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 $T:\mathbb{F}^n\rightarrow\mathbb{F}^m$ ?


Let's consider $$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix},$$ here $m=2$ and $n=3$. To be able to multiply $A$ by some vector $v$, the vector should be taken from $\mathbb{F}^3$. In other words,
$$v=\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$$ and the result $A \cdot v$ is a vector from $\mathbb{F}^2$.
