Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Textbook Question
Chapter 60, Problem 25A
Given three side, find the area of each of the triangles 19 through 26. Where necessary, round the answers to 1 decimal place.
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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…
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 60 Solutions
Mathematics For Machine Technology
Ch. 60 - Find the area of this figure. Round your answer to...Ch. 60 - Prob. 2ACh. 60 - The four holes in the figure are 0.8 cm in...Ch. 60 - Prob. 4ACh. 60 - Prob. 5ACh. 60 - Prob. 6ACh. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...
Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Find the unknown area, base, or height for each of...Ch. 60 - Given three side, find the area of each of the...Ch. 60 - Given three side, find the area of each of the...Ch. 60 - Given three side, find the area of each of the...Ch. 60 - Given three side, find the area of each of the...Ch. 60 - Given three side, find the area of each of the...Ch. 60 - Given three side, find the area of each of the...Ch. 60 - Given three side, find the area of each of the...Ch. 60 - Given three side, find the area of each of the...Ch. 60 - Solve these exercises. Find the cross-sectional...Ch. 60 - Solve these exercises. Two triangular pieces are...Ch. 60 - Solve these exercises. Determine the area of the...Ch. 60 - Solve these exercises. Find the area of the...Ch. 60 - Solve these exercises. The area of the irregularly...
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- 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.arrow_forwardDetermine the moment about the origin O of the force F4i-3j+5k that acts at a Point A. Assume that the position vector of A is (a) r =2i+3j-4k, (b) r=-8i+6j-10k, (c) r=8i-6j+5karrow_forwardNot use ai pleasearrow_forward
- Calculs Insights πT | cos x |³ dx 59 2arrow_forward2. Consider the ODE u' = ƒ (u) = u² + r where r is a parameter that can take the values r = −1, −0.5, -0.1, 0.1. For each value of r: (a) Sketch ƒ(u) = u² + r and determine the equilibrium points. (b) Draw the phase line. (d) Determine the stability of the equilibrium points. (d) Plot the direction field and some sample solutions,i.e., u(t) (e) Describe how location of the equilibrium points and their stability change as you increase the parameter r. (f) Using the matlab program phaseline.m generate a solution for each value of r and the initial condition u(0) = 0.9. Print and turn in your result for r = −1. Do not forget to add a figure caption. (g) In the matlab program phaseline.m set the initial condition to u(0) = 1.1 and simulate the ode over the time interval t = [0, 10] for different values of r. What happens? Why? You do not need to turn in a plot for (g), just describe what happens.arrow_forwardTrue or False and whyarrow_forward
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