Homework answers / question archive / Modeling and Analysis of Mechanical Systems Problem Set 1 You must show all of your work or you will not receive credit

Modeling and Analysis of Mechanical Systems Problem Set 1 You must show all of your work or you will not receive credit

Mechanical Engineering

Share With

Modeling and Analysis of Mechanical Systems Problem Set 1 You must show all of your work or you will not receive credit. Problem 1. Is the intersection of the three planes u+v+w+z = 6, u+w+z = 4, and u+w = 2 (all in four-dimensional space), a line, a point, or an empty set? What is the intersection if the fourth plane u = −1 is included? Find a fourth equation that leaves us with no solution. Problem 2. Given the system u ? ? 1 1 0 ? ?+v ? ? 1 2 1 ? ?+w ? ? 1 3 2 ? ? = b, show that the three columns on the left lie in the same plane by writing the third column as a combination of the first two. What are all the solutions (u, v,w) if b is the zero vector (0,0,0)? Problem 3. What conditions are needed on y1, y2, y3 to ensure that the points (0, y1), (1, y2), (2, y3) lie on a straight line? Problem 4. Find all numbers a such that forward-elimination stops (a) forever, and (b) briefly. ax+3y = −3 4x+6y = 6. Problem 5. Given the system x+4y−2z = 1 x+7y−6z = 6 3y+qz = t, find q such that the system is singular and then, given that q, find t such that the system has an infinite number of solutions. Problem 6. Given the system ax+2y+3z = b1 ax+ay+4z = b2 ax+ay+az = b3, 1 which three numbers a will cause elimination to fail to give three pivots? Problem 7. Which of the following matrices are guaranteed to equal (A+B) 2 ? A 2 +2AB+B 2 , A(A+B) +B(A+B), (A+B)(B+A), A 2 +AB+BA+B 2 . Problem 8. The matrix that rotates the x-y plane by an angle θ is A(θ) = cosθ −sinθ sinθ cosθ . Verify that A(θ1)A(θ2) = A(θ1 +θ2) from the identities for cos(θ1 +θ2) and sin(θ1 +θ2). What is A(θ) times A(−θ)? Problem 9. Which elimination matrices E21,E32, and E43 are needed for A = ? ? ? ? 2 −1 0 0 −1 2 −1 0 0 −1 2 −1 0 0 −1 2 ? ? ? ? . Problem 10. The function y = a + bx + cx2 goes through the points (x, y) = (1,4), (2,8), and (3,14). Find and solve a matrix equation for the unknowns (a,b, c). Problem 11. If A is m×n, how many separate multiplications are involved when (a) A multiplies a vector x with n components? (b) A multiplies an n× p matrix B? Then AB is m× p. (c) A multiplies itself to produce A 2 ? Here, m = n. Problem 12. Solve by elimination, exchanging rows when necessary: u+4v+2w = −2 −2u−8v+3w = 32 v+w = 1, Problem 13. What are L and D for this matrix A? What is U in A = LU and what is the new U in A = LDU? A = ? ? 2 4 8 0 3 9 0 0 7 ? ?. Problem 14. Compute L and U for the symmetric matrix A = ? ? ? ? a a a a a b b b a b c c a b c d ? ? ? ? . 2 Find four conditions on a,b, c,d to get A = LU with four pivots. Problem 15. Compute the symmetric LDLT factorization of A = ? ? 1 3 5 3 12 18 5 18 30 ? ?. Problem 16. (a) Suppose addition in R 2 adds an extra 1 to each component, so that (3,1) + (5,0) = (9,2) instead of (8,1). With scalar multiplication unchanged, which of the 8 rules of addition and scalar multiplication for a vector space are broken? (b) Show that the set of all positive real numbers, with x+y and cx redefined to equal the usual xy and x c , is a vector space, or indicate why it is not. What is the “zero vector”? Problem 17. Which of the following are subspaces of R ∞? (a) All sequences (x1, x2,...) with x j = 0 from some point onward. (b) All decreasing sequences: x j+1 ≤ x j for each j. (c) All convergent sequences: the x j have a limit as j → ∞. Problem 18. The functions f(x) = x 2 and g(x) = 5x are “vectors” in the vector space F of all real functions. Which rule of addition or scalar multiplication is broken if multiplying f(x) by c gives the function f(cx)? Problem 19. Describe the column spaces (lines or planes) of these particular matrices: A = ? ? 1 2 0 0 0 0 ? ? and B = ? ? 1 0 0 2 0 0 ? ? and C = ? ? 1 0 2 0 0 0 ? ?. Problem 20. Why isn’t R 2 a subspace of R 3 ? Problem 21. For the system given below, find the echelon form U, the free variables, and the special solutions: A = 0 1 0 3 0 2 0 6 , b = b1 b2 . What is the complete solution to Ax = b for this system? Problem 22. What are the special solutions to Rx = 0 for R given by R = ? ? 1 0 2 3 0 1 4 5 0 0 0 0 ? ?? 3 Problem 23. Determine if the following vectors are independent or dependent: (a) the vectors are (1,3,2), (2,1,3), and (3,2,1). (b) the vectors are (1,−3,2), (2,1,−3), and (−3,2,1). Problem 24. Describe the subspace of R 3 (is it a line or a plane or R 3 ?) spanned by (a) the two vectors (1,1,−1) and (−1,−1,1). (b) the three vectors (0,1,1), (1,1,0), and (0,0,0). (c) the columns of a 3 by 5 echelon matrix with 2 pivots. (d) all vectors with positive components. Problem 25. Suppose y1(x), y2(x), and y3(x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 for each case. 4