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We prove that for every two natural numbers M and N, if Tau is a Borel,finite, absolutely friendly measure on a compact set K of R^MN, then theintersection of K and BA(M,N) is a winning set in Schmidt's game sense playedon K, where BA(M,N) is the set of badly approximable M\times N matrices. As animmediate consequence we have the following application. If K is the attractorof an irreducible finite family of contracting similarity maps of R^(M\times N)satisfying the open set condition, (the Cantor ternary set, Koch's curve andSierpinski's gasket to name a few examples), the dimK=dimK\capBA(M,N).

Schmidt's game, Badly Approximable Linear Forms and Fractals