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Linear Algebra Proof (vector spaces and spans)

  • Thread starter jmm
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  • #1
jmm
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Homework Statement



If [itex]ℝ^{n}=span(X_{1},X_{2},...,X_{k})[/itex] and A is a nonzero m x n matrix, show that [itex]AX_{i}≠0[/itex] for some i.

Homework Equations


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The Attempt at a Solution


Hey guys, I'm way over my head here. I really don't know how to approach this problem. I would really appreciate it if someone could give a nudge in the right direction as to where to start.
 

Answers and Replies

  • #2
tiny-tim
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hi jmm! :smile:

hint: suppose every Axi = 0 :wink:

(and remember the definition of span)
 
  • #3
jmm
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If every AXi=0 that would mean that Xi=0 for all i, right? And a set of all 0 vectors would not span Rn, right? ...so that would be impossible? Haha I'm really having trouble wrapping my head around this.

Thanks for your reply by the way!
 
  • #4
tiny-tim
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If every AXi=0 that would mean that Xi=0 for all i, right?
nooo :redface:

what is the definition of span = ℝn? :smile:
 
  • #5
jmm
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Does it mean that all of the linear combinations of vectors in the span form the space Rn?
 
  • #6
tiny-tim
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what does it mean about any individual vector? :smile:
 
  • #7
jmm
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Ummmm, I really don't know :(
 
  • #8
tiny-tim
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look it up!! :rolleyes:

(remember, we're talking about vector spaces :wink:)
 
  • #9
jmm
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Oh, believe me, I've been looking it up for the past 8 hours haha. Everything I've seen has it defined in terms of combinations of many vectors.
 
  • #10
tiny-tim
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in a vector space, you can express any vector as … ?
 
  • #11
jmm
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I don't know that one either. I mean I probably do but I can't figure out where you're trying to lead me :)

And I thought another linear algebra course would be good for me!
 
  • #12
tiny-tim
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hint: basis :wink:
 

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