EBK MATHEMATICS FOR MACHINE TECHNOLOGY
8th Edition
ISBN: 9781337798396
Author: SMITH
Publisher: CENGAGE LEARNING - CONSIGNMENT
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Question
Chapter 64, Problem 3A
To determine
To find:
The area of the remaining piece.
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Problem 11 (Gram-Schmidt). Try the Gram-Schmidt procedure for the vectors,
1
0
2
with respect to the standard dot product on R4. What happens? Can you explain why you are unable to
complete the algorithm?
Problem 12 (Orthogonal Matrices Preserve Orthogonality). Suppose x, y = Rn" are orthogonal to each other
with respect to some inner product (.,.) and that A is an orthogonal matrix and B is some invertible matrix.
1. Prove that Ax and Ay are also orthogonal to each other and that ||x|| = ||Ax|| and ||y|| : = ||Ay||.
2. Is it true that Bx and By are also orthogonal to each other and that ||x|| = ||Bx|| and ||y|| = = ||By||?
Provide a proof or a counter-example.
Problem 13 (Orthogonal Complement). Let W be the subspace of R5 spanned by,
1
2
2
4
u =
3
, v=
7
2
2
Find a basis of the orthogonal complement W- of W. Verify in this particular example that WW₁ = {0}
and that dim(W) + dim(W¹) = 5.
Problem 5 (Rank-Nullity Theorem). Let T : P3 → M2×2 be defined as,
T(p(x))
P(0) p'(1)]
=
1. Prove that T is a linear transformation.
2. Find ker(T). Is T injective?
3. Find im(T). Is T surjective?
4. Verify the Rank-Nullity Theorem for T.
Problem 6 (Change of Basis). Let B₁ =
polynomials in P3.
-
-
{1, x, x², x³} and B₁ = {1, x, x(x − 1), x(x − 1)(x − 2)} be two sets of
1. Is B2 a basis for P3? Justify your answer.
2. Find SB₁→B₂ and SB2→B₁. Which one is "easier" to find?
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Problem 7 (Change of Basis). Let B₁ = {eª, sin² (x), cos² (x)} and B₁ = {e*, sin(2x)}. Recall that sin(20) =
2 sin(0) cos(0). Suppose V = span (B₁) and W = span(B2). Let T: VW be a linear transformation defined
as
T(f(x)) = f'(x).
1
1. Prove that B₁ is a basis.
2. Let g(x) = 5 - 3e. Show that g = V and find T(g(x)).
3. Find [TB₁B2
4. Is T injective?
5. Is T surjective?
Problem 14 (Orthogonal Matrices). Prove each of the following.
1. P is orthogonal
PT is orthogonal.
2. If P is orthogonal, then P-1 is orthogonal.
3. If P, Q are orthogonal, then PQ is orthogonal.
Problem 15 (Orthogonal Complement). Consider P2 with the inner product,
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(1)
Chapter 64 Solutions
EBK MATHEMATICS FOR MACHINE TECHNOLOGY
Ch. 64 - In order to make a conical duct a circular sheet...Ch. 64 - Prob. 2ACh. 64 - Prob. 3ACh. 64 - Convert 12.02 sq in. to square centimeters.Ch. 64 - Prob. 5ACh. 64 - Prob. 6ACh. 64 - Prob. 7ACh. 64 - Compute the volume of each sphere 7 through 14....Ch. 64 - Prob. 9ACh. 64 - Compute the volume of each sphere 7 through 14....
Ch. 64 - Prob. 11ACh. 64 - Compute the volume of each sphere 7 through 14....Ch. 64 - Prob. 13ACh. 64 - Compute the volume of each sphere 7 through 14....Ch. 64 - Prob. 15ACh. 64 - Solve these exercise. A vat in the shape of a...Ch. 64 - Prob. 17ACh. 64 - Solve these exercises. A company produces...Ch. 64 - Solve these exercises. Pieces in the shape of...Ch. 64 - Solve these exercises. A plastic products...Ch. 64 - Solve these exercises. Find the weight of the...Ch. 64 - Compute the capacity, in liters, of the container...Ch. 64 - A seamless brass tube and brass flange assembly is...Ch. 64 - Find the number of cubic centimeters of material...Ch. 64 - Find the weight of the cast-iron angle iron shown....Ch. 64 - Prob. 26ACh. 64 - Prob. 27ACh. 64 - Compute the number of cubic centimeters of...
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