Understanding Linear Independence: Proving Non-Zero Status of One-Element Sets

[EvLer]

Hi all,
when is a one-element set is linearly independent? Just when it's non-zero?
I am not sure how to prove this on one element set.

Thanks in advance.

 

[Hurkyl]

(Assuming you speak of a vector space)

You're right on the condition. To prove it, you need to show that if any linear combination of your one element is zero, then all the coefficients are zero.

 

[EvLer]

To prove it, you need to show that if any linear combination of your one element is zero, then all the coefficients are zero.

Ok, is that what you mean?
S = {A},

A = 3 3 3
    3 3 3

A = cA?
I do not know how to create a linear combination on one element, it doesn't make sense to me. Maybe because I started Linear Algebra course two days ago.

Thanks.

 

[matt grime]

A linear combination of one element is just some scalar multiple of it. What is your definition of linear combination? Can't you apply it to a one element set? And what is A, it looks like an array? It is a simple exercise that if v is a non-zero vector, and tv=0 for some t in the underlying field then t=0. Which is what they're asking you to prove.

 

[EvLer]

I think I got confused linear combination and linear dependency. Now, I see the difference. Thank you for the explanations.