$T({\mathbf{v}}_{\mathbf{1}})={a}_{11}{\mathbf{v}}_{\mathbf{1}}+{a}_{21}{\mathbf{v}}_{\mathbf{2}}+{a}_{31}{\mathbf{v}}_{\mathbf{3}}$

$T({\mathbf{v}}_{\mathbf{2}})={a}_{12}{\mathbf{v}}_{\mathbf{1}}+{a}_{22}{\mathbf{v}}_{\mathbf{2}}+{a}_{32}{\mathbf{v}}_{\mathbf{3}}$

$T({\mathbf{v}}_{\mathbf{3}})={a}_{13}{\mathbf{v}}_{\mathbf{1}}+{a}_{23}{\mathbf{v}}_{\mathbf{2}}+{a}_{33}{\mathbf{v}}_{\mathbf{3}}$

So that we can identify T by the matrix

$\left(\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right)$

But then when I read several linear algebra book, it said if $T({\mathbf{v}}_{\mathbf{i}})=\sum _{j}{a}_{ij}{\mathbf{v}}_{\mathbf{j}}$ , then we can identify T by the matrix $({a}_{ij})$. My problem is: isn't the matrix is $({a}_{ji})$ instead of $({a}_{ij})$?