# linear algebra – 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:

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

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?

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