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Numerical Methods for Engineers
7th Edition
ISBN: 9780073397924
Author: Steven C. Chapra Dr., Raymond P. Canale
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 16, Problem 12P
You must design a triangular open channel to carry a waste stream from a chemical plant to a waste stabilization pond (Fig. P16.12). The mean velocity increases with the hydraulic radius
FIGURE P16.12
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Your work should be submitted on Moodle, before February 7 at 5 pm.
1. True or false:
(a) if E is a subspace of V, then dim(E) + dim(E) = dim(V)
(b) Let {i, n} be a basis of the vector space V, where v₁,..., Un are all eigen-
vectors for both the matrix A and the matrix B. Then, any eigenvector of A is
an eigenvector of B.
Justify.
2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1,2,-2), (1, −1, 4), (2, 1, 1)}.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
(b) Show…
1. True or false:
(a) if E is a subspace of V, then dim(E) + dim(E+) = dim(V)
(b) Let {i, n} be a basis of the vector space V, where vi,..., are all eigen-
vectors for both the matrix A and the matrix B. Then, any eigenvector of A is
an eigenvector of B.
Justify.
2. Apply Gram-Schmidt orthogonalization to the system of vectors {(1, 2, -2), (1, −1, 4), (2, 1, 1)}.
3. Suppose P is the orthogonal projection onto a subspace E, and Q is the orthogonal
projection onto the orthogonal complement E.
(a) The combinations of projections P+Q and PQ correspond to well-known oper-
ators. What are they? Justify your answer.
(b) Show that P - Q is its own inverse.
4. Show that the Frobenius product on n x n-matrices,
(A, B) =
= Tr(B*A),
is an inner product, where B* denotes the Hermitian adjoint of B.
5. Show that if A and B are two n x n-matrices for which {1,..., n} is a basis of eigen-
vectors (for both A and B), then AB = BA.
Remark: It is also true that if AB = BA, then there exists a common…
Chapter 16 Solutions
Numerical Methods for Engineers
Ch. 16 - 16.1 Design the optimal cylindrical container...Ch. 16 - 16.2 (a) Design the optimal conical container...Ch. 16 - Design the optimal cylindrical tank with...Ch. 16 - The specific growth rate of a yeast that produces...Ch. 16 - A chemical plant makes three major products on a...Ch. 16 - 16.6 Recently chemical engineers have become...Ch. 16 - 16.7 A mixture of benzene and toluene are to be...Ch. 16 - 16.8 A compound A will be converted into B in a...Ch. 16 - In problem 16.8, only one reactor is used. If two...Ch. 16 - 16.10 For the reaction:
equilibrium can be...
Ch. 16 - A chemical plant requires 106L/day of a solution....Ch. 16 - You must design a triangular open channel to carry...Ch. 16 - 16.13 As an agricultural engineer, you must design...Ch. 16 - 16.14 Find the optimal dimensions for a heated...Ch. 16 - A finite-element model of a cantilever beam...Ch. 16 - 16.16 Suppose that you are asked to design a...Ch. 16 - TheStreeter-Phelps model can be used to compute...Ch. 16 - Prob. 18PCh. 16 - 16.19 The flow in an open channel can be predicted...Ch. 16 - A cylindrical beam carries a compression load...Ch. 16 - The Splash River has a flow rate of 2106m3/d, of...Ch. 16 - 16.22 Determine the beam cross-sectional areas...Ch. 16 - AtotalchargeQ is uniformly distributed around a...Ch. 16 - A system consist softwopower plants that must...Ch. 16 - 16.25 The torque transmitted to an induction motor...Ch. 16 - (a) A computer equipment manufacture rproduces...Ch. 16 - A manufacturer provides specialized microchips....Ch. 16 - 16.28 The total drag on an airfoil can be...Ch. 16 - Mechanical/Aerospace Engineering
16.29 Roller...Ch. 16 - 16.30 An aerospace company is developing a new...Ch. 16 - 16.31 A manufacturing firm produces four types of...Ch. 16 - In a similar fashion to the case study described...Ch. 16 - 16.33 Recent interest in competitive and...
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