We know the standard form of expressing a system of linear equations in n variables in n equations.
Where A is the coefficient matrix, X is the unknown variables (each variable maybe a real or complex number) matrix and B is the constants matrix.
Now, arriving at the question, let's say I have a set of n vectors X, each of length n and another set of n+1 vectors Y, each of length n. (Clearly, ). All elements of X are independent. The same goes for Y.
Now I want to express each element in Y, as a linear expression of all the elements in X. So that would be something like below, for some coefficient matrix K.
My question is, is it necessary that K must always exist i.e. there must be a way to express all elements in Y as a linear combination of X.
I think the answer is No. The reasoning is that, for a system of linear equations where number of equations is strictly greater than the number of unknowns, there does not exist a solution at all. In the above case, we have n unknowns from X and n+1 equations, each one for each element in Y.
Is my answer and reasoning correct?