Linear Algebra Proof (vector spaces and spans)

[jmm]

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.

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.

 

[tiny-tim]

hi jmm!

hint: suppose every Ax_i = 0

(and remember the definition of span)

 

[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.

Thanks for your reply by the way!

 

[tiny-tim]

If every AXi=0 that would mean that Xi=0 for all i, right?

nooo

what is the definition of span = ℝⁿ?

 

[jmm]

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

 

[tiny-tim]

what does it mean about any individual vector?

 

[jmm]

Ummmm, I really don't know :(

 

[tiny-tim]

look it up!

(remember, we're talking about vector spaces )

 

[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.

 

[tiny-tim]

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

 

[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!

 

[tiny-tim]

hint: basis