Suppose we have a $2\times3$ matrix which takes a $3\times1$ column vector expressed in a basis $\mathcal{B}$. The output of this multiplication is a $2\times1$ vector. That is: $$\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}_\mathcal{B} = \begin{bmatrix} x_1' \\ x_2' \end{bmatrix}_\text{basis?}$$ My question is what is the basis of this new vector?
If we take the vector: $$\begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix}_\mathcal{B}$$ then it sends it in: $$ \mathbf{x}= \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} $$ Of course we can write $\mathbf{x}$ as: $$ \mathbf{x} = a_1\begin{bmatrix} 1 \\ 0 \end{bmatrix} + a_2 \begin{bmatrix} 0 \\ 1 \end{bmatrix} $$
but we don't know the basis.