COLUMN VECTORS a=[1;2;3] b. a(1) = first.entry of a

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1 Lecture 2 vectors and matrices ROW VECTORS Enter the following in SciLab: [1,2,3] scilab notation for row vectors [8]==8 a=[2 3 4] separate entries with spaces or commas b=[10,10,10] a+b, b-a add, subtract the respective coordinates 2*a, a+1 scalar product and addition a**2, 2**a a*b wrong dimensions for matrix multiplication a.*b pointwise multiplication not matrix muliplication [a,b] a followed on the right by b., means continues on right. a(1) = first.entry of a first row a, second row b.; starts new row below. a(length(a)) = last entry of a [a;b] ; means starts new row below. a(1), a(2), a(3)... a(i)= ith entry of vector a. For the first time this semester, we are going to use a for loop. A for loop is a statement which allows the computer to execute repeatedly, for a given number of iterations, a sequence of instructions. for i=[3,1,5], disp(2*i), end In this example, the values of the entries of vector [3,1,5] will be consecutively assigned to the loop variable i. For each of these values, the machines displays (using the built-in function disp) the value 2*i. Once all the values have been assigned, the loop stops. As a result, we obtain the output : 6 =2*i for i=3 2 =2*i for i=1 10 =2*i for i=5 1 COLUMN VECTORS a=[1;2;3] b b=b transpose a+b, b-a, a.*b operations are performed component wise [a,b] column a followed on the left by column b [a;b] append b below a. OPERATIONS ON VECTORS sum(a) sum of entries,sum( [4,3,2] ) = 9 prod(a) product of entries, prod( [4,3,2] ) = 24 max(a) largest entry max( [4,3,2] ) = 4 min(a) smallest entry min( [4,3,2] ) = 2 length(a) number of entries in the vector, length( [4,3,2] )=3 not the geometric length or magnitude of the vector! PROGRESSIONS [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 0:5 abreviation for [0:1:5] [5:-1:1] Open SciNotes. Reminder: All classwork and homework Scilab code lines including testing lines are to be entered in SciNotes (not in Scilab). c2.1 Select File/New. File/Save as c2.1 Copy/paste the template lines to SciNotes. Fill in the 3 blanks. Write as a progression ([n:s:m]) the vector:[2.0,1.9,1.8,..,0.0]] (should be 9 symbols with n + s + m = 1.9). //c2.1(1) [ : : ]

2 DOT (INNER) PRODUCT [a,b,c].[a,b,c ]= a*a +b*b +c*c = the dot product. The above formula only works for vectors of length 3. The dot-product function (built into Matlab but not Scilab) must work for vectors of any length. We have to define it. function P = dot product(a,b) //Type in SciNotes P = sum(a.*b) //3 function lines dot product([1,2],[3,4]) //2 testing lines dot product([1,2,3],[4,5,6]) //Answers: 11, 32. Scilab (Matlab) has lots of built-in functions. When asked to write a function, you are to write your own. Do not use a built-in function which does the same thing. This week, use only the functions provided in this lecture. e2.1 Write a function H(a) for the sum of the cubes of the components of a vector a, i.e., a a a 3 n. Test it on input provided by the user (see lecture 1). function x=h(a) x=sum(a**3) a=input( Enter a vector. > ) disp(h(a)) MAGNITUDE (GEOMETRIC LENGTH) a 2 + b 2 + c 2 This formula only for vectors of length 3. Let s write a function which works for vectors of any length. c2.2 File/Save as c2.2. Copy/paste the commented template lines to SciNotes. Define a function mag(a) which calculates the geometric length of the vector a (a can be any vector of any length). After the function, write lines which get a vector a from the user and display mag(a). //c2.2 mag(a)=geometric length of a. //Test on user input: a = input( Enter a vector. > ) NOTATION. In Matlab we often write a1 or a 1 for a 1. Variable names may include characters, numbers or underscores. MATRICES AND MATRIX MULTIPLICATION Enter in SciLab: a=[1 2; 3 4] b=[1,1;1,1] 2*b,a+b, a-b, a.*b, a*b [n,m]=size(a) n= number of rows m= number columns Matrix ( multiplication: ) ( ) ( ) a11 a 12 b11 b 12 a11 b = 11 + a 12 b 21 a 11 b 12 + a 12 b 22 a 21 a 22 b 21 b 22 a 21 b 11 + a 22 b 21 a 21 b 21 + a 22 b 22 The entry c ij in the ith row and jth column of the product is the dot product of the ith row of the first matrix and the jth column of the second matrix. 2

3 a=[1 2;3 5] a(1,2),a(2,1) a(i,j) = entry in ith row, jth column a(2,2)=4 changes a(2,2) to 4 a,a(:,:) a,a(:,1),a(:,2) a(:,j)=all rows of jth column=jth column a,a(1,:),a(2,:) a(i,:)=all columns of ith row=ith row a a(1,2)=10 a(:,2)=[9;9] a(1,:)=[0,0] b [b,[0;0]] appends new column on left [b;[0,0]] appends new row below z=b saves b to z b(1,:)= b(1,:)+2 b=z recovers b from z b(:,1)= b(:,1)-2 What does this do? IDENTITY MATRIX, MATRIX INVERSES b=ones(1,3) b=0*b b=ones(3,1) b=8*b I=eye(2,2) I is the 2x2 identity matrix 1 is the identity for multiplication of numbers: 1x = x1 = x. I=eyes(n) is the identity for matrix multiplication: if A is n n then Ia=aI=a. a, I*a, a*i x 1 is the inverse operation of multiplication of numbers: xx 1 = x 1 x = 1. The notation is the same for matrices. inv(a) is the inverse of the matrice a. a=[1,2;3,4] inv(a) a*inv(a),inv(a)*a e2.3 Write a function J(a) which adds a row of 8 s above the matrix a and then adds a column of 9 s on the right. Test it on [1 2; 3 4] and on [1 2 3; 4 5 6; 7 8 9]. function a=j(a) [n,m]=size(a) // n= number of rows, m= number columns a=[8*ones(1,m);a] a=[a,9*ones(n+1,1)] b=[1,2; 3,4],J(b) J([1,2,3; 4,5,6; 7,8,9]) c2.3 Select File/New in SciNotes. File/Save as c2.3. Paste template. Together, one step-and-test at a time. Suppose a is a matrix with 2 or more rows. Write a function K(a) which adds 1 to a s bottom right entry. Then negates (that is to say multiply by -1) a s next-to-the-last row. Then appends a column of 0 s to the right. Test it on [1,2; 3,4],[1,1,1,1; 2,2,2,2; 3,3,3,3]. //c2.2 Function K(a) adds one to bottom right entry. // Negates next-to-last row. Appends a column of 0s on the // right. // Test on [1,2; 3,4;5,6] and on [1,1,1,1;2,2,2,2;3,3,3,3] // Answers: [1,2,0;-3,-4,0;5,7,0], // [1,1,1,1,0;-2,-2,-2,-2,0;3,3,3,4,0] 3

4 SOLVING SIMULTANEOUS EQUATIONS In mathematics, solving a linear system of equations can be performed by computing the so-called reduced row echelon form of the augmented matrix corresponding to the system. e2.4 Solve 2x + 4y = 2 x + 3y = 3 Corresponding ( augmented ) matrix a = We perform operations on the rows of the matrix a to compute its reduced row echelon form: all zero rows are at the bottom of the matrix, the the leading coefficient of a non-zero row appears to the right of the leading coefficient of the row above it, all the entries in a column below a leading entry are zeroes. The solution of the system is deduced from the last column of the reduded row echelon form of the a. In scilab enter [2,4,2;1,3,3] rref(a) Solution x = 3 y = 3 Corresponding ( augmented ) matrix a = [ ] We obtain rref(a)= which means x + 2y = = 0 Hence x = 3 2y where y is arbitrary (it can be any number). c2.4 Select File/New, File/Save as c2.4 Solve x 16y + 4z = 50 x + 12y + 2z = 30 2x 4y + 6z = 80 Enter the augmented matrix line and the rref line in SciNotes. After //, hand-write the answer to two decimals places. //c2.4 Two SciNote lines, plus comment. Dont copy answers from //SciLab. // x=0, y=... c2.5 Select File/New, File/Save as c2.5 Solve x 16y + 4z = 50 x + 12y + 2z = 30 x + 68y z = 0 Enter the augmented matrix line and the rref line in SciNotes. //c2.5 Two SciNote lines, plus comment. Dont copy answers from //SciLab. // x=1.48, y=... e2.5 Solve 2x + 4y = 6 x + 2y = 3 a=[2,4,6; 1,2,3] rref(a). In scilab enter 4

5 HOMEWORK 2 DUE BEFORE FRIDAY JANUARY 30TH s CLASS h2.1(3), h2.2(3), h2.3(3) to: gautier@math.hawaii.edu subject line: 190 h2(9) h2.1(3). The distance between 2 vectors [a 1,..., a n ] and [b 1,..., b n ] is given by (a 1 b 1 ) (a n b n ) 2. Write a Scilab function for the distance D=distance(a,b) between vectors a and b. Use it to find the distance between [1,2] and [3,3] and the distance between [1,2,3] and [3,3,3]. Your function must work for vectors of any length, it may not use See example e2.1 above. Don t send answers. Try entering the formula one step-and-test at a time. // h2.1(3)function D = distance(a,b) // Distance between [1,2],[3,3] and between // [1,2,3],[3,3,3], 3 function lines, 2 test lines. // Answers: , h2.2(3) In SciNotes, write a Scilab function F(a) which, on a matrix a, appends a row of 0 s to the bottom of a and then appends a column of 3 s on the right of the resulting matrix. Hint: 8*ones(r,1) is a column of r 8 s (see example e2.3). Test it on [1,2,3; 4,5,6] and on [1,2; 3,4; 5,6 ]. Your function must work for any matrix of any size. / h2.2(3) Function F(a) // appends a row of 0s at the bottom of a, // then appends a column of 3 s on the right. // Test on [1,2,3; 4,5,6]. //Answer: [1,2,3,3; 4,5,6,3; 0,0,0,3] // Test on [1,2; 3,4; 5,6]. //Answer: [1,2,3; 3,4,3; 5,6,3; 0,0,3] h2.3(3) In SciNotes, write a Scilab function G(a) which, on a matrix a, multiplies the first column by 10, then subtracts 100 from the last row. Hint: n=length(a(:,1))= index of the last row. (see example e2.3). Test it on [1,2,3; 4,5,6;] and on [1,2; 3,4; 5,6]. Your function must work for any matrix of any size. // h2.2(3) Function G(a) // multiplies first column by 10 then // then subtracts 100 from the last row // Test on [1,2,3; 4,5,6]. //Answer: [10,2,3; -60,-95,-94] // Test on [1,2; 3,4; 5,6]. //Answer: [10,2; 30,4; -50,-94] 5

MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra

MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra A. Vectors A vector is a quantity that has both magnitude and direction, like velocity. The location of a vector is irrelevant;