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hwsolB18

hwsolB18 - Sheet1 Page 2 i.e w1 = v1-2*w1 w2 = v2(1/2*w1 w2...

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Sheet1 Page 1 HW 18 Solutions _______________ V = [ -2.8867 4.0276 2.1607 0.0369 0.3649 -6.1858 0.0999 -6.0291 0.4916 4.0812 0.5427 2.7973 ] (i) Rounded to four significant digits: V'*V = [ 16 -32 8 -32 73 -7 8 -7 29 ] (ii) Forward elimination on V'*V: m x1 x2 x3 ___________________________ 16*-32 8 2 -32 73-7 -1/2 8-7 29 ___________________________ 16-32 8 0 9* 9 -1 0 9 25 ___________________________ 16-32 8 0 9 9 0 0 16 Hence: L = [ 1 0 0 -2 1 0 1/2 1 1 ] D = [ 16 0 0 0 9 0 0 0 16 ] U = [ 1 -2 1/2 0 1 1 0 0 1 ] As usual, U = L.' (iii) Solve ( W*U = V or [w1 w2 w3]*U = [v1 v2 v3]

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Unformatted text preview: Sheet1 Page 2 i.e., w1 = v1-2*w1 + w2 = v2 (1/2)*w1 + w2 + w3 = v3 Therefore w1 = v1 w2 = 2*v1 + v2 w3 = -(1/2)*v1 -(2*v1 + v2) + v3 = -(5/2)*v1 - v2 + v3 ( In other words, inv(U) = [ 1 2 -5/2 0 1 -1 0 0 1 ] ) The squared norms of w1, w2 and w3 are given by the diagonal entries of D, i.e., they are equal to 16, 9 and 16 respectively. Dividing each by its norm results in a new set of vectors with unit norm. Therefore w1 = (1/4)*v1 w2 = (2/3)*v1 + (1/3)*v2 w3 = -(5/8)*v1 - (1/4)*v2 + (1/4)*v3 w...
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hwsolB18 - Sheet1 Page 2 i.e w1 = v1-2*w1 w2 = v2(1/2*w1 w2...

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