Linear Algebra Proof (vector spaces and spans)

1. Jan 31, 2012

jmm

1. The problem statement, all variables and given/known data

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

2. Relevant equations
~

3. 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.

2. Jan 31, 2012

tiny-tim

hi jmm!

hint: suppose every Axi = 0

(and remember the definition of span)

3. Jan 31, 2012

jmm

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.

4. Jan 31, 2012

tiny-tim

nooo

what is the definition of span = ℝn?

5. Jan 31, 2012

jmm

Does it mean that all of the linear combinations of vectors in the span form the space Rn?

6. Jan 31, 2012

tiny-tim

what does it mean about any individual vector?

7. Jan 31, 2012

jmm

Ummmm, I really don't know :(

8. Jan 31, 2012

tiny-tim

look it up!!

(remember, we're talking about vector spaces )

9. Jan 31, 2012

jmm

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. Jan 31, 2012

tiny-tim

in a vector space, you can express any vector as … ?

11. Jan 31, 2012

jmm

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. Jan 31, 2012

hint: basis