Can all linear transformations be calculated with a matrix? For all finite vector spaces and all linear transformations to/from those spaces, how can you prove/show from the definition of a linear transformation that all linear transformations can be calculated using a matrix.

The most general way to show this is to show the dual space of linear transformations on a finite dimensional vector space V over a field F is isomorphic to the vector space of m x n matrices with coeffecients in the same field F. However, although the proof isn't difficult,it needs considerably more machinery then just the definition of a linear transformation and building to the isomorphism theorem takes a number of pages. An excellent and self contained presentation of linear transformations and matrices which ends with the isomorphism theorem can be found in Chapter 5 of the beautiful online textbook by S. Gill Williamson of The University of California at San Diego. It's Theorem 5.13. But my strongest advice to you is to work through the entire chapter-not only is it a really beautiful and clear presentation of the structure of the vector space L(U,V), you'll only fully understand the proof after working through the chapter. It's worth the effort.  
