Jtdylktuy

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:

$$S = begin{bmatrix} -1\1\0end{bmatrix},
begin{bmatrix} 0\1\1end{bmatrix},
begin{bmatrix} 1\0\0end{bmatrix}\
T = begin{bmatrix} 1\2end{bmatrix},
begin{bmatrix} 1\-1end{bmatrix}$$

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 $$L(begin{bmatrix} 2\1\-1end{bmatrix})$$ 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

$A' = BA$