# 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
$$A' = BA$$