Can I know the matrix of a linear transformation?

Question

Given T(V) = W, where T is a linear transformation in 3-dimensions, is it possible to find the matrix knowing V and W?

For instance: A[1 1 1] = [-3 5 8]

Can I find out A?

*Basis vectors are just [1 0 0], [0 1 0] and [0 0 1]

Answer 1

If you have any 3 linearly independent 3-dimensional vectors $\vec{v}_1,\vec{v}_2,\vec{v}_3 \in \mathbb{R}^3$ and we define $\vec{w}_k = T\left(\vec{v}_k\right)$ with $T$ linear, and you know $\vec{w}_1, \vec{w}_2, \vec{w}_3$, you can describe the underlying matrix exactly.

The same holds for $\mathbb{R}^n$ with $n$ vectors.

UPDATE

The reason why you need 3 vectors is because if you have less, an infinite amount of matrices can be found to satisfy your criteria. In particular, consider the example you cited in the comments to your question with $\vec{v} = (1,1,1)$ and $T(v) = \vec{w} = (-3, 5, 8)$.

One matrix that will do the trick is $$ A_1 = \begin{pmatrix} -3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 8 \end{pmatrix} $$ but another one is $$ A_2 = \begin{pmatrix} -3 & 0 & 0 \\ 5 & 0 & 0 \\ 8 & 0 & 0 \end{pmatrix} $$ or for that matter any matrix $$ A_x = \begin{pmatrix} -3 & x & -x \\ 5 & x & -x \\ 8 & x & -x \end{pmatrix}. $$ As you can see, the number of such matrices is uncountably infinite. Same thing will happen if you can only fix the actions of the map on 2 (linearly independent) vectors instead of one. But when actions of the map on 3 (linearly independent) vectors are fixed, that determines the map uniquely.