for MATLAB Enter the vectors into vector u and vector v. dot(u, v) Programming Syntax for Maple with(linalg); Enter the vectors into vector u and vector v. dotprod(u, v); Programming Syntax for Mathematica Enter the vectors into vector u and vector v. u.v Programming Syntax for Derive Enter the vectors into vector u and vector v. u.v EXAMPLE Using the Inner Product on C [, ] (Calculus) Use the inner product defined in Example and the functions f x x and g x x in C, to find (b) d f, g. Keystrokes for TI-, TI- Plus, and TI-4 Plus To find d f, g, use the following keystrokes. nd MATH X,T,,n X,T,,n x x, X,T,,n,, ENTER Keystrokes for TI- To find d f, g, use the following keystrokes. nd nd [CALC] F x-var x-var x x, x-var, ENTER,

28 Online Technology Guide Keystrokes for TI-, TI-, and Voyage To find d f, g, use the following keystrokes. nd F X X, X,, > > ENTER Programming Syntax for MATLAB MATLAB cannot calculate inner products. Programming Syntax for Maple sqrt(int((x x^)^, x..)); Programming Syntax for Mathematica Sqrt[Integrate[(x x^)^, {x,, }]] Programming Syntax for Derive sqrt(int((x x^)^, x,, )) Section.4, page EXAMPLE Solving the Normal Equations Find the solution of the least squares problem Ax b c c. Keystrokes for TI-, TI- Plus, and TI-4 Plus Enter the data into lists L and L. To find the least squares regression line, use the following keystrokes. STAT 4 nd [L], nd [L] ENTER Keystrokes for TI- Use nd [STAT] F to enter the data. There should be a in each row of the f Stat column. To find the least squares regression line, use the following keystrokes. nd [STAT] F F ENTER

29 Keystrokes and Programming Syntax for Selected Examples Keystrokes for TI- Enter the data into c and c using the Data/Matrix Editor. To find the least squares regression line, use the following keystrokes while in the Data/Matrix Editor. F alpha [c] alpha [c] ENTER ENTER Keystrokes for TI- and Voyage Enter the data into c and c using the Data/Matrix Editor. To find the least squares regression line, use the following keystrokes while in the Data/Matrix Editor. F c c ENTER ENTER Programming Syntax for MATLAB x [ ]; y [ ]; polyfit(x, y, ) Programming Syntax for Maple with(statistics); xvalues := vector([,, ]); yvalues := vector([,, ]); Fit( a*x b, xvalues, yvalues, x); Programming Syntax for Mathematica data {{, }, {, }, {, }} Fit[data, {, x}, x] Programming Syntax for Derive Fit([x, a*x b], [[, ], [, ], [, ]])

30 Online Technology Guide Section.4, page EXAMPLE Application to Astronomy Table. 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.), and the period is provided in years. Find a model for these data. (Source: CRC Handbook of Chemistry and Physics) TABLE. Planet Mercury Venus Earth Mars Jupiter Saturn Distance, x Period, y Keystrokes for TI-, TI- Plus, and TI-4 Plus Enter the data into lists L and L. To find the power model, use the following keystrokes. STAT ALPHA [A] nd [L], nd [L] ENTER Keystrokes for TI- Use nd [STAT] F to enter the data. There should be a in each row of the f Stat column. To find the power model, use the following keystrokes. nd [STAT] F MORE F ENTER Keystrokes for TI- Enter the data into c and c using the Data/Matrix Editor. To find the power model, use the following keystrokes while in the Data/Matrix Editor. F alpha [c] alpha [c] ENTER ENTER Keystrokes for TI- and Voyage Enter the data into c and c using the Data/Matrix Editor. To find the power model, use the following keystrokes while in the Data/Matrix Editor. F c c ENTER ENTER Programming Syntax for MATLAB x [ ]; y [ ]; polyfit(log(x), log(y), )

31 Keystrokes and Programming Syntax for Selected Examples Section., page Programming Syntax for Maple with(statistics); xvalues := vector([.,.,.,.,.,.4]); yvalues := vector([.4,.,.,.,.,.4]); Fit( a*x ^b, xvalues, yvalues, x); Programming Syntax for Mathematica <<Statistics`NonLinearFit` data = {{.,.4}, {.,.}, {.,.}, {.,.}, {.,.}, {.4,.4}} NonlinearFit[data, a*x^b, x, {a, b}] Programming Syntax for Derive Derive cannot calculate power models. EXAMPLE Finding the Cross Product of Two Vectors Provided that u i j k and v i j k, find (a) u v. Keystrokes for TI-, TI- Plus, and TI-4 Plus These graphing utilities cannot find the cross product. Keystrokes for TI- Enter the vectors into vector U and vector V. To find the cross product, use the following keystrokes. nd [VECTR] F F nd [M] F, F ENTER Keystrokes for TI- Store the vectors in U and V. To find the cross product, use the following keystrokes. nd [MATH] 4 alpha [L] alpha [U], alpha [V] Keystrokes for TI- and Voyage Store the vectors in U and V. To find the cross product, use the following keystrokes. nd [MATH] 4 L U, V ENTER ENTER

32 Online Technology Guide Section., page 4 Programming Syntax for MATLAB Enter the vectors into vector u and vector v. cross(u, v) Programming Syntax for Maple with(linalg); Enter the vectors into vector u and vector v. crossprod(u, v); Programming Syntax for Mathematica Enter the vectors into vector u and vector v. Cross[u, v] Programming Syntax for Derive Enter the vectors into vector u and vector v. Cross(u, v) EXAMPLE Finding Eigenvalues and Eigenvectors Find the eigenvalues of A. Keystrokes for TI-, TI- Plus, and TI-4 Plus These graphing utilities cannot find eigenvalues or eigenvectors. Keystrokes for TI- To find the eigenvalues and eigenvectors, use the following keystrokes. nd [MATRX] F F4 nd [M] F ENTER nd [M] F nd [M] F ENTER Keystrokes for TI- To find the eigenvalues and eigenvectors, use the following keystrokes. nd [MATH] 4 alpha [A] ENTER nd [MATH] 4 alpha [A] alpha [A] ENTER

33 Keystrokes and Programming Syntax for Selected Examples Keystrokes for TI- and Voyage To find the eigenvalues and eigenvectors, use the following keystrokes. nd [MATH] 4 A ENTER nd [MATH] 4 A A ENTER Programming Syntax for MATLAB [V, D] eig(a) Programming Syntax for Maple with(linalg); eigenvalues(a); eigenvectors(a); Programming Syntax for Mathematica Eigensystem[A] // MatrixForm Programming Syntax for Derive Eigenvalues(A) Derive cannot find eigenvectors.