# Linear Algebra Proof (vector spaces and spans)

## Homework Statement

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.

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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.

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tiny-tim
Homework Helper
hi jmm! hint: suppose every Axi = 0 (and remember the definition of span)

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.

tiny-tim
Homework Helper
If every AXi=0 that would mean that Xi=0 for all i, right?
nooo what is the definition of span = ℝn? Does it mean that all of the linear combinations of vectors in the span form the space Rn?

tiny-tim
Homework Helper
what does it mean about any individual vector? Ummmm, I really don't know :(

tiny-tim
Homework Helper
look it up!! (remember, we're talking about vector spaces )

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.

tiny-tim
Homework Helper
in a vector space, you can express any vector as … ?

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!

tiny-tim
hint: basis 