If W is a subspace of R^3 then any basis of W must consist of vectors in R^3. The correct procedure is to write the given vectors in W as therowsof a matrix and then reduce it to echelon form. In other words, the prof is correct to use the transpose of A. In the row-reduced form of the matrix, the nonzero rows form a basis for W.

A subspace W of R^n has infinitely many bases, but they must all consist of vectors in W, and therefore in R^n. You can never have basis vectors in a different-dimensional space from the one you started out in.