If $v$ is a row vector and $A$ a matrix, the product $w = v A$ can be seen as a vector containing a number of linear combinations of the columns of vector $v$. For instance, if $ v = \begin{bmatrix}1, 2\end{bmatrix}, \quad A = \begin{bmatrix}0 & 0 & 0 \\ 1 & 1 & 1\end{bmatrix}, \quad w = vA = \begin{bmatrix}2, 2, 2\end{bmatrix} $ read by columns, the matrix $A$ is saying: make 3 combinations of the columns of vector $v$, each of which consists of taking 0 times the first column and 1 time the second column.
Now, the goal is to reconstruct, to the extent of possible, vector $v$ from $A$ and $w$, in other words to find a vector $v'$ such that $v'A = w .$
Two things to consider:
The matrix $A$ can have any number of columns and may or may not be square or invertible.
There are times when elements of the original vector can't be known, because $w$ contains no information about them. In the previous example, this would be the case of $v_1$. In this case, we would accept any value of $v'_1$ as correct.
How would you approach this problem? Can $v'$ be found doing simple operations with $w$ and $A$ or do I have to invent an algorithm specifically for the purpose?