Suppose I start with a data matrix $A=\begin{pmatrix}
1 &2 &1 \\ 
 -2&-3 &1 \\ 
 3& 5&0 
\end{pmatrix}$ and I replace every row by its more compact representation in the basis $
\{(1,2,1), (2,3,-1)\}$ of the row space.

That way I obtain a new matrix $B=\begin{pmatrix}
1 &0 \\ 
 0& -1\\ 
 1& 1
\end{pmatrix}$.

* 1. What is the relation between the matrix $B$ and the original matrix $A$, in terms of matrix multiplication?
* 2. Conceptually, I want to say that what was done is to replace the original columns (features) by new features derived from the previous ones. But is that really what was done? How do I write the new columns as a linear combination of the old columns? (In general)