For a transformation
For a transformation
on , apparently the matrix of is a matrix. How can this be possible? Isn't the definition that
, so if is the matrix, we can't multiply a matrix with a , can we?
Answer & Explanation
For any linear transformation over finite dimensional real vector spaces if we consider fixed ordered bases of and respectively, then we can find the coordinate vectors/expansion in terms of these bases.
This directly follows from the definition of the bases. Given and clearly is a unique way to express in terms of the . If this way was not unique then linear independence of will be contradicted. So we may associate with in a one to one way. Likewise coordinates can be associated with every vector in . This association essentially allows us to identify an element of the vector space with the vector space . The same can be done in too with respect to .
Now if is the matrix of with respect to these bases, then a very beautiful result (which is not difficult to prove also) says that as , the corresponding coordinates of , say , go to the corresponding coordinates of , which are precisely given by . Essentially, the role of the action of is played by multiplication by in the coordinate world. So the correct interpretation is not that but . It is not that the matrix and the transformation are identical, but under the identification of vectors by their coordinates, the behavior of the transformation matches the action of multiplication by the matrix. The same holds if we are working over any arbitrary field instead of .
As an illustration in your case let and be the fixed ordered basis of where is the matrix which has at the place and 's elsewhere.
It is now clear that the matrix of with respect to this basis is
If we have some matrix then its coordinates will be
Now since , under , so it must happen that their coordinates also change in the same fashion under .
So we must have
which can be verified by direct multiplication.
Also, in the same vein, if you compose linear transformations then their effect is the same as multiplying the corresponding matrices. This is the real reason behind defining matrix multiplication in the fashion that we do so.