SOLVING SIMULTANEOUS EQUATIONS. Builtin rref(a) EXAMPLE E4.1 Solve Corresponding augmented matrix 2x 4y 2 a 242 x 3y 3

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1 Lecture 4 matrices and progressions Open Lecture 4 on class website: ELEMENTARY ROW OPERATIONS swap mult add_mult Given a matri, there are three types of elementary row operations: You may v switch (permute) rows, -- swap v multiply a row by a nonzero constant, -- mult v add a multiple of one row to another row -- add_mult To pivot on the i-jth entry of a matri (assumed nonzero) means using elementary row operations to make that entry 1 and all other entries in the jth column 0. The leading coefficient of a row is the first nonzero entry. A matri is in refuced row echelon form (rref) iff the leading coefficient of each row is 1 and it is to the right of the leading coefficient of the previous row CLASSWORK C4.1(3) pivot Together. Write a function pivot(a,i,j) which pivots matri a on row i and column j. Assume a(i,j)= 0. List a sequence of pivots and swaps (see H3.2(3)) which converts a=[0,2,3,5;2,3,4,6;5,6,6,7] to reduced row echelon form (rref) Copy lines to SciNotes, paste, File/Save as c4.1(3)pivot //c4.1(3)pivot pivot(a,i,j) pivots on a(i,j) //delete this line, write the function swap //delete this line, write the function pivot a=[0,2,3,5;2,3,4,6;5,6,6,7] //convert a to rref. clc;disp(a) //delete this line, add a sequence of pivots and swaps to get rref SOLVING SIMULTANEOUS EQUATIONS. Builtin rref(a) EXAMPLE E4.1 Solve Corresponding augmented matri 2 4y 2 a 242 3y Solution Reduced row echelon form: 3 y 2 rref a a=[2,4,2; 1,3,3] rref(a) Eecute, get the solution from the rref matri, hand-write the general answer as a comment. // =-3, y=2

2 2y 3z EXAMPLE E4.2 Solve 4 5y 6z 7 rref y 7z Hence z 2, hence 2 z where z is arbitrary. In this case, there are infinitely many solutions, one for each choice of the arbitrary parameter z. a=[1,2,3,4; 4,5,6,7; 5,6,7,8] rref(a) Eecute, get the solution from the rref matri, hand-write the general answer as a comment. // =-2+z, y=3-2, z is arb CLASSWORK C4.2(2) rref 6y 4z 5 12y 2z 3 8y z 0 Enter the augmented matri line and the rref line in SciNotes, not Scilab. Copy lines, File/New in SciNotes, paste, File/Save as c4.2(2)rref. // c4.2(2)rref Two SciNote lines, plus comment. Don t copy answers from SciLab. //delete this, fill in the matri and rref(a) lines, need disp(a), disp(rref(a)). [0,1,5] row vector [0;1;5] column vector [0:1:5] start with 0, step size 1, end with 5 0:2:8 step size 2, [ ] may be omitted. [1:.1:2] step size.1 0:45 abbreviation for [0:1:4]=[0,1,2,3,4,5] [8:1:2] [8:-1:2] for i=[3,1,5]; disp(i); end for i=3:5; disp(i); end CLASSWORK C4.3(1) progression Write [2.0,1.9,1.8,...,0.0]as a progression: [n:s:m] Copy lines, paste into SciNotes, File/Save as c4.3(1), fill the 3 spaces. //c4.3(1)progression [1.0,.9,.8,...,0.0] disp([ : : ]) AREAS AND INTEGRALS Builtin intg(f,a,b) b a f is the integral of function f from a to b, it is the signed area between the function and the -ais over the interval a, b. // =1.49,y=,z= //Write answer in the blanks above, rounded to two decimal places. ARITHMETIC PROGRESSIONS [n:s:m].

3 1 a f() -1-1 Area below the -ais is considered negative. In Scilab we write intg(a,b,f) for b a f where f must be a user-defined function. EXAMPLE 4.3 Find 1 cos 2 function y=f() y=1-(cos())^2; disp(intg(-%pi,%pi,f)) // should get = CLASSWORK C4.4(2) semicircle Together. Calculate the area of the unit semicircle, the top half of 2 y 2 1. For a circle of radius is 1, the area is r The semicircle area is half of this /2 3.14/ b 1 In 2 y 2 1, solving for y gives y 1 2 f 1 2 is the upper semicircle. Define the upper semicircle function f. Use the integral intg(a,b,f) to get upper semicircle area. Copy lines, File/New in SciNotes, paste, File/Save as c4.4(2)semicircle. //c4.4(2)semicircle Define upper semicircle function. //Use intg to find the circle area. Ans: 3.14 b To approimate the integral, divide the line segment a, b into equal-width segments. Above each segment draw a rectangle whose height is the function s value. The sum of the (signed) areas of these rectangles is a Riemann sum. The smaller the rectangle width, the more accurate the approimation. f() a b CLASSWORK C4.5(4) my_intg Together. Write a function my_intg(a,b) which uses Riemann sums with rectangles of width, 1/10 5 to calculate the integral b a f d of a user-defined function f.

4 Define the function f 1 2 Find the integral using your function my_intg(a,b) Copy lines, File/New in SciNotes, paste, File/Save as c4.5(4)my_intg. //c4.5(4)my_intg Riemann sums. Ans: 1.57 function y=f() y=sqrt(1-^2) function rsum=my_intg(a,b) //delete this, write the function my_intg here clc;printf("\n") printf("my area = %f \n", my_intg(-1,1)) printf("scilab''s = %f \n",intg(-1,1,f)) CLASSWORK C4.6(3) pos_prod Together. Write a function pos_prod(a) which finds the product of the positive entries in the matri a. Test with the given lines. Copy lines, File/New in SciNotes, paste, File/Save as c3.4(2)pos_prod(a). //c4.6(3)pos_prod pos_prod(a)=product positive entries //delete this, write the function pos_prod(a) a = [1,0,-2; 3,0,-4; -5,6,0]; //ans=18 disp(a),printf(" pos_prod = %i \n",pos_prod(a)) a = [1,-2; 3,-4]; //ans=3 disp(a),printf(" pos_prod = %i \n",pos_prod(a)) CLASSWORK PROBLEMS DUE END OF CLASS C4.1(3)pivot, C4.2(2)rref, C4.3(1)progression, C4.4(2)semicircle, C4.5(4)my_intg, C4.6(3)pos_count to: dale@math.hawaii.edu subject line: 190 c4(15) HOMEWORK H4.1(2) pivot Use the classwork function pivot(a,i,j)to convert a=[3,0,3,5;2,0,4,6;5,6,6,7] to reduced row echelon form. //h4.1(2)pivot add lines to convert //a=[3,0,3,5; 2,0,4,6; 5,6,6,7] HOMEWORK H4.2(2) rref 16y 4z 50 12y 2z y 6z 80 Enter the augmented matri line and the rref line in SciNotes, not Scilab. // h4.2(2)rref SciNote lines, plus comment. Don t copy answers from SciLab. //delete this, fill in the matri and rref(a) lines // =,y=, z= //Write answer in the blanks above, rounded to two decimal places. HOMEWORK H4.3(2) my_intg Define the function f sin Find the integral from 0 to using your function my_intg(a,b). You need to have function code for f and function code for my_intg. No credit for using the built-in intg. //h4.3(2)my_intg integral sin() from 0 to pi //Ans: 2

5 HOMEWORK H4.4(3) countup In SciNotes, write a function countup(n)which generates the nn matri whose entries, when read in the row-column order are 1,2,3,...,n*n. For n=2, countup(2)=[1,2; 3,4]. For n=3, countup(3)=[1,2,3;4,5,6;7,8,9] Fill in each of the two blanks with a single letter/digit. //h4.4(3)countup function b=countup(n) C=1 for i=1:n, for j=1:n b(i,j)= //hint: single letter C = C+ // hint: single digit end; end for i=2:4 disp (countup(i)) end HOMEWORK H4.6(3) pos_count Write a function pos_count(a) which counts the number of positive entries in the matri a. Test with the given lines. Copy lines, File/New in SciNotes, paste, File/Save as h4.6(3)pos_count(a). //h4.6(3)pos_count pos_count(a)=# positive entries //delete this write the function pos_count(a) here a = [1,0,-2;3,0,-4; -5,6,0]; //ans=3 disp(a),printf(" pos_count = %i \n",pos_count(a)) a = [1,-2;3,-4]; //ans=2 disp(a),printf(" pos_count = %i \n",pos_count(a)) HOMEWORK DUE BEFORE NEXT CLASS - QUIZ ON ONE OF THESE H4.1(2)pivot, H4.2(2)rref, H4.3(2)my_intg, H4.4(3)countup, H4.5(3)pos_sum, H4.6(3)pos_count to: dale@math.hawaii.edu subject line: 190 h4(15) HOMEWORK H4.5(3) pos_sum Write a function pos_sum(a) which totals the positive entries in the matri a. Test with these lines. Like classwork pos_prod(a) but a growing sum instead of a growing product. //h4.5(3)pos_sum pos_sum(a)=sum of positive entries //delete this. write the function pos_sum (a) here, a = [1,0,-2;3,0,-4; -5,6,0]; //ans=10 disp(a),printf(" pos_sum = %i \n",pos_sum(a)) a = [1,-2;3,-4]; //ans=4 disp(a),printf(" pos_sum = %i \n",pos_sum(a))