Suppose that I have a set of k values s_1,…,s_k in K and k vectors u_1,…,u_k in K^r for a field K not necessarily closed and k<r . Under which very general circumstances can I find a vector w in K^r such that w * u_k=s_k for each k? Observe that orthogonality would correspond to s_k=0 for all k.

janine83fz 2022-08-13 Answered
Suppose that I have a set of k values s 1 , , s k K and k vectors u 1 , , u k K r for a field K not necessarily closed and k<r . Under which very general circumstances can I find a vector w K r such that w u k = s k for each k? Observe that orthogonality would correspond to s k = 0 for all k.
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Answers (1)

choltas5j
Answered 2022-08-14 Author has 13 answers
It is sufficient that u 1 , , u k are linearly independent. You can extend this set of vectors to a basis u 1 , , u r K r and write w = k = 1 r w k u k with w k K. Then you can define a diagonal inner product , via u i , u j = a i δ j i , where δ j i = 0if i j and δ j i = 1 if i = j. The initial equations w , u k = s k will generate you a system of linear equations, where you have the freedom to choose arbitrary s k + 1 , , s r K to solve it for a 1 , , a r
If u 1 , , u k are not linearly independent the set of equations might already be overdetermined. For instance if u 1 = u 2 and s 1 s 2 there won't be an inner product satisfying the set of equations.
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