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Math 307: Problems for section 3.3

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Question;1. Show that for v, w? Cnv+w2=v2+w2+ 2 Re(v, w)and use this to prove the polarization identityv, w =14v+w2? v?w2+ i v? iw2? i v + iw22. Show that if q1, q2,..., qn form a basis in Rn, then they also form a basis when regarded asvectors in Cn. In other words, show that 1.) if the only linear combination c1 q1 + ? ? ? + cn qnusing real numbers c1,..., cn that equals zero has c1 = ? ? ? = cn = 0, then the same is true forcomplex numbers, and 2.) if every vector in Rn can be written as c1 q1 + ? ? ? + cn qn for somereal numbers c1,..., cn then every vector in Cn can be written as a linear combination usingcomplex numbers. If the basis q1, q2,..., qn is orthonormal in Rn is is also orthonormal inCn?3. Show that any 2 ? 2 orthogonal matrix is either a rotation matrix or a re?ection matrix.4. Let Q = q1 |q2 | ? ? ? |qkwhere q1, q2,... qk? Rn form an orthonormal set. (That is, theysatisfy qi = 1 for i = 1,..., k and qi ? qj = 0 if i = j, but there might not be enough vectorsto form a basis, i.e., possibly k < n). Identify the matrices QT Q and QQT. Show thatthe projection p of a vector v onto the subspace spanned by q1, q2,... qk can be writtenp = k=1 qi qT v = k=1 (qi ? v)qi.iii5. For an m?n matrix A with linearly independent columns there is a factorization (called theQR factorization) A = QR where Q is an m ? n matrix whose columns form an orthonormalset, and R is an upper triangular matrix. For every k = 1, 2,... n the?rst k columns of Qspans the same subspace as the?rst k columns of A. In MATLAB/Octave the matricesQ and R in the QR decomposition of A are computed using [Q R] = qr(A,0). (Withoutthe second argument 0 a related but di?erent decomposition is computed.)(For those ofyou who have learned about Gram-Schmidt: The columns of Q are the vectors obtainedby applying the Gram-Schmidt procedure to the columns of A.Using MATLAB/Octave, compute and orthonormal basis q1, q2 for the plane in? 4 spanned?????R1?11?1??1??1?by a1 =?? and a2 =?? Compute the projection p of the vector v =?? onto the?1??1??1?11?1plane. What are the coe?cients of p when expanded in the basis q1, q2?16. Using MATLAB/Octave and the discussion in the previous problem,?nd an orthonormal??????110?1??0??0????????2??1??1?set of vectors q1, q2 and q3 with the same span as???? and??. Provide the commands?0??0??1????????0??1??1?000that you used.7. Do the following computational experiment. First start with a random symmetric 10 ? 10matrix A (for example B=rand(10,10), A=B?*B, will produce such a matrix) and computeits QR factorization. Call the factors Q1 and R1. Now multiply Q1 and R1 in the ?wrong?order to obtain A2 = R1 Q1 and compute the QR factorization of the resulting matrix A2.Repeat this step to obtain a sequence of matrices Qk, Rk and Ak. Do these sequencesconverge? If so can you identify the limit? (Hint: eig(C) computes the eigenvalues of C).8. If U1 and U2 are unitary matrices, is U1 U2 a unitary matrix too?9. If q1,..., qn is an orthonormal basis for Cn do the complex conjugated vectors q1,..., qnform an orthonormal basis as well? Give a reason.

 

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