Find basis vectors of the vector space of all $4 \times 4$ magic squares I'm taking a course in linear algebra and I need to solve this problem:

Let's define a magic square as a matrix whose sums of all the
numbers on a line, a column and on both the main diagonal and the main
anti-diagonal are the same.

*

*Prove that $4 \times 4$ magic squares form a vector space.


*Find the basis vectors of this vector space.

There are more questions in the exercise, but I guess these are the most important ones that it will help me solve other questions.
I have already searched almost the whole Internet, but I'm not able to find the answer. Thank you!

Here's another basis, with an easy proof of linear independence: the entry marked with a star is the only nonzero entry in that location in any of the eight matrices. 
$$\pmatrix{1*&0&0&0\cr0&0&0&1\cr0&1&0&0\cr0&0&1&0\cr}\quad\pmatrix{0&1*&0&0\cr0&0&0&1\cr0&0&1&0\cr1&0&0&0\cr}\quad\pmatrix{0&0&1*&0\cr0&0&0&1\cr0&0&1&0\cr1&1&-1&0\cr}\quad\pmatrix{0&0&0&1*\cr0&0&0&1\cr0&-1&2&0\cr1&2&-1&-1\cr}\quad\pmatrix{0&0&0&0\cr1*&0&0&-1\cr0&1&-1&0\cr-1&-1&1&1\cr}\quad\pmatrix{0&0&0&0\cr0&1*&0&-1\cr0&0&-1&1\cr0&-1&1&0\cr}\quad\pmatrix{0&0&0&0\cr0&0&1*&-1\cr0&-1&0&1\cr0&1&-1&0\cr}\quad\pmatrix{0&0&0&0\cr0&0&0&0\cr1*&1&-1&-1\cr-1&-1&1&1\cr}$$
A similar approach is taken in Ward, Vector spaces of magic squares, Math Mag 53 (1980) 108-111. 
