How to find $[T]^m _m$ for $T: \mathbb R^3 \rightarrow \mathbb R^3$ s.t $T(x) = Ax$ when only $A$ and $m$ are provided$?$

Question

I am presented with the following problem:

Let $T: \mathbb R^3 \rightarrow \mathbb R^3$ be linear and define it as such: $T(x) = Ax$ for some $3\times 3$ matrix $A$.

Let $m_1, m_2, m_3 \in \mathbb R^3$ be linearly independent, call this basis "$m$".

Find $[T]^m _m$.

In general terms how can I find the matrix of transformation $[T]^m _m$, given that only $A$ and $m$ are provided.

How could I approach this$?$

I didn't post the values of $A$ or of the vectors in $m$ because I don't want people to compute an answer. I would like to get it myself... I just need to be pointed in the right direction...

The way this is presented to me is what confuses me.

Answer 1

The matrix $[T]_m^m$ representing $T$ with respect to the basis $m$ (used both for the domain and range) is the matrix

$$ [T]_m^m = ([Tm_1]_m \,| \, [Tm_2]_m \, | \, [Tm_3]_m). $$

Here, $[Tm_i]_m$ is the coordinates of the vector $T(m_i)$ with respect to the basis $m$. This is a $3 \times 1$ column vector.

More explicitly, to find the matrix you need to calculate $Tm_i$, then calculate the coordinates of $T(m_i)$ with respect to the basis $m$ (obtaining $[Tm_i]_m$) and then the vectors $[Tm_i]_m$ will be the columns of the matrix $[T]_m^m$.