1. The given set with the given operations is not a vector space. Check all the properties and list which properties fail to hold. The set of all ordered pairs of real numbers with the operations (x,, y) (x,,y;) = (x, +x,, y, +y,) and rO(x, y) = (0,0).
1. The given set with the given operations is not a vector space. Check all the properties and list which properties fail to hold. The set of all ordered pairs of real numbers with the operations (x,, y) (x,,y;) = (x, +x,, y, +y,) and rO(x, y) = (0,0).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 48E
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![1. The given set with the given operations is not a vector space. Check all the properties and list which
properties fail to hold.
The set of all ordered pairs of real numbers with the operations (x,, y,) O (x,, y2) = (x, +x,,y, + y,) and
rO(x, y) = (0,0).
2. Prove that a vector u in a vector space has only one negative -u. [Hint: Assume that there are two
negatives and conclude that they must be equal.]
3. C(-00, 00) denotes the vector space of real-valued continuous functions.
Which of the following subsets are subspaces of the vector space C(-x,0) ? Explain.
a) All nonnegative functions.
b) All constant functions.
c) All functions f such that f(0) = 0.
d) All functions f such that f(0) = 5.
e) All differentiable functions.
4. a) Prove that proj (proj-v) = proj,v
b) Prove that proj (v – proj v) =0.
c) Explain parts a) and b) geometrically.
5. a) Prove that u-v2u + v for all vectors in R".
b) Explain parts a) geometrically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78fa799e-dd0d-498f-bbc7-16121e7aabca%2F4e1791e1-bdee-49bc-8c00-0171cf1aa0c3%2Furt3mpj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. The given set with the given operations is not a vector space. Check all the properties and list which
properties fail to hold.
The set of all ordered pairs of real numbers with the operations (x,, y,) O (x,, y2) = (x, +x,,y, + y,) and
rO(x, y) = (0,0).
2. Prove that a vector u in a vector space has only one negative -u. [Hint: Assume that there are two
negatives and conclude that they must be equal.]
3. C(-00, 00) denotes the vector space of real-valued continuous functions.
Which of the following subsets are subspaces of the vector space C(-x,0) ? Explain.
a) All nonnegative functions.
b) All constant functions.
c) All functions f such that f(0) = 0.
d) All functions f such that f(0) = 5.
e) All differentiable functions.
4. a) Prove that proj (proj-v) = proj,v
b) Prove that proj (v – proj v) =0.
c) Explain parts a) and b) geometrically.
5. a) Prove that u-v2u + v for all vectors in R".
b) Explain parts a) geometrically.
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