EBK MATHEMATICS FOR MACHINE TECHNOLOGY
8th Edition
ISBN: 9781337798396
Author: SMITH
Publisher: CENGAGE LEARNING - CONSIGNMENT
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Textbook Question
Chapter 49, Problem 10AR
In Exercises 9 and 10, solve for the unknown in each of the equations using one of the six principles of equality. Check each answer. Round the answers to 3 decimal places where necessary.
a.
b.
c.
d.
e.
f.
g.
h.
i.
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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…
Chapter 49 Solutions
EBK MATHEMATICS FOR MACHINE TECHNOLOGY
Ch. 49 - Prob. 1ARCh. 49 - a. Twice M minus the square of P. b. The sum of A...Ch. 49 - Prob. 3ARCh. 49 - Prob. 4ARCh. 49 - Prob. 5ARCh. 49 - Prob. 6ARCh. 49 - Prob. 7ARCh. 49 - Prob. 8ARCh. 49 - Prob. 9ARCh. 49 - In Exercises 9 and 10, solve for the unknown in...
Ch. 49 - In Exercises 11 and 12, solve for the unknown and...Ch. 49 - In Exercises 11 and 12, solve for the unknown and...Ch. 49 - Prob. 13ARCh. 49 - Prob. 14ARCh. 49 - Prob. 15ARCh. 49 - Prob. 16ARCh. 49 - Prob. 17ARCh. 49 - Prob. 18ARCh. 49 - Prob. 19ARCh. 49 - Prob. 20ARCh. 49 - Prob. 21ARCh. 49 - Prob. 22AR
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