Can you name a basis of a vector space belonging to V and a linear transformation T:V->W such that the linear

transformation of the element(s) of the basis forms the basis of the vector space W?

For example, the basis for V is {e1, e2, ..., e_n} and the basis for W under the linear map is {T(e1), T(e2)..., T(e_n)}. Name the basis for V, W (V and W being either real or complex) and one such linear map.

If you pick any isomorphism between spaces and any basis, you can get exactly what you want. However, here's a slightly more interesting example:

V = R^3
W = R
T(a, b, c) = a

Basis: {(1, 0, 0), (1, 1, 0), (1, 1, 1)}

Under T, this basis maps to the basis {1}.