How to show something is a subspace

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Discussion Overview

The discussion revolves around how to demonstrate that the set {(x,y,z) € R^3 :√11x - √13z=0} is a subspace of R^3. Participants explore the necessary conditions and definitions related to subspaces, including vector addition and scalar multiplication.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • Some participants emphasize the need to verify that the set satisfies the definition of a subspace by proving that the sum of two elements in the subspace yields another element in the subspace and that scalar multiplication of an element in the subspace also yields another element in the subspace.
  • There is a question regarding the definition of addition and scalar multiplication in the vector space, with references to the axioms governing these operations.
  • Some participants suggest that understanding the axioms is crucial for confirming that the set meets the criteria for being a subspace.

Areas of Agreement / Disagreement

Participants generally agree on the importance of checking the axioms related to vector addition and scalar multiplication to establish whether the set is a subspace. However, there is no consensus on the specific steps to demonstrate this using the given equation.

Contextual Notes

Participants have not provided specific definitions or assumptions regarding the vector space operations, which may affect the clarity of the discussion.

HappyN
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How would you show that {(x,y,z) € R^3 :√11x - √13z=0} is a subspace of R^3?

I know you have to make sure it fits the definition of a subspace, i.e prove
u+v € W
and alpha(v) € W

but I am not sure how you would do this using √11x - √13z=0 ?
 
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HappyN said:
How would you show that {(x,y,z) € R^3 :√11x - √13z=0} is a subspace of R^3?

I know you have to make sure it fits the definition of a subspace, i.e prove
u+v € W
and alpha(v) € W

but I am not sure how you would do this using √11x - √13z=0 ?

What is the definition of addition and scalar multiplication in your vector space?
 
chiro said:
What is the definition of addition and scalar multiplication in your vector space?

Do you mean the 7 axioms?
like: v+w=w+v for all v,w € V
(v+w)+z=v+(w+z) etc?
 
I think that Chiro means think about the axioms and make sure the subspace satisfy them.
In particular check that the sum of two elements in the subspace yield another element in the subspace and that the scalar multiple of an element in the subspace yields another element in the subspace. The rest of the axioms will then follow automatically.
 
Simon_Tyler said:
I think that Chiro means think about the axioms and make sure the subspace satisfy them.
In particular check that the sum of two elements in the subspace yield another element in the subspace and that the scalar multiple of an element in the subspace yields another element in the subspace. The rest of the axioms will then follow automatically.

Yep that's pretty much what I was trying to get the OP to think about.
 

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