Math_235-Assig_4-F09

Math_235-Assig_4-F09 - Math 235 Assignment 4 Due:...

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Unformatted text preview: Math 235 Assignment 4 Due: Wednesday, Oct 14th 1. Determine which sets of vectors are orthonormal in R3 under the standard inner product. If a set is only orthogonal, normalize the vectors to produce an orthonormal set. 1/3 —2/3 1 —1 1 a) 2/3, 2/3. b) 2, o , —1 2/3 —1/3 1 1 1 2. Determine which sets of vectors are orthonormal in M (2,2) under the inner product < A, B >2 tr(ATB). If a set is only orthogonal, normalize the vectors to produce an orthonormal set. a) [3, 2], [31 32]. b) [s 1H; :3], [3 33]. 3. Find the coordinates of (5,0, —2, 2) and (4, 2, —2,3) in R4 under the standard inner product with respect to the orthonormal basis 1 1 1 1 B = 1, la 1), _1’ 17 ‘1), E(—1!03 130)) 1,0: 4. Prove that the product. of two orthogonal matrices is an orthogonal matrix. 5. Prove that if R is an orthogonal matrix, then det R = 21:1. Give an example of a matrix A that has det A = 1, but is not orthogonal. 6. Observe that the dot product of two vectors 15,17 6 R" can be written as —o —oT—v f-y=2: J. Use this fact to prove that if an n x 12 matrix R is orthogonal, then = for every 23’ 6 IR". —o 7. Let {171, . . .,17n} be an orthonormal basis for V, and let :3 = Clo] + + one}, and g‘ = (11171 + - - - + dnfin. Show that < 5,27>= eidi + - - - +cndn, and “a”? = cf + . . . + 03V 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., 51, 32, S3) and then construct any required matrices from these vectors (eg, A = [51 s2 33]). This will make it easier for you to work with the various elements. Orthogonal Vectors Are the following sets of vectors in R10 orthogonal? (a) a1 = (310)—2asa-11_7,210:0) a2 = (2,—8,7,2,6,1,3,4, 1,5) (13 = (5, —2, 3, 1, —3, 0, —6, 2, 4, —5) (b) b1 = (5, ——6,4, —6, —2,2,5, —8,8,5) (22 — (0,—1,—1,—4,0,0,6,6,3,—2) ()3 = (6, 1, —3,s,7, 1,2,2, —6, —4) ()4 = (—1, —5,7, —6, —5,—6, —5, —1,—4,8) [\3 ...
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Math_235-Assig_4-F09 - Math 235 Assignment 4 Due:...

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