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Span and vector space

  • #1
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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.
 

Answers and Replies

  • #2
PeroK
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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?
 
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  • #3
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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.
 
  • #4
PeroK
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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.
 
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  • #5
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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. :)
 

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