Meaning of a transformation with respect to 1 or 2 bases? So let's say I have:
$$A =\begin{bmatrix}
    1 & 2 & 1\\
    -1 & 1 & 0
\end{bmatrix}$$
$A$ represents a transformation $L: R^3 \rightarrow R^2$ with respect to bases $S$ and $T$ where:
$$S = \begin{bmatrix} -1\\1\\0\end{bmatrix},
      \begin{bmatrix} 0\\1\\1\end{bmatrix},
      \begin{bmatrix} 1\\0\\0\end{bmatrix}\\
  T = \begin{bmatrix} 1\\2\end{bmatrix},
      \begin{bmatrix} 1\\-1\end{bmatrix}$$
So I take that to mean that $A$ has the form:
$$A =\begin{bmatrix}
    L[(S_1)]_T & [L(S_2)]_T & [L(S_3)]_T\\
\end{bmatrix}$$
So each column is the transformation applied to vector from S then with respect to T basis. Now my question is:, what does $A$ actually do? I know it applies a transformation to some vector through multiplication but what vectors does it accept as "proper" input? Should the vector it multiplies with be in a certain basis? 
And what if I said to compute a matrix $A$ that represents L with respect to S (and only S)? Would the columns be: 
$$A =\begin{bmatrix}
    L(S_1) & L(S_2) & L(S_3)\\
\end{bmatrix}$$
And what vectors would you feed that transformation then?
For example: If I just said compute $$L(\begin{bmatrix} 2\\1\\-1\end{bmatrix})$$ then what would you do? I don't think I can just multiply the vector by one of the matrices without changing basis first but is there a way to know which basis I need and which matrix to use?
Edit: Here is a link to the problem: https://gyazo.com/ae66b2896d1249026f1bc8757a0c88dc

From the description you have provided, the matrix $A$ is intended to represent a linear mapping $L$ from $\mathbb{R}^3$ to $\mathbb{R}^2$ that takes as input a vector of co-ordinates relative to basis $S$ in $\mathbb{R}^3$ and outputs a vector of co-ordinates relative to basis $T$ in $\mathbb{R}^2$. There is nothing about the properties of the matrix $A$ itself that tells you this - it is only the surrounding description that tells you this is how the matrix $A$ is meant to be interpreted.
I am not sure what you mean when you say "a matrix $A$ that represents $L$ with respect to S (and only S)". You have to specify what basis the output vector will be relative to. If this basis $T'$ is different from $T$ then the matrix representing $L$ with respect to $S$ and $T'$ will still be a $2 \times 3$ matrix but it will have different values from $A$. Specifially, if the transformation from basis $T$ to basis $T'$ is represented by the $2 \times 2$ matrix $B$, and if the matrix representing $L$ relative to $S$ and $T'$ is $A'$ then
$A' = BA$
