A matrix corresponds to a linear mapping between (finite dimensional) vector spaces. It can for example be used to model rotations or reflections. Here is a video that explains it better than I ever could:
Like QED said, repeating because I think it is an important point in avoiding confusion. A matrix describes or represents a transformation, that is what is usually meant by a matrix transformation.
we might need more info from the OP, as to whether he means the linear transformation defined by the matrix, or the transformation of the matrix, i.e.changing it to some other matrix, defined by a change of bases, say.
It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC).
Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory, might there be examples of vector spaces in which an Hamel basis actually doesn't exist ?