I used to be questioning if there may be an inbuilt Mathematica operate that may take a vector $v$ whose parts are sums of variables $x_1, x_2, dots x_{n}$ and switch it right into a matrix-vector product, the place the matrix is only a coefficient matrix and vector is correct and composed solely of the variables? For instance, is there a Mathematica operate $F$ such that

$$start{pmatrix} x_{1} + 2x_{2} 3x_{1} + 4x_{2}finish{pmatrix} stackrel{F}{to} start{pmatrix} 1 & 2 3 & four finish{pmatrix} start{pmatrix} x_{1} x_{2} finish{pmatrix}$$

Basically, I am in search of a operate that ‘undoes’ the matrix-vector multiplication. Apologies if this has been requested earlier than or may be very easy to implement, I am not overly acquainted with Mathematica and so I do not even know what to seek for.