Coordinate Vectors
When you have a Subspace $W$ which is the span of some Basis Vectors $\mathcal{B}$. You can calculate the coordinates of a vector $\mathbf{x}$ relative to the subspace $W$, which is represented by $[\mathbf{x}]\mathcal{B}$ by solving the linear system $A[\mathbf{x}]\mathcal{B} = \mathbf{x}$. You can also calculate the original vector of a coordinate relative to a subspace by doing vector matrice multiplication $A[\mathbf{x}]_\mathcal{B} = \mathbf{x}$.