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You are probably aware that the special properties of a magic square have to do with the various ways of summing its elements . If you take the sum along any column or row, or along either of the two diagonals, you will always get the same number. Let us verify that using Matlab. The first statement to try is sum(M)

Assuming the entries of your magic square belong to some field of characteristic zero to avoid complications, the given relations tell us that a generic magic square looks like. i The map compute the sum of elements in row i or in column j , or in one of the diagonal is a linear map from the vector space of 3 times 3 matrices to the base field, hence so is each difference of such maps.

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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.

$\begingroup$ You need to specify what sort of "numbers" the squares contain. Magical squares are more usually thought of as containing integers, and in that case they don't form a vector space. $\endgroup$

$\begingroup$ The usual way to prove some set $W$ is a vector space is to prove that it is a subspace of some set $V4 that you already know to be a vector space. So, it there any set containing the one you're asked about that you know is a vector space? And do you know how to prove something is a subspace? $\endgroup$

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bath and vody works