Computing the matrix of a Linear Transformation

Question

For a matrix $A\in M_n(\mathbb{F})$ consider the linear transformation $T_A:\mathbb{F}^n\rightarrow \mathbb{F}^n$. Denote the $B_{st}=\{e_1,...,e_n\}$ the standard basis of $\mathbb{F}^n.$

Compute the matrix $[T_A]_{B_{st}}$.

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So I don't really want anyone to solve the problem for me. I'm posting this because I do not understand whats being asked for in the problem.

So I know $A\in M_n(\mathbb{F})$ means that $A$ is an $n\times n$ matrix with $\mathbb{F}$-valued entries. I was taught that every linear transformation $T:V\rightarrow V$ has an associated matrix $[T]_B\in M_n(\mathbb{F})$ which "mirrors" the transformation in $\mathbb{F}^n$ with respect to basis $B$. But I don't understand what $T_A$ is in this problem.

Is it just denoting that $A$ is a linear transformation? Also, if $T_A$ is a map from $\mathbb{F}^n\rightarrow \mathbb{F}^n$, then what is $[T_A]_{B_{st}}$? I was taught that $[\cdot]_{B_{st}}$ is a coordinate map from $V\rightarrow \mathbb{F}^n$ where if $x\in V$ and $x=a_1e_1+...+a_ne_n \mapsto (a_1,...,a_n)\in \mathbb{F}^n$. But if $T_A$ is a matrix already I don't know what $[\cdot]_{B_{st}}$ is doing.

Any help with understanding is much appreciated.

Answer 1

The map $T_A$ is$$\begin{array}{rccc}T_A\colon&\mathbb{F}^n&\longrightarrow&\mathbb{F}^n\\&v&\mapsto&Av.\end{array}$$

Answer 2

$T_A:\mathbb{F}^n\rightarrow \mathbb{F}^n$ is given by $T_Ax=Ax$ for $x \in \mathbb{F}^n.$ Since $B_{st}$ is the standard basis, we have $[T_A]_{B_{st}}=A$.