We can reverse a linear transformation using an inverse matrix to find the input vector used in the original linear transformation:

$$A^{-1}A\vec{x} = A^{-1}\vec{v}$$

Let's say we have a 2D vector that we transform into a 1D "vector":
$$\begin{pmatrix} 
a & b 
\end{pmatrix}\*\begin{pmatrix} 
x \\ 
y 
\end{pmatrix} = \begin{pmatrix} 
ax + by
\end{pmatrix} $$

**Isn't there some matrix that can then convert that 1D vector back into the original 2D vector?** Something like this:

$$\begin{pmatrix} 
c \\ 
d 
\end{pmatrix}\*\begin{pmatrix} 
ax + by
\end{pmatrix} = \begin{pmatrix} 
x \\ 
y 
\end{pmatrix} $$

My TA says that the original transformation does not have an inverse because **a single input vector would map to multiple output vectors**. But I'm not seeing those multiple outputs in my algebra.