Proving Nonempty Sets are Subspaces & Examples

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Homework Help Overview

The discussion revolves around proving properties of nonempty sets as subspaces within vector spaces, specifically addressing the conditions under which a set W qualifies as a subspace of a vector space V. Additionally, an example is sought to illustrate that the union of two subspaces may not itself be a subspace.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of a subspace and the implications of the "if and only if" condition. Some suggest that proving the conditions for a subspace requires demonstrating both directions of the statement. Others question the clarity of the definitions and seek simpler examples to illustrate the concepts.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts on the definitions and requirements for subspaces. Some guidance has been offered regarding the approach to the first part of the problem, while there is acknowledgment of the challenges faced by participants, particularly regarding language barriers and comprehension of the material.

Contextual Notes

One participant notes difficulties with language and understanding the textbook, which may impact their ability to engage with the problem fully. There is an emphasis on the need for clear examples to aid understanding.

zxc210188
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(1)Prove that a nonempty set W is a subspce of a vector space V iff a x+b y is an element of W for all scalars a and b and all vectors x and y in W

(2)Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspaces of V
 
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Have you had any thoughts at all on the problem? Really, (1) nothing more than an exercise in definitions, and you almost have to try in order to get (2) wrong by guessing at an answer.
 
What is the DEFINITION of "subspace"? Show that if "ax+ by is in W for all numbers a and b and vectors x and y in W" is true then each of the conditions for a subspace are satisfied. Because this is an "if and only if" statement, you need to then turn around and show that, if W satisfies all the conditions for a subspace, then "ax+ by is in W for all numbers a and b and vectors x and y in W" is true.

As for the second part, as Hurkyl said, it's almost impossible to get it wrong! To make it as easy as possible use R2 as V and choose 2 very easy subspaces.
 
I know it should be prove in two direction, but I have no idea about how to prove it.
I am a student in Taiwan, so my English is not very well. I cannot understand what the textbook talk about compeletely, so I learn it from doing questions and problems, I did try to think on the problems, but I failed.It may be easy for you but hard for a foreign student.
 
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