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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) (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).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
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Linear algebra
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.
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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