5) An affine subset of V is a set of the form v + U = {v+ u|u e U} for some vector v E V and some subspace U of V. Prove that a nonempty subset A of a vector space V is an affine subset of V if and only if lv + (1 – 2)w e A for all v,w e A and all 2 e F. Here is a hint for one direction. Suppose A is a subset with the property above and a E A, and let U = A – a = {x – a|x € A}. Prove that U is a subspace of V. Once you have proved that, then it is easy to see that A = a + U.
5) An affine subset of V is a set of the form v + U = {v+ u|u e U} for some vector v E V and some subspace U of V. Prove that a nonempty subset A of a vector space V is an affine subset of V if and only if lv + (1 – 2)w e A for all v,w e A and all 2 e F. Here is a hint for one direction. Suppose A is a subset with the property above and a E A, and let U = A – a = {x – a|x € A}. Prove that U is a subspace of V. Once you have proved that, then it is easy to see that A = a + U.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![5) An affine subset of V is a set of the form v + U = {v+u]u E U} for some vector v e V
and some subspace U of V. Prove that a nonempty subset A of a vector space V is an
affine subset of V if and only if Av + (1 – 1)w E A for all v,w E A and all 2 E F.
Here is a hint for one direction. Suppose A is a subset with the property above and a E A,
and let U = A – a = {x – a|x € A}. Prove that U is a subspace of V. Once you have
proved that, then it is easy to see that A = a + U.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d0c464a-3b23-4ba9-ab5d-6d0abfc5dcf1%2Fd6079d27-1ca0-4cac-bb78-a17c9e638a29%2F6ya3ndj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5) An affine subset of V is a set of the form v + U = {v+u]u E U} for some vector v e V
and some subspace U of V. Prove that a nonempty subset A of a vector space V is an
affine subset of V if and only if Av + (1 – 1)w E A for all v,w E A and all 2 E F.
Here is a hint for one direction. Suppose A is a subset with the property above and a E A,
and let U = A – a = {x – a|x € A}. Prove that U is a subspace of V. Once you have
proved that, then it is easy to see that A = a + U.
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