Solving Vector Subspace Questions: A & B in V

[flon]

Hey guys this is the question

. Let A and B be vector subspaces of a vector space V .
The intersection of A and B, A ∩ B, is the set {x ∈ V | x ∈ A and x ∈ B}.
The union of A and B, A ∪ B, is the set {x ∈ V | x ∈ A or x ∈ B}.
a) Determine whether or not A ∩ B is a vector subspace of V . Prove your answer.
b) Determine whether or not A ∪ B is a vector subspace of V . Prove your answer.


My strategy for this is to find two subspaces in V and find a counter claim so that the union of A and B is not a subspace and similarly for the intersection of A and B would this be strategy be enough to answer the question?

thanks so much.

 

[lanedance]

for the union pick arbitrary elements a in A, b in B, is a+b in A U B?

 

[lanedance]

for int pick arbitrary elements in A int B, and test the subspace closure requirements

 

[HallsofIvy]

Hey guys this is the question

. Let A and B be vector subspaces of a vector space V .
The intersection of A and B, A ∩ B, is the set {x ∈ V | x ∈ A and x ∈ B}.
The union of A and B, A ∪ B, is the set {x ∈ V | x ∈ A or x ∈ B}.
a) Determine whether or not A ∩ B is a vector subspace of V . Prove your answer.
b) Determine whether or not A ∪ B is a vector subspace of V . Prove your answer.


My strategy for this is to find two subspaces in V and find a counter claim so that the union of A and B is not a subspace and similarly for the intersection of A and B would this be strategy be enough to answer the question?

thanks so much.

It would be sufficient if they are both not subspaces. But are you sure of that?

 

[flon]

sorry if the intersection and union are both NOT subspaces?

 

[lanedance]

yeah, you'll only find a counter example if they are not a subspace