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265Matlab10.3Ex

# 265Matlab10.3Ex - LAB 1o i 1g Exercises 10.3 1 Let V 2 R3...

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Unformatted text preview: LAB 1o , i - 1g Exercises 10.3 1. Let V 2 R3 with the standard inner product and let 1 1 1 S = {114,112, 1‘3}: 2 1 O , O 0 0 1 Use routine gschmidt in MATLAB to obtain an orthonormal basis T and then ﬁnd the 1 coordinates of a: = 2 3 relative to T. Record the orthonormal basis and the coordinates ii _,L: of rebelow. 2. Let V = R4‘with‘lthe Standard inner product and let —1 2 0 1 2 1 1 1 S =, {"111‘2’ “'39 “4}: 0 7 1 7 0 1 0 1 0 1 1 Use routine gschmidt in MATLAB to obtain an orthonormal basis T and then ﬁnd the .4 coordinates of a: = (2) relative to T. Record the orthonormal basis and the coordinates 1 of :1: below. 3. Let V = R4 with the standard inner product and let LAB: 10 WWﬂW/mwwﬂxwmw .5 . 5 .5 5 .5 .5 —.5 — 5 S = {u1,u2,u3,u4} = '5 , _ 5 , _‘5 7 5 .5 — 5 .5 — 5 a) Is S an orthonormal basis? Circle one: Yes No Explain your answer. b) In MATLAB form the matrix T whose columns are the vectors in S. Generate a random vector in R4 using command 3: == rand(4,1) and then compute H at H and M Tu: H. How are the values of the norms related? Repeat the experiment for another arbitrary vector. ' 1 1 4. Let '01 = 2 and '02 = 3 . In MATLAB form the matrix A = [v1 v2] and then use 2 1 command gschmidt(A). Explain the meaningof the display generated. ' 1 i 0 5. Let A — z' 0 1‘ a) In MATLAB use command A’. Record the result. . A’ = b) In MATLAB use command C = A'*A. Record the result. C = c) What is the relation between C and C"? LAB 10 i d) Experiment with other complex matrices A to conﬁrm err part c). - t Circle one: conﬁrmed not confirmed; 6. A complex matrix A is called Hermitian if it is equal to its conjugate transpose The command A’ gives the conjugate transpose in MATLAB . ' a) How can you use MATLAB to determine if the matrix A below is Hermitian? 2 " 3—32' 3+3z' 5 b) Compute r = az’ * A a: a: for the complex vector below. m— i r— — l—z' Is r a real number? (Circle onez) YES NO 0) Experiment With Other complex vectors a: to determine whether :12’ Am will always be a real number.- - (Circle one:) Always a real number for this matrix A. Not always a real number. d) Experiment with another Hermitian matrix A and arbitrary vector :13 to see if r = az’ * A an a: is always a real number. (Circle onez) Always a real number. Not always a real number. 7. Let V = R4 with the standard inner product and let 3 1 0 1 —1 —2 '01 = 2 , '02 “ _1 , ’U3 “' 1 . 0 1 —1 a) Find an orthonormal basis for R4 containing scalar multiples of the vectors v1 and v2. Record your result below. LAB 10 I b) Find an orthonormal basis for R4 containing scalar multiples of the Vectors '01, '02, '03. Record your result below. << NOTES; COMMENTS; IDEAS >> LAB 1o ...
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265Matlab10.3Ex - LAB 1o i 1g Exercises 10.3 1 Let V 2 R3...

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