Let $V$ be the vector space of all $4$x$4$ matrices such that the sum of the elements in any row or any column is the same. What is the dimension of $V$?
Sol: I thought of this matrix where every row and column sums to $s$ and since it has $10$ variables I think the dim is 10. By separating and taking out the variables I could come up with a $10$ element basis. Through an obvious but lengthy process I could show its linear independence and the fact that it's a spanning set is obvious from the construction. Is this correct?
a & b & c & s-(a+b+c)\\
d & e & f & s-(d+e+f)\\
g & h & i & s-(g+h+i)\\
s-(a+d+g) & s-(b+e+h) & s-(c+f+i) &-2s+(a+b+c+d+e+f+g+h+i)