How to find the transformation matrix of this linear transformation?

Question

Does transformation here mean change of basis or simply that T(ei) = ei' 

ei, ei' being the given bases.

Thanks

Answer 1

I think you understand the question correctly: you must find a linear transformation (matrix) $T$ such that $T\beta_i=\beta'_i$. (and $\beta'_1$ is indeed missing an element, presumably a typo in the question).

A straightforward way of solving this is to construct a $3\times 3$ matrix $T$ with unknown elements, then demand $T\beta_i=\beta'_i$, which gives you 9 linear equations in the 9 unknowns $T_{ij}$, which you can solve.