Hi everyone how can I solve this question ?

 

Hi everyone how can I solve this question ?

View attachment 221375

This is not a question of "solving", but of articulating a sequence a steps that constitute a "proof". In this case "iff" ("if and only if") implies it is a two part proof. The first part is that the condition is necessary for the conclusion to be true. The second part of the proof is that the condition is sufficient for the conclusion to be true. That means there are no other conditions that are needed. How you do such proofs is a matter of experience and this is Homework Help, not homework done for you. show us your best attempt at this two part proof.

 

This is not a question of "solving", but of articulating a sequence a steps that constitute a "proof". In this case "iff" ("if and only if") implies it is a two part proof. The first part is that the condition is necessary for the conclusion to be true. The second part of the proof is that the condition is sufficient for the conclusion to be true. That means there are no other conditions that are needed. How you do such proofs is a matter of experience and this is Homework Help, not homework done for you. show us your best attempt at this two part proof.

Hi @Papabravo . Thanks for your help but I didnt understand what to do. How to show it is satisfy conditions what should I check or which criteria should apply. Can you help me please ?

 

Intuitively what do you know about vector spaces and linear combinations in a vector spaces?
What are some possible subsets of ℝ^3, the space of 3 dimensional vectors with real components?

 

normally it should be like Ax=b equations format. Should I apply the closure property to show that it is satisfy the condition?

 

normally it should be like Ax=b equations format. Should I apply the closure property to show that it is satisfy the condition?

That is not the direction I would go in. BTW I'm not very intuitive when it comes to constructing mathematical proofs of abstract concepts.

One subset of ℝ^3 is the null vector. this would be the 0-dimensional subset
Another subset is one of the coordinate axes that includes the origin. All vectors on the axis can be scaled by a real number and they remain on the line. If you add any two vectors they remain on the line.
Now consider a plane that goes through the origin. Any vector can be scaled by a real number and remain in the plane. Any two vectors can be added and remain in the plane.

Are you getting the flavor of what the theorem is about?