Proving Subset and Subspace Properties | V is a Subspace of Rn

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Homework Help Overview

The discussion centers around the properties of subspaces in the context of vector spaces, specifically focusing on the requirement that a subspace must contain the zero vector. Participants are exploring the implications of this property within the framework of Rn.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Some participants attempt to reason through the definitions of subspaces and question the necessity of proving that a subspace contains the zero vector if it is already included in the definition. Others express a desire to deepen their understanding of the concept of subspaces.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts on the clarity of the definition of subspaces and the adequacy of the problem as a practice question. There is no explicit consensus, but a recognition of the need for further exploration of the topic is evident.

Contextual Notes

Participants note that the definition of a subspace may inherently include the zero vector, leading to questions about the necessity of proving this property. There is an acknowledgment of the potential for more substantial practice questions to enhance understanding.

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Homework Statement



Show that if V is a subspace of R n, then V must contain the zero vector.


The Attempt at a Solution



If a set V of vectors is a subspace of Rn, then, V must contain the zero vector, must be closed under addition, and, closed under scalar multiplication.

Let u = (u1,u2,u3...un), Let w = (w1,w2,w3...wn) and k scalar is a real number.

for ku = (0,0,0,...0n), k = 0 (is this a valid?)

for u+w = 0, (u1+w1, u2+w2,u3+w3...un+wn) = (0,0,0,...0n)

i.e., u1+w1 = 0, then u1=-w1
if u1 = 1,then w1=-1 (is this valid?)
 
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negation said:

Homework Statement



Show that if V is a subspace of R n, then V must contain the zero vector.


The Attempt at a Solution



If a set V of vectors is a subspace of Rn, then, V must contain the zero vector, must be closed under addition, and, closed under scalar multiplication.

Let u = (u1,u2,u3...un), Let w = (w1,w2,w3...wn) and k scalar is a real number.

for ku = (0,0,0,...0n), k = 0 (is this a valid?)

for u+w = 0, (u1+w1, u2+w2,u3+w3...un+wn) = (0,0,0,...0n)

i.e., u1+w1 = 0, then u1=-w1
if u1 = 1,then w1=-1 (is this valid?)

If your definition of V being a 'subspace' includes that V contains the zero vector, then there's not much to show. Is there?
 
Dick said:
If your definition of V being a 'subspace' includes that V contains the zero vector, then there's not much to show. Is there?

Well, yea. But I thought it was a good way for the practice question to brush up my conceptual understanding of subspace. It's pretty nebulous at the moment.
 
negation said:
Well, yea. But I thought it was a good way for the practice question to brush up my conceptual understanding of subspace. It's pretty nebulous at the moment.

Ok. But you might want to pick a little more substantial practice question.
 
Dick said:
Ok. But you might want to pick a little more substantial practice question.

Looking forward to doing so.
 

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