EBK MATHEMATICS FOR MACHINE TECHNOLOGY
8th Edition
ISBN: 9781337798396
Author: SMITH
Publisher: CENGAGE LEARNING - CONSIGNMENT
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Textbook Question
Chapter 13, Problem 10A
As indicated in Figure 13−3, rack sizes are given according to diametral pitch. Given four different diametral pitches, find the linear pitch and the whole depth of each rack to 4 decimal places. All dimensions are in inches.
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marked on the quality of your explanations.
You can discuss the problems with classmates, but you should write your solutions sepa-
rately (meaning that you cannot copy the same solution from a joint blackboard, for exam-
ple).
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…
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.
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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