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 is 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?