the vector space of Magic Squares Can anyone offer help? I have no clue how to do this problem.
Magical squares are 3 by 3 matrices with the following properties: the sum of all numbers in each row, and in each column, and in each diagonal is equal. This number is called the magical number. 
(i)Prove that the set of magical squares forms a vector space with the usual matrix addition and scalar-matrix product. 
(ii) Find a basis of the vector space of magical squares and determine its dimension.

The given relations correspond to a homogeneous system of linear equations for the $9$ entries (ordered left to right, top to bottom) and the negated sum, with the following matrix:
$$
\pmatrix{
1&1&1&0&0&0&0&0&0&1\\
0&0&0&1&1&1&0&0&0&1\\
0&0&0&0&0&0&1&1&1&1\\
1&0&0&1&0&0&1&0&0&1\\
0&1&0&0&1&0&0&1&0&1\\
0&0&1&0&0&1&0&0&1&1\\
1&0&0&0&1&0&0&0&1&1\\
0&0&1&0&1&0&1&0&0&1
}
$$
This matrix has rank $7$ over the real numbers (as you can establish by hand or using Wolfram|Alpha), so the solution set is a vector space of dimension $10-7=3$. If the squares are meant to contain real numbers, their addition and scalar multiplication correspond to those of these solution vectors, so the squares also form such a vector space.
