A5 - Math 235 Assignment 5 Due Wednesday June 15th 1 Find a...

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Unformatted text preview: Math 235 Assignment 5 Due: Wednesday, June 15th 1. Find a basis for the orthogonal complement of S 2 Span { [i g] , E 31] } under the standard inner product for A42X2(R). 2. On P2 define the inner product (p,q) :2 p(-l)q(——l) + p(0)q(0) + p(l)q(l) and let S = Span{1,x —— $62}. (a) Use the Gram—Schmidt procedure to determine an orthogonal basis for S. (b) Use the orthogonal basis you found in part (a) to determine proj5(1 + a: + 51:2). 1 3. Define the inner product (5, g) 2 2x1y1 + 5023/2 + 3x3y3 on R3. Extend the set 0 —1 to an orthogonal basis for R3. 4. Suppose that W is a Ic—dirnensional subspace of an inner product space V. Prove that projW is a linear operator on \V with kernel Wt. 5. Let V be an n dimensional inner product space and let S be a subspace of V. Prove that projgi a? : perpS 53’ for all a? E V. 6. If {171, . . . ,fik} is an orthonormal basis for a subspace S of R", verify that the standard matrix of projS can be written - _. ~+T —» 4’ —» 4“ [prOJS] = @1711 + @2212 + - - - + vkvk Use MATLAB to complete the following questions. You do not need to submit a printout of your work. Simply use MATLAB to solve the problems, and submit written answers to the questions along with the rest of your assignment. For questions that involve a set of vectors, enter each vector separately, giving it a name (e.g., 31, S2, s3) and then construct any required matrices from these vectors (e.g., A = [$1 $2 33]). This will make it easier for you to work with the various elements. Orthogonality and Gram—Schmidt Gram-Schmdit MATLAB can help us quickly find an orthonormal basis for a set of vectors. Complete the following steps to find an orthonormal basis for the column space of the matrix 3 ——5 1 1 1 1 A- ——1 5 ——2 3 —7 8 1. Enter the matrix A and create vectors $1, $2, and 5233 representing the columns of A. >> A = [3 -5 1; 1 1 1; *1 5 -2; 3 '7 8] >> X1 = A(:,1) >> x2 = A(:,2) >> x3 = A(:,3) 2. Begin the Grain—Schmidt process by setting 121 = 5131. >> v1 = x1 3. Calculate 02 and v3. x2 — (x2’*v1)/(v1’*v1)*v1 X3 - (X3’*v1)/(v1’*v1)*v1 - (x3’*v2)/(v2’*v2)*v2 >> v2 >> v3 II 4. Verify that {121,122,213} is an orthogonal set. >> V = [v1 v2 v3] >> V’*V The non~diagonal entries of the resultant matrix should be 0’s, showing that the vectors are mutally orthogonal. What are the diagonal entries? 5. Normalize {121, 122, 03} to create an orthonormal basis. >> n1 = v1/norm(v1) >> n2 = v2/norm(v2) >> n3 = v3/norm(v3) 6. Verify that {711,712,713} is an orthonormal set. >> N = [n1 n2 n3] >> N’*N The diagonal entries of the resultant matrix should be 1’s with 0’s elsewhere (i.e., the identity matrix), showing that the vectors are mutally orthonormal. Question 1 (a) Using MATLAB and the Gram—Schmidt process, find an orthonormal basis for the column space of —3 2 3 —1 7 1 —3 2 —6 2 0 5 1 O 4 4 O —5 A = 0 ~53 M7 2 ~41 3 2 O 5 2 —4 3 0 3 ~1 2 1 3 1 1 2 2 —2 -1 (b) Verify that the basis you found is orthonormal. Finding an Orthonormal Basis—the Easy Way! Using MATLAB to help you complete the steps of the Gram~Schmidt process certainly makes things faster and easier (and avoids errors), however, there is an even better way to find an orthonormal basis using MATLAB! The orth command returns an orthonormal basis for the range (i.e., column space) of a matrix. 3 ~33 1 For example, an orthonormal basis for the column space of the matrix A x .j g _; 3 ——7 8 can be found as follows: >>A=[3‘51;111;-15-2;3~78] >>N=orth(A) The columns of the matrix N are the orthonormal basis vectors. To verify that these vectors are indeed orthonormal, check that: >> N’*N Which should be the identity matrix. W Question 2 (a) Use the orth command to find an orthonormal basis for the matrix in Question 1. (b) Verify that the basis you found is orthonormal. ...
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