Can two n-dimensional vectors that are not scalar multiples of each other be added to make any point on an n-dimensional coordinate plan?
If we have u u , v v R n and c u u v v where c R , then is
a 1 , , a n R ( m , n R ( m u u + n v v = [ a 1 a n ] ) )  true?
Note: Bolded letters represent vectors.

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2022-07-16Added 14 answers

We know that R n has a basis of n vectors (such as the standard unit basis { e 1 , e 2 , . . . . e n }). By a known theorem from linear algebra, we then have that any set of less than n vectors in R n does not span R n . Hence, as long as n>2, vectors u and v could not span R n . That is, a 1 . . . a n such that m , n R , m u + n v ( a 1 . . . a n ), which means the given statement cannot be true.

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