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&lt;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.

Question

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&amp;lt;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.

choltas5j · Accepted Answer

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&#039;t be an inner product satisfying the set of equations.

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.

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