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.

Question

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.

Selden1f · Accepted Answer

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&amp;gt;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.

If we have bbu,bbv in RR^n and c bbu != bbv where c in RR, then is AA_(a_1,...,a_n) in RR (EE_(m,n in RR)(mbbu+n bbv=[[a_1],[vdots],[a_n]])) true?

Answered question

Answer & Explanation

New Questions in Precalculus