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Linear Algebra- Spanning Sets

  • Thread starter Roni1985
  • Start date
  • #1
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Homework Statement


Let {x1,x2,.....,xk} be a spanning set for a vector space V.
a) if we add another vector, xk+1 to the set, will we still have a spanning set ? Explain


Homework Equations





The Attempt at a Solution



I think 'yes', but I am not sure if my explanation is correct.
It doesn't matter if they are independent or dependent, because we know that the first k vectors are independent and that's what we need to create a spanning set for V.
If the kth+1 vector is independent, the set is going to span a bigger vector space, a vector space that includes V. So, it's going to be a spanning set for V regardless of xk+1

Is my reasoning correct ?

Thanks.
 

Answers and Replies

  • #2
radou
Homework Helper
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Yes, it doesn't matter if they're independent or not, the set which contains xk+1 still spans V. And you don't know if the first k vectors are independent, as a matter of fact. Suppose dimV < k, for example.
 

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