Thread: Linear Dependent/Spanning System?

Is there a difference between a linear dependent system and a spanning one? I've been confused about this for a while. Something is linearly dependent if it has a non trivial solution, and something is spanning if one of the vectors can be represented as a linear combination of another - implying there is a non-trivial solution- Some of my classmates told me i'm wrong, and I don't understand why. An explanation would be awesome. Can't seem to find anything online.

A Spanning set for a Vector space V can be linearly dependent but not every linearly dependent set is a spanning set for V.

How is that so?

The set of linearly dependent vectors spans . Now the set of linearly dependent vectors does not span

A system of vectors v1; v2;...; vn in V is called a generating system (also a spanning system, or a complete system) in V if any vector v in V admits representation as a linear combination.

This is the definition from the book I am using. isn't v2 = 3*v1. Therefore it is a spanning system?

Any vector in V must be admitted as a linear combination.

So the vector lives in but give me a linear combination of to produce r , cant be done so this linearly dependent system does not span

Ahhh. Thank you very much. I appreciate the help. I guess i have to pay more attention to key words!