# Help with a basis vectors proof

• Nov 26th 2012, 07:35 AM
HannahEdwards55
Help with a basis vectors proof
Hi, I am struggling with this proof:
If a,b,c are vectors in R3 then show that {bxc, cxa, axb} is a basis of R3 if and only if {a,b,c} is
Thanks!
• Nov 26th 2012, 09:54 AM
Re: Help with a basis vectors proof
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Note that for {a,b,c} to be a basis for R3, a,b,c must be linearly independent.
Next consider what happens if you have a product of linearly independent vectors.

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Note that for {bxc,cxa,axb} to be a basis for R3, bxc,cxa,axb must be linearly independent and argue that a,b,c must be independent as well. (you can prove this by contradiction. Suppose at least one vector, say a, is not linearly independent, in other words linearly dependent, and then show that at least one of the vectors bxc,cxa,axb must be linearly dependent as well for a contradiction)

EDIT: forgot to mention that since we are working with R3, three linearly independent vectors are enough to span R3, and hence are a basis. The key here is to verify linear independence.