Span and Vector Space: Understanding Vectors in Linear Algebra

[Poetria]

Homework Statement

The question is:
if vectors v1, v2, v3 belong to a vector space V does it follow that:

span (v1, v2, v3) = V

span (v1, v2, v3) is a subset of V.[/B]

2. The attempt at a solution:

If I understand it correctly the answer to both questions is yes.
The first: the linear combinations of these three vectors fill the space V.

Am I correct? I would like to make sure if I have understood the definitions.

 

[PeroK]

Homework Statement

The question is:
if vectors v1, v2, v3 belong to a vector space V does it follow that:

span (v1, v2, v3) = V

span (v1, v2, v3) is a subset of V.[/B]

2. The attempt at a solution:

If I understand it correctly the answer to both questions is yes.
The first: the linear combinations of these three vectors fill the space V.

Am I correct? I would like to make sure if I have understood the definitions.

Why must the span of three vectors be the full vector space? If so, then why not the span of just two vectors?

 

[Poetria]

Why must the span of three vectors be the full vector space? If so, then why not the span of just two vectors?

Yes, I know, they should be linearly independent to be a spanning set. Well, we don't know if they are independent. :( I guess there may be also other vectors needed for a spanning set for the vector space V: we don't know this either. In this case the answer to the first question would be negative. But the second would hold. Have I got it?

I have read the definitions so many times that I am somewhat dizzy.

 

[PeroK]

Yes, I know, they should be linearly independent to be a spanning set. Well, we don't know if they are independent. :( I guess there may be also other vectors needed for a spanning set for the vector space V: we don't know this either. In this case the answer to the first question would be negative. But the second would hold. Have I got it?

I have read the definitions so many times that I am somewhat dizzy.

There are two reasons why the span of $v_1, v_2, v_3$ may not be all of $V$. If $V$ is 3-dimensional, the vectors may be linearly dependent. And, $V$ may be more than 3-dimensional in the first place.

 

[Poetria]

There are two reasons why the span of $v_1, v_2, v_3$ may not be all of $V$. If $V$ is 3-dimensional, the vectors may be linearly dependent. And, $V$ may be more than 3-dimensional in the first place.

Thank you very much. Everything is clear to me now. :)