Find basis vectors of V (subspace of a vector space of matrices) [on hold]

I edited the question so that it is more clear

Be VVV subspace of the vector space of all matrices Tm,nT^{m,n} over algebraic number field TT:

V={A∈Tm,n|∃(x1x2⋮xm)∈Tm,∃(y1y2⋮yn)∈Tn,(Aij=xi+yj)}V=\left \{ \mathbb{A}\in T^{m,n} \Bigg| \exists \begin{pmatrix}
x_{1}\\
x_{2}\\
\vdots \\
x_{m}
\end{pmatrix}\in T^{m}, \exists \begin{pmatrix}
y_{1}\\
y_{2}\\
\vdots\\
y_{n}
\end{pmatrix} \in T^{n}, \left ( \mathbb{A_{{ij}}} = x_{i}+y_{j} \right ) \right \}

where Tm,nT^{m,n} is a vector space of matrices over algebraic number field TT and TnT^{n} is a vector of nn numbers.

Example: We have vectors(12)\begin{pmatrix}
1\\
2
\end{pmatrix} and (123)\begin{pmatrix}
1\\
2\\
3
\end{pmatrix}. Then the matrix is: (1+11+21+32+22+22+3)\begin{pmatrix}
1+1 &1+2 &1+3 \\
2+2&2+2 &2+3
\end{pmatrix}

Find the basis of V and prove it is a basis (= prove linear independence and spanning properties).

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