Lets say $v,w \in \mathbb{R}^n $ and $v = A w$, where $A \in M_{n \times n}$. We are given all entries of $A$.
Now, If I am given all the components of $w$ it is very easy to find $v$, and if I am given $v$ I can find $w$ by inverting $A$.
The question is if I am given $m$ components of $v$ and $l$ components of $w$, how can I find $v,w$ ?
I know this cannot be done for all possible inputs, but I want an algorithmic solution method similar to finding the inverse of a matrix.
So far, I would solve problems like this by converting to systems of equations and then solving them in a case by case basis.
I would like to be able to do this with linear algebra, or tensors.
*Note, when I say components, I refer to the standard basis coordinates.