Let M2 be the vector space of 2 x 2 matrices.How to find a basis for the subspace of M2 consisting of symmetric matrices.
The problem it creates for me is that i ca guess the solution but i don't have any symstematic procedure in mind...
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For Mn, you take the n matrices that are all zeroes except have a single 1 on the diagonal, plus the n(n-1)/2 matrices that have zeroes everywhere except a 1 in the i-j position and a 1 in the j-i position, where i and j are unequal. I don't think you can get any more "systematic" than this.
better maybe just take your guess and try to prove it is independent and spans.
or here ios an idea: try to write down amap from some standard vector space R^t to the symmetric amtrices, in such a way that your maop is linear and an isomorphism. then it trakes a basis of the standard space to a basis of those matrices.
i.e. map say (1,0,0) to a symmetric 2by2 matrix, and (0,1,0) to another one and (0,0,1) to another one.
i.e. try mapping "upper triangular" matrices isomorphically to symmetricm ones.
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