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Vectors - Spanning sets

  • #1

Homework Statement


Consider a set of vectors:

S = {v[itex]_{1}[/itex], v[itex]_{2}[/itex], v[itex]_{3}[/itex], v[itex]_{4}[/itex][itex]\subset[/itex] ℝ[itex]^{3}[/itex]

a) Can S be a spanning set for ℝ[itex]^{3}[/itex]? Give reasons for your answer.
b) Will all such sets S be spanning sets? Give a reason for your answer.


The Attempt at a Solution



a) Yes, because a linear combination of these vectors can form any given vector in ℝ[itex]^{3}[/itex].

b) Yes, don't really know a reason besides something similar to the one above, I can't see why not.

I'm not really sure of these answers, can anyone confirm this, or given any insight to understand this better?

Cheers
 

Answers and Replies

  • #2
HallsofIvy
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What about, {(1, 1, 0), (1, 2, 0), (3, 1, 0)}?
 
  • #3
Well I guess since the 3rd elements are zero, you can't form a vector in ℝ[itex]^{3}[/itex] which has a non-zero 3rd element.

So part b is definitely a no. However part a "can" be since they haven't explicitly defined the vectors, so it's possible given that zero/non-zero condition, is that the only reason?
 
  • #4
HallsofIvy
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In general, a set with fewer than n vectors cannot span a vector space of dimenson n but a set with n or more vectors may. A set with more than n vectors cannot be independent but a set with n or fewer may. Only with sets with exactly n vectors is it possible to both span and be independent (a basis).
 

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