We shall talk about the algebraic extension . We shall assume that both and are algebraic over the field .
Assume . Hence, the basis for the vector space over is . Now say . Then the basis of over is .
It is known that the basis of over is (arranged as a matrix).
None of these can be dependant on each other (by definition). Also, they are in number.
Now let us first construct from . Say . Then the basis of over is . Now let us assume over is . The basis matrix of over will hence be over is .
It is possible that not a single term in the two matrices are the same. However, .