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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.