EBK MATHEMATICS FOR MACHINE TECHNOLOGY
8th Edition
ISBN: 9781337798396
Author: SMITH
Publisher: CENGAGE LEARNING - CONSIGNMENT
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Textbook Question
Chapter 13, Problem 14A
A shaft is being cut in a lathe. The tool feeds (advances) 0.015 inch each time the shaft turns once (1 revolution). How many revolutions will the shaft turn when the tool advances 3.120 inches? Round the answer to 2 decimal places.
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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…
Question 1. Let f: XY and g: Y Z be two functions. Prove that
(1) if go f is injective, then f is injective;
(2) if go f is surjective, then g is surjective.
Question 2. Prove or disprove:
(1) The set X = {k € Z} is countable.
(2) The set X = {k EZ,nЄN} is countable.
(3) The set X = R\Q = {x ER2
countable.
Q} (the set of all irrational numbers) is
(4) The set X = {p.√2pQ} is countable.
(5) The interval X = [0,1] is countable.
Question 3. Let X = {f|f: N→ N}, the set of all functions from N to N. Prove
that X is uncountable.
Extra practice (not to be submitted).
Question. Prove the following by induction.
(1) For any nЄN, 1+3+5++2n-1 n².
(2) For any nЄ N, 1+2+3++ n = n(n+1).
Question. Write explicitly a function f: Nx N N which is bijective.
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.
Chapter 13 Solutions
EBK MATHEMATICS FOR MACHINE TECHNOLOGY
Ch. 13 - Prob. 1ACh. 13 - Express the common fraction 44125 as a decimal...Ch. 13 - Prob. 3ACh. 13 - Use Figure 132 to answer Exercises 4 through 6....Ch. 13 - Use Figure 132 to answer Exercises 4 through 6....Ch. 13 - Use Figure 132 to answer Exercises 4 through 6....Ch. 13 - Prob. 7ACh. 13 - Divide the numbers in Exercises 7 through 9....Ch. 13 - Divide the numbers in Exercises 7 through 9....Ch. 13 - As indicated in Figure 133, rack sizes are given...
Ch. 13 - Four sets of equally spaced holes are shown in the...Ch. 13 - A cross-sectional view of a bevel gear is shown in...Ch. 13 - How many complete bushings each 14.60 millimeters...Ch. 13 - A shaft is being cut in a lathe. The tool feeds...Ch. 13 - How much stock per stroke is removed by the wheel...Ch. 13 - An automatic screw machine is capable of producing...Ch. 13 - The bolt in Figure 136 has 7.7 threads. Determine...Ch. 13 - The block in Figure 13-7 has a threaded hole with...Ch. 13 - The length of a side of a square equals the...Ch. 13 - All sections of the block in Figure 13-9 are equal...
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