Definition of an image of a linear transformation I have the following definition of an image of a linear transformation, but I think that there's a mistake in the size of the field (confusion between m and n).
$$\begin{align*}
& A\in M_{m \times n}(\mathbb{F})\\
& T:\mathbb{F}^n\rightarrow\mathbb{F}^m \\
& T(v) = A\cdot v
\end{align*}$$
$$v = \begin{pmatrix}
\lambda_1 \\
\vdots \\
\lambda_n
\end{pmatrix},\quad\quad
A = \begin{pmatrix}
\mid & & \mid \\
 u_1 & \ldots & u_n \\
\mid & & \mid
\end{pmatrix}$$
$$
\begin{align*}
ImT &= \left\{w\in W \mid \exists v\in V \;.\; w=T(v)\right\} = \\
 &= \left\{w\in W \mid \exists v\in V \;.\; w=A\cdot v\right\} = \\
 &= \left\{w\in W \mid \text{ w is a linear combination of } u_1,\ldots,u_n\in \mathbb{F}^m \right\} = \\
 &= sp\left\{ u_1,\ldots,u_n\right\} = C(A)
\end{align*}
$$


*

*Notice in "w is a linear ...". Should it be a linear combination of $u_1,\ldots,u_n\in \mathbb{F}^n$?

*Shouldn't the transformation be $T:\mathbb{F}^m\rightarrow\mathbb{F}^n$ instead of $T:\mathbb{F}^n\rightarrow\mathbb{F}^m$ ?


Re 1.: No the map's image is part of $\mathbb{F}^m$.
Re 2.: Review how matrix multiplication is done:
$$
w = A v \iff w_i = \sum_{j=1}^n a_{ij} v_j \quad (i \in \{1, \ldots, m \})
$$
It is kind of mathematician's karate, the column vector $v$ gets multiplicated with the $i$-th row vector from $A$ to yield the component $w_i$.
$$
m\left\{
\left(
\begin{matrix}
\vdots \\
w_i \\
\vdots
\end{matrix}
\right)
\right.
=
\left(
\begin{matrix}
* & \cdots & * \\
a_{i1} & \cdots & a_{in} \\
* & \cdots & *
\end{matrix}
\right)
\left(
\begin{matrix}
v_1 \\
\vdots \\
v_n
\end{matrix}
\right)
$$
