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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 GaussJordan. (b) Obtain the elementary matrices for the row operations in (a), and hence give an explanation on why A can be calculated using GaussJordan. (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 ndimensional 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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 Spring '10
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