assign2_08 - The Chinese University of Hong Kong Department...

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Unformatted text preview: The Chinese University of Hong Kong Department of Systems Engineering & Engineering Management ERG 2018 Advanced Engineering Mathematics (2008) Assignment 2 Question 1 Referring to the given A−1 in Q. 16 of Section 2.3 (Kolman & Hill, 2008), p.125 (a) Find A using the procedure of Gauss-Jordan. (b) Obtain the elementary matrices for the row operations in (a), and hence give an explanation on why A can be calculated using Gauss-Jordan. (Hint: note that row interchange may be necessary for handling zero pivot element) (c) Using (a) and (b) to find the matrices P and U such that A−1 = P U , where U is an upper triangular matrix. Is the matrix P a lower triangular matrix? Question 2 (a) Find the dot (inner) product of the two vectors u1 = [1, −2, 1, 3]T and u2 = [0, 1, −1, 2]T and determine the angle between them. (b) Discuss and comment on the validity of the cosine law and sine law for general n-dimensional vectors. (c) For two sets of real numbers {x1 , x2 , ..., xn } and {y1 , y2 , ..., yn }, prove the following two inequalities: n n n 2 yi ≥ ( i=1 n 2 yi i=1 i=1 n x2 i i=1 n xi yi )2 x2 i i=1 + ≥ i=1 (xi − yi )2 (Hint: Use vector approach) Question 3 (a) A vector p can be expressed as the sum of two orthogonal vectors, one parallel (p1 ) and one perpendicular (p2 ) to another vector q . The vector p1 is often referred to as the orthogonal projection of p on q . Obtain expressions for p1 and p2 using p, q amd evaluate p1 · p2 . (b) The concepts of linear combination (LC) and linear dependence (LD) of a set of vectors are said to be intimately related to the solutions of a homogenous set of linear equations (Ax = 0). Apply the method of Gauss elimination to find the vector space spanned by the two vectors u1 = [1, −2, 1, 3]T and u2 = [0, 1, −1, 2]T . Then use Gauss elimination again to test whether the two veectors are LD or LI (linearly independent). (c) Consider two additional vectors u3 = [1, 1, −1, 0]T and u4 = [0, 1, 0, −1]T to (b), and find the span of these four vectors. Are they LI or LD? (d) Solve Q. 3 of Section 4.5 (Kolman & Hill, 2008), p. 227. Question 4 Show that the matrix C = AAT is symmetric for a general rectangular A matrix using (a) The matrix multiplication rule for multiplying two conformable matrices to contain matrix elements, i.e., n cij = k=1 aik bkj (b) A vectored approach in considering the column and row vectors of a matrix ...
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