
#1
Mar2806, 03:28 PM

P: 1

Is it possible to prove 2 vectors are linearly independent with just the following information?:
A is an nxn matrix. V1 and V2 are nonzero vectors in Rn such that A*V1=V1 and A*V2 = 2*V2. Is this enough information, or is more needed to prove the LI of the 2 vectors? 



#2
Mar2806, 03:56 PM

Sci Advisor
HW Helper
P: 2,004

Yes, this is sufficient. Since A is linear it should scale scalar multiples of a vector by the same factor.




#3
Mar2906, 03:49 AM

P: 696

This is a special case of a more general theorem that states that any set of eigenvectors of a matrix (linear transformation) are linearly independent if the eigenvectors "belong" to different eigenvalues.




#4
Mar2906, 06:19 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,879

Linear Independence Proof
Specifically, suppose CV_{1}+ DV_{2}= 0 and apply A to both sides: A(CV_{1}+ DV_{2})= CAV_{1}+ DAV_{2}= CV_{1}+ 2DV_{2}= 0. Now subtract the first equation from that one.



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